Hausdorff density: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
No edit summary
en>Novn
m Added link
 
Line 1: Line 1:
This lists the [[character table]]s for the more common [[point groups in three dimensions|molecular point groups]] used in the study of [[molecular symmetry]]. These tables are based on the [[group theory|group-theoretical]] treatment of the [[symmetry]] operations present in common [[molecule]]s, and are useful in molecular [[spectroscopy]] and [[quantum chemistry]].  Information regarding the use of the tables, as well as more extensive lists of them, can be found in the references.<ref>{{cite book | last = Drago | first = Russell S. | title = Physical Methods in Chemistry | publisher = W.B. Saunders Company | year = 1977 |  isbn = 0-7216-3184-3}}</ref><ref>{{cite book | last=Cotton | first = F. Albert | title = Chemical Applications of Group Theory | publisher = John Wiley & Sons: New York | year = 1990 | isbn = 0-471-51094-7}}</ref><ref>{{cite web | last = Gelessus  | first = Achim  | title = Character tables for chemically important point groups | publisher = Jacobs University, Bremin; Computational Laboratory for Analysis, Modeling, and Visualization | date=2007-07-12 | url=http://symmetry.jacobs-university.de/ | accessdate=2007-07-12 }}</ref><ref name="ShirtsFixJCE">{{cite journal | last=Shirts | first=Randall B. | title=Correcting Two Long-Standing Errors in Point Group Symmetry Character Tables | journal=[[Journal of Chemical Education]] | volume=84 | issue=1882 | publisher=[[American Chemical Society]] | year=2007 | url=http://jchemed.chem.wisc.edu/Journal/Issues/2007/Nov/abs1882.html | accessdate= 2007-10-16 | doi=10.1021/ed084p1882 | pages=1882|bibcode = 2007JChEd..84.1882S }}</ref><ref>{{cite web | url=http://www.webqc.org/symmetry.php | title=POINT GROUP SYMMETRY CHARACTER TABLES | last= Vanovschi | first=Vitalii | accessdate=2008-10-29 | publisher=WebQC.Org}}</ref>
Hi there. Let me begin by introducing the writer, her name is Sophia. Doing ballet is some thing she would by no means give up. Distributing manufacturing is where my main income arrives from and  psychic readings [http://test.jeka-nn.ru/node/129 online psychic reading] ([http://conniecolin.com/xe/community/24580 conniecolin.com]) it's some thing I truly enjoy. For many years he's been living in Alaska and he doesn't strategy on altering it.<br><br>Also visit my weblog [http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review free online tarot card readings]
 
== Notation ==
For each non-linear group, the tables give the most standard notation of the finite group isomorphic to the point group, followed by the [[Order (group theory)|order of the group]] (number of invariant symmetry operations). The finite group notation used is: Z<sub>n</sub>: [[cyclic group]] of order ''n'', D<sub>n</sub>: [[dihedral group]] isomorphic to the symmetry group of an ''n''&ndash;sided regular polygon, S<sub>n</sub>: [[symmetric group]] on ''n'' letters, and A<sub>n</sub>: [[alternating group]] on ''n'' letters.
 
The character tables then follow for all groups.  The rows of the character tables correspond to the irreducible representations of the group, with their conventional names in the left margin.  The naming conventions are as follows:
 
* ''A'' and ''B'' are singly degenerate representations, with the former transforming symmetrically around the principal axis of the group, and the latter asymmetrically. ''E'', ''T'', ''G'', ''H'', ... are doubly, triply, quadruply, quintuply, ... degenerate representations.
* ''g'' and ''u'' subscripts denote symmetry and antisymmetry, respectively, with respect to a center of inversion.  Subscripts "1" and "2" denote symmetry and antisymmetry, respectively, with respect to a nonprincipal rotation axis.  Higher numbers denote additional representations with such asymmetry.
* Single prime ( ' ) and double prime ( <nowiki>''</nowiki> ) superscripts denote symmetry and antisymmetry, respectively, with respect to a horizontal mirror plane σ<sub>h</sub>, one perpendicular to the principal rotation axis.
 
All but the two rightmost columns correspond to the [[symmetry operation]]s which are invariant in the group.  In the case of sets of similar operations with the same characters for all representations, they are presented as one column, with the number of such similar operations noted in the heading.
 
The body of the tables contain the characters in the respective irreducible representations for each respective symmetry operation, or set of symmetry operations.
 
The two rightmost columns indicate which irreducible representations describe the symmetry transformations of the three Cartesian coordinates (''x'',&nbsp;''y''&nbsp;and&nbsp;''z''), rotations about those three coordinates (''R<sub>x</sub>'',&nbsp;''R<sub>y</sub>''&nbsp;and&nbsp;''R<sub>z</sub>''), and functions of the quadratic terms of the coordinates(''x''<sup>2</sup>,&nbsp;''y''<sup>2</sup>,&nbsp;''z''<sup>2</sup>,&nbsp;''xy'',&nbsp;''xz'',&nbsp;and&nbsp;''yz'').
 
The symbol ''i'' used in the body of the table denotes the [[imaginary unit]]: ''i''<sup>&nbsp;2</sup>&nbsp;=&nbsp;&minus;1.  Used in a column heading, it denotes the operation of inversion. A superscripted uppercase "C" denotes [[Complex conjugate|complex conjugation]].
 
== Character tables ==
 
=== Nonaxial symmetries ===
These groups are characterized by a lack of a proper rotation axis, noting that a <math>C_1</math> rotation is considered the identity operation.  These groups have [[Involution (mathematics)|involutional]] symmetry: the only nonidentity operation, if any, is its own inverse.
 
In the group <math>C_1</math>, all functions of the Cartesian coordinates and rotations about them transform as the <math>A</math> irreducible representation.
 
{| class="wikitable" centered
 
! Point Group !! Canonical Group !! Order !! Character Table
|-
| <math>C_1</math> || <math>Z_1</math> || <math>1</math>
||
{| class="wikitable" centered
  |  || <math>E</math>
  |- 
  | <math>A</math> || <math>1</math>
|}
 
|-
| <math>C_i</math> || <math>Z_2</math> || 2
||
{| class="wikitable" centered
  |  || <math>E</math> || <math>i</math> ||  ||
  |-
  | <math>A_g</math> || <math>1</math> ||  <math>1</math>
  | <math>R_x</math>, <math>R_y</math>, <math>R_z</math>
  | <math>x^2</math>, <math>y^2</math>, <math>z^2</math>, <math>xy</math>, <math>xz</math>, <math>yz</math>
  |-
  | <math>A_u</math> || <math>1</math> || <math>-1</math> || <math>x</math>, <math>y</math>, <math>z</math> ||
|}
|-
| <math>C_s</math> || <math>Z_2</math> ||  <math>2</math>
||
{| class="wikitable" centered
  |  || <math>E</math> || <math>\sigma_h</math> ||  ||
  |-
  | <math>A'</math> || <math>1</math> ||  <math>1</math>
  | <math>x</math>, <math>y</math>, <math>R_z</math>
  | <math>x^2</math>, <math>y^2</math>, <math>z^2</math>, <math>xy</math>
  |-
  | <math>A''</math> || <math>1</math> || <math>-1</math>
  | <math>z</math>, <math>R_x</math>, <math>R_y</math> || <math>yz</math>, <math>xz</math>
|}
|-
|}
 
=== Cyclic symmetries ===
The families of groups with these symmetries have only one rotation axis.
 
==== Cyclic groups (''C''<sub>n</sub>) ====
The cyclic groups are denoted by ''C''<sub>n</sub>.  These groups are characterized by an ''n''-fold proper rotation axis ''C''<sub>n</sub>.  The ''C''<sub>1</sub> group is covered in the [[#Nonaxial groups|nonaxial groups]] section.
 
{| class="wikitable"  style="text-align:center"
! Point<br>Group !! Canonical<br>Group !! Order !! Character Table
|-
|  ''C''<sub>2</sub> || Z<sub>2</sub> || 2
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E || ''C''<sub>2</sub><sup>&nbsp;</sup> || colspan="2" | &nbsp;
  |-
  | A || 1 || 1 || ''R<sub>z</sub>'', ''z''
  | ''x''<sup>2</sup>, ''y''<sup>2</sup>, ''z''<sup>2</sup>, ''xy''
  |-
  | B || 1 || &minus;1 || ''R<sub>x</sub>'', ''R<sub>y</sub>'', ''x'', ''y''
  | ''xz'', ''yz''
|}
|-
|  ''C''<sub>3</sub> || Z<sub>3</sub> ||  3
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E || ''C''<sub>3</sub><sup>&nbsp;</sup> || ''C''<sub>3</sub><sup>2</sup>
  | colspan="2"  | ''θ'' = ''e''<sup>2π''i'' /3</sup>
  |-
  | A || 1 ||  1 || 1 || ''R<sub>z</sub>'', ''z''
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>
  |-
  |  E  ||  1 <br> 1 
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>''), <br> (''x'', ''y'')
  |  (''x''<sup>2</sup> - ''y''<sup>2</sup>, ''xy''), <br> (''xz'', ''yz'')
  |-
|}
|-
|  ''C''<sub>4</sub> || Z<sub>4</sub> ||  4
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E || ''C''<sub>4</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || ''C''<sub>4</sub><sup>3</sup> 
  | colspan="2"  | &nbsp;
  |-
  | A  || 1 ||  1 || 1 || 1 ||''R<sub>z</sub>'', ''z''
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | B  || 1 || &minus;1 || 1 || &minus;1 || &nbsp;
  | ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy''
  |-
  |  E  ||  1 <br> 1  ||  ''i'' <br> &minus;''i'' ||  &minus;1 <br> &minus;1  ||    &minus;''i'' <br> ''i''
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>''), <br> (''x'', ''y'')  ||  (''xz'', ''yz'')
  |-
|}
|-
|  ''C''<sub>5</sub> || Z<sub>5</sub> ||  5
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | ''C''<sub>5</sub><sup>&nbsp;</sup> || ''C''<sub>5</sub><sup>2</sup>
  | ''C''<sub>5</sub><sup>3</sup> || ''C''<sub>5</sub><sup>4</sup>
  | colspan="2" | ''θ'' = ''e''<sup>2π''i'' /5</sup>
  |-
  | A  || 1 || 1 || 1 || 1 || 1 || ''R<sub>z</sub>'', ''z''
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  |  E<sub>1</sub> 
  |  1 <br> 1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  ''θ''<sup>2</sup> <br> (''θ''<sup>2</sup>)<sup>C</sup>
  |  (''θ''<sup>2</sup>)<sup>C</sup> <br> ''θ''<sup>2</sup>
  |    ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>''), <br> (''x'', ''y'') ||  (''xz'', ''yz'')
  |-
  | E<sub>2</sub>
  |  1 <br> 1
  |  ''θ''<sup>2</sup> <br> (''θ''<sup>2</sup>)<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup> 
  |    (''θ''<sup>2</sup>)<sup>C</sup> <br> ''θ''<sup>2</sup>
  |  &nbsp; ||  (''x''<sup>2</sup> - ''y''<sup>2</sup>, ''xy'')
  |-
|}
|-
|  ''C''<sub>6</sub> || Z<sub>6</sub> ||  6
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | ''C''<sub>6</sub><sup>&nbsp;</sup> || ''C''<sub>3</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || ''C''<sub>3</sub><sup>2</sup>
  | ''C''<sub>6</sub><sup>5</sup>
  | colspan="2"  | ''θ'' = ''e''<sup>2π''i'' /6</sup>
  |-
  | A  || 1 ||  1 || 1 ||  1 || 1 ||  1 || ''R<sub>z</sub>'', ''z''
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | B  || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || &nbsp; || &nbsp;
  |-
  |  E<sub>1</sub> 
  |  1 <br> 1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;1 <br> &minus;1
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>''), <br> (''x'', ''y'') 
  |  (''xz'', ''yz'')
  |-
  |  E<sub>2</sub> 
  |  1 <br> 1
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  1 <br> 1
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  &nbsp;  ||  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
|}
|-
|  ''C''<sub>8</sub> || Z<sub>8</sub> ||  8
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | ''C''<sub>8</sub><sup>&nbsp;</sup> || ''C''<sub>4</sub><sup>&nbsp;</sup>
  | ''C''<sub>8</sub><sup>3</sup> || ''C''<sub>2</sub><sup>&nbsp;</sup>
  | ''C''<sub>8</sub><sup>5</sup> || ''C''<sub>4</sub><sup>3</sup>
  | ''C''<sub>8</sub><sup>7</sup>
  | colspan="2"  | ''θ'' = ''e''<sup>2π''i'' /8</sup>
  |-
  | A  || 1 ||  1 || 1 ||  1 || 1 ||  1 || 1 || 1  || ''R<sub>z</sub>'', ''z''
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | B  || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || &nbsp; || &nbsp;
  |-
  | E<sub>1</sub> 
  |  1 <br> 1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  ''i'' <br> &minus;''i''
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;1 <br> &minus;1
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  &minus;''i'' <br> ''i''
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>''), <br> (''x'', ''y'') 
  |  (''xz'', ''yz'')
  |-
  |  E<sub>2</sub> 
  |  1 <br> 1
  |  ''i'' <br> &minus;''i''
  |  &minus;1 <br> &minus;1
  |  &minus;''i'' <br> ''i''
  |  1 <br> 1
  |  ''i'' <br> &minus;''i''
  |  &minus;1 <br> &minus;1
  |  &minus;''i'' <br> ''i''
  | &nbsp;  ||  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
  |  E<sub>3</sub> 
  |  1 <br> 1
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  ''i'' <br> &minus;''i''
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  &minus;1 <br> &minus;1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  &minus;''i'' <br> ''i''
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &nbsp;  || &nbsp;
  |-
|}
|-
|}
 
==== Reflection groups (''C''<sub>nh</sub>) ====
The reflection groups are denoted by  ''C''<sub>nh</sub>.  These groups are characterized by i) an ''n''-fold proper rotation axis ''C''<sub>n</sub>; ii) a mirror plane ''σ<sub>h</sub>'' normal to  ''C''<sub>n</sub>. The ''C''<sub>1''h''</sub> group is the same as the ''C''<sub>s</sub> group in the [[#Nonaxial groups|nonaxial groups]] section.
 
{| class="wikitable"  style="text-align:center"
! Point<br>Group !! Canonical<br>group !! Order !! Character Table
|-
|  ''C''<sub>2''h''</sub> || Z<sub>2</sub> &times; Z<sub>2</sub> ||  4
|  align="left" |
{|  class="wikitable" style="text-align:center"
  | &nbsp; || E || ''C''<sub>2</sub><sup>&nbsp;</sup>
  | ''i'' || ''σ<sub>h</sub><sup>&nbsp;</sup>'' || colspan="2" | &nbsp;
  |-
  | A<sub>g</sub> || 1 ||  1 ||  1 ||  1 || ''R<sub>z</sub>''
  | ''x''<sup>2</sup>, ''y''<sup>2</sup>, ''z''<sup>2</sup>, ''xy''
  |-
  | B<sub>g</sub> || 1 || &minus;1 ||  1 || &minus;1 || ''R<sub>x</sub>'', ''R<sub>y</sub>''
  | ''xz'', ''yz''
  |-
  | A<sub>u</sub> || 1 ||  1 || &minus;1 || &minus;1 || ''z'' || &nbsp;
  |-
  | B<sub>u</sub> || 1 || &minus;1 || &minus;1 ||  1 || ''x'', ''y'' || &nbsp;
  |-
|}
|-
|  ''C''<sub>3''h''</sub> || Z<sub>6</sub> || 6
|  align="left" |
{|  class="wikitable" style="text-align:center"
  | &nbsp; || E || ''C''<sub>3</sub><sup>&nbsp;</sup>
  | ''C''<sub>3</sub><sup>2</sup> || ''σ<sub>h</sub><sup>&nbsp;</sup>''
  | ''S''<sub>3</sub><sup>&nbsp;</sup> || ''S''<sub>3</sub><sup>5</sup>
  | colspan="2"  | ''θ'' = ''e''<sup>2π''i'' /3</sup>
  |-
  | A' || 1 || 1 || 1 || 1 || 1 || 1 || ''R<sub>z</sub>''
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  |  E'  || align="center" | 1 <br> 1 
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  1 <br> 1 
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  (''x'', ''y'')  || (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
  | A<nowiki>''</nowiki> || 1 || 1 || 1 || &minus;1 || &minus;1 || &minus;1 || ''z'' || &nbsp;
  |-
  |  E<nowiki>''</nowiki>  ||  align="center" | 1 <br> 1 
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  &minus;1 <br> &minus;1 
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>'') || (''xz'', ''yz'')
  |-
|}
|-
|  ''C''<sub>4''h''</sub> || Z<sub>2</sub> &times; Z<sub>4</sub> || 8
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E || ''C''<sub>4</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || ''C''<sub>4</sub><sup>3</sup> 
  | ''i'' || ''S''<sub>4</sub><sup>3</sup> || ''σ<sub>h</sub><sup>&nbsp;</sup>''
  | ''S''<sub>4</sub><sup>&nbsp;</sup> || colspan="2" | &nbsp;
  |-
  | A<sub>g</sub>  || 1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1
  | ''R<sub>z</sub>'' || ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | B<sub>g</sub>  || 1 || &minus;1 ||  1 || &minus;1 ||  1 || &minus;1 ||  1 || &minus;1
  | &nbsp; || ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy''
  |-
  |  E<sub>g</sub>  ||  1 <br> 1  ||  ''i'' <br> &minus;''i''  ||  &minus;1 <br> &minus;1 
  |  &minus;''i'' <br> ''i''  ||  1 <br> 1  ||  ''i'' <br> &minus;''i''
  |  &minus;1 <br> &minus;1  ||  &minus;''i'' <br> ''i''
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>'')  || (''xz'', ''yz'')
  |-
  | A<sub>u</sub>  ||  1 ||  1 ||  1 ||  1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || ''z'' || &nbsp;
  |-
  | B<sub>u</sub>  ||  1 || &minus;1 ||  1 || &minus;1 || &minus;1 ||  1 || &minus;1 ||  1 || &nbsp; || &nbsp;
  |-
  |  E<sub>u</sub>  || 1 <br> 1  ||  ''i'' <br> &minus;''i'' ||  &minus;1 <br> &minus;1 
  |  &minus;''i'' <br> ''i''  ||  &minus;1 <br> &minus;1  ||  &minus;''i'' <br> ''i''
  |  1 <br>  1  ||  ''i'' <br> &minus;''i''
  |  (''x'', ''y'')  || &nbsp;
  |-
|}
|-
|  ''C''<sub>5''h''</sub> || Z<sub>10</sub> || 10
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | ''C''<sub>5</sub><sup>&nbsp;</sup> || ''C''<sub>5</sub><sup>2</sup>
  | ''C''<sub>5</sub><sup>3</sup> || ''C''<sub>5</sub><sup>4</sup>
  | ''σ<sub>h</sub><sup>&nbsp;</sup>'' || ''S''<sub>5</sub><sup>&nbsp;</sup>
  | ''S''<sub>5</sub><sup>7</sup> || ''S''<sub>5</sub><sup>3</sup>
  | ''S''<sub>5</sub><sup>9</sup>
  | colspan="2"  | ''θ'' = ''e''<sup>2π''i'' /5</sup>
  |-
  | A'  || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || ''R<sub>z</sub>''
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  |  E<sub>1</sub>' 
  |  1 <br> 1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  ''θ''<sup>2</sup> <br> (''θ''<sup>2</sup>)<sup>C</sup>
  |  (''θ''<sup>2</sup>)<sup>C</sup> <br> ''θ''<sup>2</sup>
  |    ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  1 <br> 1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  ''θ''<sup>2</sup> <br> (''θ''<sup>2</sup>)<sup>C</sup>
  |  (''θ''<sup>2</sup>)<sup>C</sup> <br> ''θ''<sup>2</sup>
  |    ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  (''x'', ''y'') || &nbsp;
  |-
  |  E<sub>2</sub>'
  |  1 <br> 1
  |  ''θ''<sup>2</sup> <br> (''θ''<sup>2</sup>)<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup> 
  |  (''θ''<sup>2</sup>)<sup>C</sup> <br> ''θ''<sup>2</sup>
  |  1 <br> 1
  |  ''θ''<sup>2</sup> <br> (''θ''<sup>2</sup>)<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup> 
  |    (''θ''<sup>2</sup>)<sup>C</sup> <br> ''θ''<sup>2</sup>
  |  &nbsp;  ||  (''x''<sup>2</sup> - ''y''<sup>2</sup>, ''xy'')
  |-
  | A<nowiki>''</nowiki>  || 1 || 1 || 1 || 1 || 1
  | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1
  | ''z'' ||  &nbsp;
  |-
  |  E<sub>1</sub><nowiki>''</nowiki> 
  |    1 <br> 1
  |    ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |    ''θ''<sup>2</sup> <br> (''θ''<sup>2</sup>)<sup>C</sup>
  |    (''θ''<sup>2</sup>)<sup>C</sup> <br> ''θ''<sup>2</sup>
  |    ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  &minus;1 <br> &minus;1
  |  &minus;''θ''<sup>&nbsp;</sup> <br> -''θ''<sup>C</sup>
  |  &minus;''θ''<sup>2</sup> <br> &minus;(''θ''<sup>2</sup>)<sup>C</sup>
  |  &minus;(''θ''<sup>2</sup>)<sup>C</sup> <br> &minus;''θ''<sup>2</sup>
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>'')  || (''xz'', ''yz'')
  |-
  |  E<sub>2</sub><nowiki>''</nowiki>
  |    1 <br> 1
  |    ''θ''<sup>2</sup> <br> (''θ''<sup>2</sup>)<sup>C</sup>
  |    ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |    ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup> 
  |    (''θ''<sup>2</sup>)<sup>C</sup> <br> ''θ''<sup>2</sup>
  |  &minus;1 <br> &minus;1
  |  &minus;''θ''<sup>2</sup> <br> &minus;(''θ''<sup>2</sup>)<sup>C</sup>
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup> 
  |  &minus;(''θ''<sup>2</sup>)<sup>C</sup> <br> &minus;''θ''<sup>2</sup>
  |  &nbsp;  || &nbsp;
  |-
|}
|-
|  ''C''<sub>6''h''</sub> || Z<sub>2</sub> &times; Z<sub>6</sub> || 12
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | ''C''<sub>6</sub><sup>&nbsp;</sup> || ''C''<sub>3</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || ''C''<sub>3</sub><sup>2</sup>
  | ''C''<sub>6</sub><sup>5</sup> || ''i'' || ''S''<sub>3</sub><sup>5</sup>
  | ''S''<sub>6</sub><sup>5</sup> || ''σ<sub>h</sub><sup>&nbsp;</sup>''
  | ''S''<sub>6</sub><sup>&nbsp;</sup> || ''S''<sub>3</sub><sup>&nbsp;</sup>
  | colspan="2" | ''θ'' = ''e''<sup>2π''i'' /6</sup>
  |-
  | A<sub>g</sub>  || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
  | ''R<sub>z</sub>''  || ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | B<sub>g</sub>  || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1
  | 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1
  | &nbsp; || &nbsp;
  |-
  |  E<sub>1g</sub> 
  |  1 <br> 1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;1 <br> &minus;1
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  1 <br> 1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;1 <br> &minus;1
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>'')  ||  (''xz'', ''yz'')
  |-
  |  E<sub>2g</sub> 
  |  1 <br> 1
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  1 <br> 1
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup> 
  |  1 <br> 1
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  1 <br> 1
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  &nbsp; ||  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
  | A<sub>u</sub>  || 1 || 1 || 1 || 1 || 1 || 1
  | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1
  | ''z'' || &nbsp;
  |-
  | B<sub>u</sub>  || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1
  | &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1
  | &nbsp; || &nbsp;
  |-
  |  E<sub>1u</sub> 
  |  1 <br> 1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;1 <br> &minus;1
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  &minus;1 <br> &minus;1
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |    ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |    1 <br> 1
  |    ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  (''x'', ''y'')  ||  &nbsp;
  |-
  |  E<sub>2u</sub> 
  |  1 <br> 1
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  1 <br> 1
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup> 
  |  &minus;1 <br> &minus;1
  |    ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |    ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  &minus;1 <br> &minus;1
  |    ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |    ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  &nbsp; || &nbsp;
|}
|-
|}
 
==== Pyramidal groups (''C''<sub>nv</sub>) ====
The pyramidal groups are denoted by ''C''<sub>nv</sub>. These groups are characterized by i) an ''n''-fold proper rotation axis ''C''<sub>n</sub>; ii) ''n'' mirror planes ''σ<sub>v</sub>'' which contain ''C''<sub>n</sub>. The ''C''<sub>1''v''</sub> group is the same as the ''C''<sub>s</sub> group in the [[#Nonaxial groups|nonaxial groups]] section.
 
{| class="wikitable"  style="text-align:center"
! Point<br>Group !! Canonical<br>group !!Order !! Character Table
|-
|  ''C''<sub>2''v''</sub> || Z<sub>2</sub> &times; Z<sub>2</sub><br> (=D<sub>2</sub>) || 4
|  align="left" |
{|  class="wikitable" style="text-align:center"
  | &nbsp; || E || ''C''<sub>2</sub><sup>&nbsp;</sup>
  | ''σ<sub>v</sub><sup>&nbsp;</sup>''
  | ''σ<sub>v</sub>'<sup>&nbsp;</sup>''
  | colspan="2" | &nbsp;
  |-
  | A<sub>1</sub> || 1 ||  1 ||  1 ||  1 || ''z''
  | ''x''<sup>2</sup>, ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub> || 1 ||  1 || &minus;1 || &minus;1 || ''R<sub>z</sub>'' || ''xy''
  |-
  | B<sub>1</sub> || 1 || &minus;1 ||  1 || &minus;1 || ''R<sub>y</sub>'', ''x'' || ''xz''
  |-
  | B<sub>2</sub> || 1 || &minus;1 || &minus;1 ||  1 || ''R<sub>x</sub>'', ''y'' || ''yz''
  |-
|}
|-
|  ''C''<sub>3''v''</sub> || D<sub>3</sub> || 6
|  align="left" |
{|  class="wikitable" style="text-align:center"
  | &nbsp; || E || 2 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | 3 ''σ<sub>v</sub><sup>&nbsp;</sup>''
  | colspan="2" | &nbsp;
  |-
  | A<sub>1</sub> || 1 || 1 ||  1 || ''z''
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub> || 1 || 1 || &minus;1 || ''R<sub>z</sub>'' || &nbsp;
  |-
  | E  || 2 || &minus;1 || 0 || (''R<sub>x</sub>'', ''R<sub>y</sub>''), (''x'', ''y'')
  | (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy''), (''xz'', ''yz'')
  |-
|}
|-
|  ''C''<sub>4''v''</sub> || D<sub>4</sub> || 8
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E || 2 ''C''<sub>4</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || 2 ''σ<sub>v</sub><sup>&nbsp;</sup>''
  | 2 ''σ<sub>d</sub><sup>&nbsp;</sup>'' || colspan="2"  | &nbsp;
  |-
  | A<sub>1</sub>  || 1 ||  1 ||  1 ||  1 ||  1 
  | ''z'' || ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub>  || 1 ||  1 ||  1 || &minus;1 || &minus;1 || ''R<sub>z</sub>'' || &nbsp;
  |-
  | B<sub>1</sub>  || 1 || &minus;1 ||  1 ||  1 || &minus;1
  | &nbsp; || ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>
  |-
  | B<sub>2</sub>  || 1 || &minus;1 ||  1 || &minus;1 ||  1 || &nbsp; || ''xy''
  |-
  | E || 2 || 0 || &minus;2 || 0 || 0
  | (''R<sub>x</sub>'', ''R<sub>y</sub>''), (''x'', ''y'') || (''xz'', ''yz'')
  |-
|}
|-
|  ''C''<sub>5''v''</sub> || D<sub>5</sub> || 10
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | 2 ''C''<sub>5</sub><sup>&nbsp;</sup> || 2 ''C''<sub>5</sub><sup>2</sup>
  | 5 ''σ<sub>v</sub><sup>&nbsp;</sup>'' || colspan="2" | ''θ'' = 2π/5
  |-
  | A<sub>1</sub>  || 1 || 1 || 1 ||  1 || ''z''
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub>  || 1 || 1 || 1 || &minus;1 || ''R<sub>z</sub>'' || &nbsp;
  |-
  | E<sub>1</sub> || 2 || 2 cos(''θ'') || 2 cos(2''θ'') || 0 
  | (''R<sub>x</sub>'', ''R<sub>y</sub>''), (''x'', ''y'') || (''xz'', ''yz'')
  |-
  | E<sub>2</sub> || 2 || 2 cos(2''θ'') || 2 cos(''θ'') || 0
  | &nbsp; ||  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
|}
|-
|  ''C''<sub>6''v''</sub> || D<sub>6</sub> || 12
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | 2 ''C''<sub>6</sub><sup>&nbsp;</sup> || 2 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || 3 ''σ<sub>v</sub><sup>&nbsp;</sup>''
  | 3 ''σ<sub>d</sub><sup>&nbsp;</sup>'' || colspan="2"  | &nbsp;
  |-
  | A<sub>1</sub>  || 1 || 1 || 1 || 1 ||  1 ||  1 
  | ''z'' || ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub>  || 1 || 1 || 1 || 1 || &minus;1 || &minus;1 || ''R<sub>z</sub>'' || &nbsp;
  |-
  | B<sub>1</sub>  || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || &nbsp; || &nbsp;
  |-
  | B<sub>2</sub>  || 1 || &minus;1 || 1 || &minus;1 || &minus;1 || 1 || &nbsp; || &nbsp;
  |-
  | E<sub>1</sub> ||2 || 1 || &minus;1 || &minus;2 || 0 || 0
  | (''R<sub>x</sub>'', ''R<sub>y</sub>''), (''x'', ''y'') || (''xz'', ''yz'')
  |-
  | E<sub>2</sub> ||2 || &minus;1 || &minus;1 || 2 || 0 || 0 || &nbsp;
  |  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |}
|-
|}
 
==== Improper rotation groups (''S''<sub>n</sub>) ====
The improper rotation groups are denoted by ''S''<sub>n</sub>.  These groups are characterized by an ''n''-fold improper rotation axis ''S''<sub>n</sub>, where ''n'' is necessarily even.  The ''S''<sub>2</sub> group is the same as the ''C''<sub>s</sub> group in the [[#Nonaxial groups|nonaxial groups]] section.
 
The S<sub>8</sub> table reflects the 2007 discovery of errors in older references.<ref name="ShirtsFixJCE"/> Specifically,  (''R<sub>x</sub>'', ''R<sub>y</sub>'') transform not as E<sub>1</sub> but rather as E<sub>3</sub>.
 
{| class="wikitable"  style="text-align:center"
! Point<br>Group !! Canonical<br>group !! Order !! Character Table
|-
|  ''S''<sub>4</sub> || Z<sub>4</sub> || 4
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E || ''S''<sub>4</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || ''S''<sub>4</sub><sup>3</sup> 
  | colspan="2"  | &nbsp;
  |-
  | A  || 1 ||  1 || 1 || 1 ||''R<sub>z</sub>'', &nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | B  || 1 || &minus;1 || 1 || &minus;1 || ''z''
  | ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy''
  |-
  | E  ||  1 <br> 1 || ''i'' <br> &minus;''i'' || &minus;1 <br> &minus;1 
  | &minus;''i'' <br> ''i''
  | (''R<sub>x</sub>'', ''R<sub>y</sub>''), <br> (''x'', ''y'')  ||  (''xz'', ''yz'')
  |-
|}
|-
|  ''S''<sub>6</sub> || Z<sub>6</sub> ||  6
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | ''S''<sub>6</sub><sup>&nbsp;</sup> || ''C''<sub>3</sub><sup>&nbsp;</sup>
  | ''i'' || ''C''<sub>3</sub><sup>2</sup> || ''S''<sub>6</sub><sup>5</sup>
  | colspan="2"  | ''θ'' = ''e''<sup>2π''i'' /6</sup>
  |-
  | A<sub>g</sub> || 1 ||  1 || 1 ||  1 || 1 ||  1 || ''R<sub>z</sub>''
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  |  E<sub>g</sub> 
  |  1 <br> 1
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  1 <br> 1
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>'')
  |  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy''), <br> (''xz'', ''yz'')
  |-
  | A<sub>u</sub> || 1 || &minus;1 ||  1 || &minus;1 || 1 || &minus;1 || ''z'' || &nbsp;
  |-
  |  E<sub>u</sub>
  |  1 <br> 1
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  &minus;1 <br> &minus;1
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  (''x'', ''y'')  || &nbsp;
  |-
|}
|-
|  ''S''<sub>8</sub> || Z<sub>8</sub> ||  8
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | ''S''<sub>8</sub><sup>&nbsp;</sup> || ''C''<sub>4</sub><sup>&nbsp;</sup>
  | ''S''<sub>8</sub><sup>3</sup> || ''i''
  | ''S''<sub>8</sub><sup>5</sup> || ''C''<sub>4</sub><sup>2</sup>
  | ''S''<sub>8</sub><sup>7</sup>
  | colspan="2"  | ''θ'' = ''e''<sup>2π''i'' /8</sup>
  |-
  | A  || 1 ||  1 || 1 ||  1 || 1 ||  1 || 1 || 1  || ''R<sub>z</sub>''
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | B  || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || ''z'' || &nbsp;
  |-
  |  E<sub>1</sub> 
  |    1 <br> 1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  ''i'' <br> &minus;''i''
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;1 <br> &minus;1
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  &minus;''i'' <br> ''i''
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  (''x'', ''y'') || (''xz'', ''yz'')
  |-
  |  E<sub>2</sub> 
  |  1 <br> 1  ||  ''i'' <br> &minus;''i''  ||  &minus;1 <br> &minus;1
  |  &minus;''i'' <br> ''i''  ||  1 <br> 1  ||  ''i'' <br> &minus;''i''
  |  &minus;1 <br> &minus;1  ||  &minus;''i'' <br> ''i''
  |  &nbsp; ||  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
  |  E<sub>3</sub> 
  |  1 <br> 1
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;''i'' <br> ''i''
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  &minus;1 <br> &minus;1
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  ''i'' <br> &minus;''i''
  |  &minus;''θ''<sup>  </sup> <br> &minus;''θ''<sup>C</sup>
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>'') || (''xz'', ''yz'')
  |-
|}
|-
|}
 
=== Dihedral symmetries ===
The families of groups with these symmetries are characterized by 2-fold proper rotation axes normal to a principal rotation axis.
 
==== Dihedral groups (''D''<sub>n</sub>) ====
The dihedral groups are denoted by ''D''<sub>n</sub>.  These groups are characterized by i) an ''n''-fold proper rotation axis ''C''<sub>n</sub>; ii) ''n'' 2-fold proper rotation axes ''C''<sub>2</sub> normal to ''C''<sub>n</sub>.  The ''D''<sub>1</sub> group is the same as the ''C''<sub>2</sub> group in the [[cyclic groups]] section.
 
{| class="wikitable"  style="text-align:center"
! Point<br>Group !! Canonical<br>group !!Order !! Character Table
|-
|  ''D''<sub>2</sub> || Z<sub>2</sub> &times; Z<sub>2</sub><br>(=D<sub>2</sub>) ||  4
|  align="left" |
{|  class="wikitable" style="text-align:center"
  | &nbsp; || E || ''C''<sub>2</sub><sup>&nbsp;</sup>(''z'')
  | ''C''<sub>2</sub><sup>&nbsp;</sup>(''x'')
  | ''C''<sub>2</sub><sup>&nbsp;</sup>(''y'') || colspan="2" | &nbsp;
  |-
  | A || 1 ||  1 ||  1 ||  1 || &nbsp;
  | ''x''<sup>2</sup>, ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | B<sub>1</sub> || 1 ||  1 || &minus;1 || &minus;1 || ''R<sub>z</sub>'', ''z'' || ''xy''
  |-
  | B<sub>2</sub> || 1 || &minus;1 || &minus;1 ||  1 || ''R<sub>y</sub>'', ''y'' || ''xz''
  |-
  | B<sub>3</sub> || 1 || &minus;1 ||  1 || &minus;1 || ''R<sub>x</sub>'', ''x'' || ''yz''
  |-
|}
|-
|  ''D''<sub>3</sub> || D<sub>3</sub> ||  6
|  align="left" |
{|  class="wikitable" style="text-align:center"
  | &nbsp; || E || 2 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | 3 ''C''<sub>2</sub><sup>&nbsp;</sup> || colspan="2"  |  &nbsp;
  |-
  | A<sub>1</sub> || 1 || 1 ||  1 || &nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub> || 1 || 1 || &minus;1 || ''R<sub>z</sub>'', ''z'' || &nbsp;
  |-
  | E  || 2 || &minus;1 || 0 || (''R<sub>x</sub>'', ''R<sub>y</sub>''), (''x'', ''y'')
  | (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy''), (''xz'', ''yz'')
  |-
|}
|-
|  ''D''<sub>4</sub> || D<sub>4</sub> ||  8
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E || 2 ''C''<sub>4</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || 2 ''C''<sub>2</sub>'<sup>&nbsp;</sup>
  | 2 ''C''<sub>2</sub><nowiki>''</nowiki><sup>&nbsp;</sup>
  | colspan="2"  | &nbsp;
  |-
  | A<sub>1</sub>  || 1 ||  1 ||  1 ||  1 ||  1  || &nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub>  || 1 ||  1 ||  1 || &minus;1 || &minus;1 || ''R<sub>z</sub>'', ''z'' || &nbsp;
  |-
  | B<sub>1</sub>  || 1 || &minus;1 ||  1 ||  1 || &minus;1
  | &nbsp; || ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>
  |-
  | B<sub>2</sub>  || 1 || &minus;1 ||  1 || &minus;1 ||  1 || &nbsp; || ''xy''
  |-
  | E || 2 || 0 || &minus;2 || 0 || 0
  | (''R<sub>x</sub>'', ''R<sub>y</sub>''), (''x'', ''y'') || (''xz'', ''yz'')
  |-
|}
|-
|  ''D''<sub>5</sub> || D<sub>5</sub> ||  10
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | 2 ''C''<sub>5</sub><sup>&nbsp;</sup> || 2 ''C''<sub>5</sub><sup>2</sup>
  | 5 ''C''<sub>2</sub><sup>&nbsp;</sup> || colspan="2"  | ''θ''=2π/5
  |-
  | A<sub>1</sub>  || 1 || 1 || 1 ||  1 || &nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub>  || 1 || 1 || 1 || &minus;1 || ''R<sub>z</sub>'', ''z'' || &nbsp;
  |-
  | E<sub>1</sub> || 2 || 2 cos(''θ'') || 2 cos(2''θ'') || 0 
  | (''R<sub>x</sub>'', ''R<sub>y</sub>''), (''x'', ''y'') || (''xz'', ''yz'')
  |-
  | E<sub>2</sub> || 2 || 2 cos(2''θ'') || 2 cos(''θ'') || 0
  | &nbsp; ||  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
|}
|-
|  ''D''<sub>6</sub> || D<sub>6</sub> ||  12
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | 2 ''C''<sub>6</sub><sup>&nbsp;</sup> || 2 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || 3 ''C''<sub>2</sub>'<sup>&nbsp;</sup>
  | 3 ''C''<sub>2</sub><nowiki>''</nowiki><sup>&nbsp;</sup>
  | colspan="2"  | &nbsp;
  |-
  | A<sub>1</sub>  || 1 ||  1 || 1 ||  1 ||  1 ||  1 || &nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub>  || 1 ||  1 || 1 ||  1 || &minus;1 || &minus;1
  | ''R<sub>z</sub>'', ''z'' || &nbsp;
  |-
  | B<sub>1</sub>  || 1 || &minus;1 || 1 || &minus;1 ||  1 || &minus;1 || &nbsp; || &nbsp;
  |-
  | B<sub>2</sub>  || 1 || &minus;1 || 1 || &minus;1 || &minus;1 ||  1 || &nbsp; || &nbsp;
  |-
  | E<sub>1</sub> ||2 || 1 || &minus;1 || &minus;2 || 0 || 0
  | (''R<sub>x</sub>'', ''R<sub>y</sub>''), (''x'', ''y'') || (''xz'', ''yz'')
  |-
  | E<sub>2</sub> ||2 || &minus;1 || &minus;1 || 2 || 0 || 0 || &nbsp;
  |  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |}
|-
|}
 
==== Prismatic groups (''D''<sub>nh</sub>) ====
The prismatic groups are denoted by ''D''<sub>nh</sub>.  These groups are characterized by i) an ''n''-fold proper rotation axis ''C''<sub>n</sub>; ii) ''n'' 2-fold proper rotation axes ''C''<sub>2</sub> normal to ''C''<sub>n</sub>; iii) a mirror plane ''σ<sub>h</sub>'' normal to ''C''<sub>n</sub> and containing the ''C''<sub>2</sub>s.  The ''D''<sub>1''h''</sub> group is the same as the ''C''<sub>2''v''</sub> group in the [[#Pyramidal groups|pyramidal groups]] section.
 
The D<sub>8''h''</sub> table reflects the 2007 discovery of errors in older references.<ref name="ShirtsFixJCE"/> Specifically, symmetry operation column headers 2S<sub>8</sub> and 2S<sub>8</sub><sup>3</sup> were reversed in the older references.
 
{| class="wikitable"  style="text-align:center"
! Point<br>Group !! Canonical<br>group !!Order !! Character Table
|-
|  ''D''<sub>2''h''</sub>
| Z<sub>2</sub>&times;Z<sub>2</sub>&times;Z<sub>2</sub><br>(=Z<sub>2</sub>&times;D<sub>2</sub>) ||  8
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E || ''C''<sub>2</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup>(x)
  | ''C''<sub>2</sub><sup>&nbsp;</sup>(y)  || ''i''
  | ''σ(xy)<sub>&nbsp;</sub><sup>&nbsp;</sup>''
  | ''σ(xz)<sub>&nbsp;</sub><sup>&nbsp;</sup>'' 
  | ''σ(yz)<sub>&nbsp;</sub><sup>&nbsp;</sup>''  || colspan="2" | &nbsp;
  |-
  | A<sub>g</sub> || 1 ||  1 ||  1 ||  1 || 1 ||  1 ||  1 ||  1 || &nbsp; 
  | ''x''<sup>2</sup>, ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | B<sub>1g</sub> || 1 ||  1 || &minus;1 || &minus;1 ||  1 ||  1 || &minus;1 || &minus;1
  | ''R<sub>z</sub>''  || ''xy''
  |-
  | B<sub>2g</sub> || 1 || &minus;1 || &minus;1 ||  1 ||  1 || &minus;1 ||  1 || &minus;1
  | ''R<sub>y</sub>'' || ''xz''
  |-
  | B<sub>3g</sub> || 1 || &minus;1 ||  1 || &minus;1 ||  1 || &minus;1 || &minus;1 ||  1
  | ''R<sub>x</sub>'' || ''yz''
  |-
  | A<sub>u</sub> || 1 ||  1 ||  1 ||  1
  | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &nbsp; || &nbsp;
  |-
  | B<sub>1u</sub> || 1 ||  1 || &minus;1 || &minus;1
  | &minus;1 || &minus;1 ||  1 ||  1 || ''z'' || &nbsp;
  |-
  | B<sub>2u</sub> || 1 || &minus;1 || &minus;1 ||  1
  | &minus;1 ||  1 || &minus;1 ||  1 || ''y'' || &nbsp;
  |-
  | B<sub>3u</sub> || 1 || &minus;1 ||  1 || &minus;1
  | &minus;1 ||  1 ||  1 || &minus;1 || ''x'' || &nbsp;
  |-
|}
|-
|  ''D''<sub>3''h''</sub> || D<sub>6</sub> || 12
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E || 2 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | 3 ''C''<sub>2</sub><sup>&nbsp;</sup> || ''σ<sub>h</sub><sup>&nbsp;</sup>''
  | 2 ''S''<sub>3</sub><sup>&nbsp;</sup> || 3 ''σ<sub>v</sub><sup>&nbsp;</sup>''
  | colspan="2"  | &nbsp;
  |-
  | A<sub>1</sub>' || 1 || 1 || 1 || 1 || 1 || 1 || &nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub>' || 1 || 1 || &minus;1 || 1 || 1 || &minus;1 || ''R<sub>z</sub>'' || &nbsp;
  |-
  |  E'  || 2  || &minus;1 ||  0 ||  2 || &minus;1 ||  0 ||  (''x'', ''y'') 
  | (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
  | A<sub>1</sub><nowiki>''</nowiki> || 1 || 1 || 1 || &minus;1 || &minus;1 || &minus;1
  | &nbsp; || &nbsp;
  |-
  | A<sub>2</sub><nowiki>''</nowiki> || 1 || 1 || &minus;1 || &minus;1 || &minus;1 || 1
  | ''z'' || &nbsp;
  |-
  |  E<nowiki>''</nowiki>  ||  2 || &minus;1 ||  0 || &minus;2 ||  1 ||  0
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>'') || (''xz'', ''yz'')
  |-
|}
|- 
|  ''D''<sub>4''h''</sub> || Z<sub>2</sub>&times;D<sub>4</sub> || 16
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E || 2 ''C''<sub>4</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || 2 ''C''<sub>2</sub>'<sup>&nbsp;</sup> 
  | 2 ''C''<sub>2</sub><nowiki>''</nowiki><sup>&nbsp;</sup> || ''i''
  | 2 ''S''<sub>4</sub><sup>&nbsp;</sup> || ''σ<sub>h</sub><sup>&nbsp;</sup>''
  | 2 ''σ<sub>v</sub><sup>&nbsp;</sup>''
  | 2 ''σ<sub>d</sub><sup>&nbsp;</sup>'' || colspan="2"  | &nbsp;
  |-
  | A<sub>1g</sub>  || 1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1
  | &nbsp; || ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2g</sub>  || 1 ||  1 ||  1 || &minus;1 || &minus;1
  |  1 ||  1 ||  1 || &minus;1 || &minus;1
  | ''R<sub>z</sub>'' || &nbsp;
  |-
  | B<sub>1g</sub>  || 1 || &minus;1 ||  1 ||  1 || &minus;1
  |  1 || &minus;1 ||  1 ||  1 || &minus;1
  | &nbsp; || ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>
  |-
  | B<sub>2g</sub>  || 1 || &minus;1 ||  1 || &minus;1 ||  1
  |  1 || &minus;1 ||  1 || &minus;1 ||  1
  | &nbsp; ||  ''xy''
  |-
  |  E<sub>g</sub>  ||    2 || 0 || &minus;2 || 0 || 0 || 2 || 0 || &minus;2 || 0 || 0
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>'')  || (''xz'', ''yz'')
  |-
  | A<sub>1u</sub>  || 1 ||  1 ||  1 ||  1 ||  1
  | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1
  | &nbsp; || &nbsp;
  |-
  | A<sub>2u</sub>  || 1 ||  1 ||  1 || &minus;1 || &minus;1
  | &minus;1 || &minus;1 || &minus;1 ||  1 ||  1
  | ''z'' || &nbsp;
  |-
  | B<sub>1u</sub>  || 1 || &minus;1 ||  1 ||  1 || &minus;1
  | &minus;1 ||  1 || &minus;1 || &minus;1 ||  1
  | &nbsp; || &nbsp;
  |-
  | B<sub>2u</sub>  || 1 || &minus;1 ||  1 || &minus;1 ||  1
  | &minus;1 ||  1 || &minus;1 || 1 || &minus;1
  | &nbsp; || &nbsp;
  |-
  |  E<sub>u</sub>  || 2 || 0 || &minus;2 || 0 || 0 || &minus;2 || 0 || 2 || 0 || 0 
  |  (''x'', ''y'')  || &nbsp;
  |-
|}
|-
|  ''D''<sub>5''h''</sub> || D<sub>10</sub> || 20
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | 2 ''C''<sub>5</sub><sup>&nbsp;</sup> || 2 ''C''<sub>5</sub><sup>2</sup>
  | 5 ''C''<sub>2</sub><sup>&nbsp;</sup>
  | ''σ<sub>h</sub><sup>&nbsp;</sup>'' || 2 ''S''<sub>5</sub><sup>&nbsp;</sup>
  | 2 ''S''<sub>5</sub><sup>3</sup> || 5 ''σ<sub>v</sub><sup>&nbsp;</sup>''
  | colspan="2"  | ''θ''=2π/5
  |-
  | A<sub>1</sub>'  || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 
  | &nbsp; || ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub>'  || 1 || 1 || 1 || &minus;1 || 1 || 1 || 1 || &minus;1 
  | ''R<sub>z</sub>'' || &nbsp;
  |-
  |  E<sub>1</sub>' || 2 || 2 cos(''θ'') || 2 cos(2''θ'') || 0 || 2
  | 2 cos(''θ'') || 2 cos(2''θ'') || 0 ||  (''x'', ''y'') || &nbsp;
  |-
  |  E<sub>2</sub>' || 2 || 2 cos(2''θ'') || 2 cos(''θ'') || 0 || 2
  | 2 cos(2''θ'') || 2 cos(''θ'') || 0
  |  &nbsp;  ||  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
  | A<sub>1</sub><nowiki>''</nowiki>  || 1 || 1 || 1 || 1
  | &minus;1 || &minus;1 || &minus;1 || &minus;1
  | &nbsp; ||  &nbsp;
  |-
  | A<sub>2</sub><nowiki>''</nowiki>  || 1 || 1 || 1 || &minus;1
  | &minus;1 || &minus;1 || &minus;1 || 1
  | ''z'' ||  &nbsp;
  |-
  |  E<sub>1</sub><nowiki>''</nowiki> || 2 || 2 cos(''θ'')
  | 2 cos(2''θ'') || 0 || &minus;2 ||  &minus;2 cos(''θ'')
  | &minus;2 cos(2''θ'') || 0 
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>'')  || (''xz'', ''yz'')
  |-
  |  E<sub>2</sub><nowiki>''</nowiki> || 2 || 2 cos(2''θ'')
  | 2 cos(''θ'') || 0 || &minus;2 || &minus;2 cos(2''θ'')
  | &minus;2 cos(''θ'') || 0 || &nbsp; || &nbsp;
  |-
|}
|-
|  ''D''<sub>6''h''</sub>
| Z<sub>2</sub>&times;D<sub>6</sub> ||  24
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | 2 ''C''<sub>6</sub><sup>&nbsp;</sup> || 2 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || 3 ''C''<sub>2</sub>'<sup>&nbsp;</sup>
  | 3 ''C''<sub>2</sub><nowiki>''</nowiki><sup>&nbsp;</sup> || ''i''
  | 2 ''S''<sub>3</sub><sup>&nbsp;</sup> || 2 ''S''<sub>6</sub><sup>&nbsp;</sup>
  | ''σ<sub>h</sub><sup>&nbsp;</sup>'' || 3 ''σ<sub>d</sub><sup>&nbsp;</sup>''
  | 3 ''σ<sub>v</sub><sup>&nbsp;</sup>'' || colspan="2"  | &nbsp;
  |-
  | A<sub>1g</sub>  || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
  | &nbsp; || ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2g</sub>  || 1 || 1 || 1 || 1 || &minus;1 || &minus;1
  | 1 || 1 || 1 || 1 || &minus;1 || &minus;1
  | ''R<sub>z</sub>''  || &nbsp;
  |-
  | B<sub>1g</sub>  || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1
  | 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1
  | &nbsp; || &nbsp;
  |-
  | B<sub>2g</sub>  || 1 || &minus;1 || 1 || &minus;1 || &minus;1 || 1
  | 1 || &minus;1 || 1 || &minus;1 || &minus;1 || 1
  | &nbsp; || &nbsp;
  |-
  |  E<sub>1g</sub> || 2 || 1 || &minus;1 || &minus;2 || 0 || 0
  | 2 || 1 || &minus;1 || &minus;2 || 0 || 0
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>'')  ||  (''xz'', ''yz'')
  |-
  |  E<sub>2g</sub> || 2 || &minus;1 || &minus;1 || 2 || 0 || 0
  | 2 || &minus;1 || &minus;1 || 2 || 0 || 0
  |  &nbsp; ||  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
  | A<sub>1u</sub>  || 1 || 1 || 1 || 1 || 1 || 1
  | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1
  | &nbsp; || &nbsp;
  |-
  | A<sub>2u</sub>  || 1 || 1 || 1 || 1 || &minus;1 || &minus;1
  | &minus;1 || &minus;1 || &minus;1 || &minus;1 || 1 || 1
  | ''z'' || &nbsp;
  |-
  | B<sub>1u</sub>  || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1
  | &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1
  | &nbsp; || &nbsp;
  |-
  | B<sub>2u</sub>  || 1 || &minus;1 || 1 || &minus;1 || &minus;1 || 1
  | &minus;1 || 1 || &minus;1 || 1 || 1 || &minus;1
  | &nbsp; || &nbsp;
  |-
  |  E<sub>1u</sub> || 2 || 1 || &minus;1 || &minus;2 || 0 || 0
  | &minus;2 || &minus;1 || 1 || 2 || 0 || 0
  |  (''x'', ''y'')  ||  &nbsp;
  |-
  |  E<sub>2u</sub> || 2 || &minus;1 || &minus;1 || 2 || 0 || 0
  | &minus;2 || 1 || 1 || &minus;2 || 0 || 0
  |  &nbsp; || &nbsp;
|}
|-
|  ''D''<sub>8''h''</sub> || Z<sub>2</sub>&times;D<sub>8</sub> || 32
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | 2 ''C''<sub>8</sub><sup>&nbsp;</sup> || 2 ''C''<sub>8</sub><sup>3</sup>
  | 2 ''C''<sub>4</sub><sup>&nbsp;</sup> || ''C''<sub>2</sub><sup>&nbsp;</sup>
  | 4 ''C''<sub>2</sub>'<sup>&nbsp;</sup>
  | 4 ''C''<sub>2</sub><nowiki>''</nowiki><sup>&nbsp;</sup> || ''i''
  | 2 ''S''<sub>8</sub><sup>3</sup> || 2 ''S''<sub>8</sub><sup>&nbsp;</sup>
  | 2 ''S''<sub>4</sub><sup>&nbsp;</sup>
  | ''σ<sub>h</sub><sup>&nbsp;</sup>''
  | 4 ''σ<sub>d</sub><sup>&nbsp;</sup>''
  | 4 ''σ<sub>v</sub><sup>&nbsp;</sup>''
  | colspan="2"  | ''θ''=2<sup>1/2</sup>
  |-
  | A<sub>1g</sub>  || 1 || 1 || 1 || 1 || 1 || 1 || 1
  | 1 || 1 || 1 || 1 || 1 || 1 || 1
  | &nbsp; || ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2g</sub>  || 1 || 1 || 1 || 1 || 1 || &minus;1 || &minus;1
  | 1 || 1 || 1 || 1 || 1 || &minus;1 || &minus;1 || ''R<sub>z</sub>''  || &nbsp;
  |-
  | B<sub>1g</sub>  || 1 || &minus;1 || &minus;1 || 1 || 1 || 1 || &minus;1
  | 1 || &minus;1 || &minus;1 || 1 || 1 || 1 || &minus;1 || &nbsp; || &nbsp;
  |-
  | B<sub>2g</sub>  || 1 || &minus;1 || &minus;1 || 1 || 1 || &minus;1 || 1
  | 1 || &minus;1 || &minus;1 || 1 || 1 || &minus;1 || 1 || &nbsp; || &nbsp;
  |-
  |  E<sub>1g</sub> || 2 || ''θ'' || &minus;''θ'' || 0 || &minus;2 || 0 || 0
  | 2 || ''θ'' || &minus;''θ'' || 0 || &minus;2 || 0 || 0
  |  (''R<sub>x</sub>'', ''R<sub>y</sub>'')  ||  (''xz'', ''yz'')
  |-
  |  E<sub>2g</sub> || 2 || 0 || 0 || &minus;2 || 2 || 0 || 0
  | 2 || 0 || 0 || &minus;2 || 2 || 0 || 0
  |  &nbsp; ||  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
  |  E<sub>3g</sub> || 2 || &minus;''θ'' || ''θ'' || 0 || &minus;2 || 0 || 0
  | 2 || &minus;''θ'' || ''θ'' || 0 || &minus;2 || 0 || 0
  |  &nbsp;  ||  &nbsp;
  |-
  | A<sub>1u</sub>  || 1 || 1 || 1 || 1 || 1 || 1 || 1
  | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &nbsp; || &nbsp;
  |-
  | A<sub>2u</sub>  || 1 || 1 || 1 || 1 || 1 || &minus;1 || &minus;1
  | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || 1 || 1 || ''z'' || &nbsp;
  |-
  | B<sub>1u</sub>  || 1 || &minus;1 || &minus;1 || 1 || 1 || 1 || &minus;1
  | &minus;1 || 1 || 1 || &minus;1 || &minus;1 || &minus;1 || 1 || &nbsp; || &nbsp;
  |-
  | B<sub>2u</sub>  || 1 || &minus;1 || &minus;1 || 1 || 1 || &minus;1 || 1
  | &minus;1 || 1 || 1 || &minus;1 || &minus;1 || 1 || &minus;1
  | &nbsp; || &nbsp;
  |-
  |  E<sub>1u</sub> || 2 || ''θ'' || &minus;''θ'' || 0 || &minus;2 || 0 || 0
  | &minus;2 || &minus;''θ'' || ''θ'' || 0 || 2 || 0 || 0
  |  (''x'', ''y'')  ||  &nbsp;
  |-
  |  E<sub>2u</sub> || 2 || 0 || 0 || &minus;2 || 2 || 0 || 0
  | &minus;2 || 0 || 0 || 2 || &minus;2 || 0 || 0 || &nbsp; || &nbsp;
  |-
  |  E<sub>3u</sub> || 2 || &minus;''θ'' || ''θ'' || 0 || &minus;2 || 0 || 0
  | &minus;2 || ''θ'' || &minus;''θ'' || 0 || 2 || 0 || 0
  |  &nbsp;  ||  &nbsp;
|}
|-
|}
 
==== Antiprismatic groups (''D''<sub>nd</sub>) ====
The antiprismatic groups are denoted by  ''D''<sub>nd</sub>. These groups are characterized by i) an ''n''-fold proper rotation axis ''C''<sub>n</sub>; ii) ''n'' 2-fold proper rotation axes ''C''<sub>2</sub> normal to ''C''<sub>n</sub>; iii) ''n'' mirror planes ''σ<sub>d</sub>'' which contain ''C''<sub>n</sub>.  The ''D''<sub>1''d''</sub> group is the same as the ''C''<sub>2''h''</sub> group in the [[#Reflection groups|reflection groups]] section.
 
{| class="wikitable"  style="text-align:center"
! Point<br>Group !! Canonical<br>group !! Order !! Character Table
|-
|  ''D''<sub>2''d''</sub> || D<sub>4</sub> || 8
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sup>&nbsp;</sup> || 2 ''S''<sub>4</sub><sup>&nbsp;</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || 2 ''C''<sub>2</sub>'<sup>&nbsp;</sup>
  | 2 ''σ<sub>d</sub><sup>&nbsp;</sup>'' || colspan="2" | &nbsp;
  |-
  | A<sub>1</sub> || 1 ||  1 ||  1 ||  1 ||  1 || &nbsp;
  | ''x''<sup>2</sup>, ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub> || 1 ||  1 ||  1 || &minus;1 || &minus;1 || ''R<sub>z</sub>'' || &nbsp;
  |-
  | B<sub>1</sub> || 1 || &minus;1 ||  1 ||  1 || &minus;1 || &nbsp;
  | ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>
  |-
  | B<sub>2</sub> || 1 || &minus;1 ||  1 || &minus;1 ||  1 || ''z'' || ''xy''
  |-
  | E            || 2 ||  0 || &minus;2 ||  0 ||  0
  | (''R<sub>x</sub>'', ''R<sub>y</sub>''), (''x'', ''y'') || (''xz'', ''yz'')
  |-
|}
|-
|  ''D''<sub>3''d''</sub> || D<sub>6</sub> || 12
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sup>&nbsp;</sup> || 2 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | 3 ''C''<sub>2</sub><sup>&nbsp;</sup> || ''i''<sup>&nbsp;</sup>
  | 2 ''S''<sub>6</sub><sup>&nbsp;</sup>
  | 3 ''σ<sub>d</sub><sup>&nbsp;</sup>''
  | colspan="2"  |  &nbsp;
  |-
  | A<sub>1g</sub> ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  &nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2g</sub> ||  1 ||  1 || &minus;1 ||  1 ||  1 || &minus;1
  |  ''R<sub>z</sub>'' || &nbsp;
  |-
  | E<sub>g</sub>  ||  2 || &minus;1 ||  0 ||  2 || &minus;1 ||  0
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'')
  | (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy''), (''xz'', ''yz'')
  |-
  | A<sub>1u</sub> ||  1 ||  1 ||  1 || &minus;1 || &minus;1 || &minus;1 || &nbsp; || &nbsp;
  |-
  | A<sub>2u</sub> ||  1 ||  1 || &minus;1 || &minus;1 || &minus;1 ||  1 ||  ''z'' || &nbsp;
  |-
  | E<sub>u</sub>  ||  2 || &minus;1 ||  0 || &minus;2 ||  1 ||  0 || (''x'', ''y'') || &nbsp;
  |-
|}
|-
|  ''D''<sub>4''d''</sub> || D<sub>8</sub> || 16
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sup>&nbsp;</sup> || 2 ''S''<sub>8</sub><sup>&nbsp;</sup>
  | 2 ''C''<sub>4</sub><sup>&nbsp;</sup> || 2 ''S''<sub>8</sub><sup>3</sup>
  | ''C''<sub>2</sub><sup>&nbsp;</sup> || 4 ''C''<sub>2</sub>'<sup>&nbsp;</sup>
  | 4 ''σ<sub>d</sub><sup>&nbsp;</sup>'' 
  | colspan="2"  | ''θ''=2<sup>1/2</sup>
  |-
  | A<sub>1</sub>  || 1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1
  | &nbsp; || ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub>  || 1 ||  1 ||  1 ||  1 ||  1 || &minus;1 || &minus;1
  | ''R<sub>z</sub>'' || &nbsp;
  |-
  | B<sub>1</sub>  || 1 || &minus;1 ||  1 || &minus;1 ||  1 ||  1 || &minus;1 || &nbsp; || &nbsp;
  |-
  | B<sub>2</sub>  || 1 || &minus;1 ||  1 || &minus;1 ||  1 || &minus;1 ||  1 || ''z'' || &nbsp;
  |-
  | E<sub>1</sub> || 2 || ''θ'' || 0 || &minus;''θ'' || &minus;2 || 0 || 0
  | (''x'', ''y'') || &nbsp;
  |-
  | E<sub>2</sub> || 2 || 0 || &minus;2 || 0 || 2 || 0 || 0
  | &nbsp; || (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
  | E<sub>3</sub> || 2 || &minus;''θ'' || 0 || ''θ'' || &minus;2 || 0 || 0
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'') || (''xz'', ''yz'')
  |-
|}
|-
|  ''D''<sub>5''d''</sub> || D<sub>10</sub> || 20
|  align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | 2 ''C''<sub>5</sub><sup>&nbsp;</sup> || 2 ''C''<sub>5</sub><sup>2</sup>
  | 5 ''C''<sub>2</sub><sup>&nbsp;</sup> || ''i''<sup>&nbsp;</sup>
  | 2 ''S''<sub>10</sub><sup>&nbsp;</sup> || 2 ''S''<sub>10</sub><sup>3</sup>
  | 5 ''σ<sub>d</sub><sup>&nbsp;</sup>'' 
  | colspan="2"  | ''θ''=2π/5
  |-
  | A<sub>1g</sub>  ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 || &nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2g</sub>  ||  1 ||  1 ||  1 || &minus;1 ||  1 ||  1 ||  1 || &minus;1
  | ''R<sub>z</sub>'' || &nbsp;
  |-
  | E<sub>1g</sub> || 2 || 2 cos(''θ'') || 2 cos(2''θ'') || 0
  | 2 || 2 cos(2''θ'') || 2 cos(''θ'') || 0
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'') || (''xz'', ''yz'')
  |-
  | E<sub>2g</sub> || 2 || 2 cos(2''θ'') || 2 cos(''θ'') || 0
  | 2 || 2 cos(''θ'') || 2 cos(2''θ'') || 0
  | &nbsp; ||  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
  | A<sub>1u</sub>  ||  1 ||  1 ||  1 ||  1
  | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &nbsp; || &nbsp;
  |-
  | A<sub>2u</sub>  ||  1 ||  1 ||  1 || &minus;1
  | &minus;1 || &minus;1 || &minus;1 ||  1 || ''z'' || &nbsp;
  |-
  | E<sub>1u</sub> || 2 || 2 cos(''θ'') || 2 cos(2''θ'') || 0
  | &minus;2 || &minus;2 cos(2''θ'') || &minus;2 cos(''θ'') || 0
  | (''x'', ''y'') || &nbsp;
  |-
  | E<sub>2u</sub> || 2 || 2 cos(2''θ'') || 2 cos(''θ'') || 0
  | &minus;2 || &minus;2 cos(''θ'') || &minus;2 cos(2''θ'') || 0
  | &nbsp; ||  &nbsp;
  |-
|}
|-
| ''D''<sub>6''d''</sub> || D<sub>12</sub> || 24
| align="left" |
{| class="wikitable" style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | 2 ''S''<sub>12</sub><sup>&nbsp;</sup> || 2 ''C''<sub>6</sub><sup>&nbsp;</sup>
  | 2 ''S''<sub>4</sub><sup>&nbsp;</sup> || 2 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | 2 ''S''<sub>12</sub><sup>5</sup> || ''C''<sub>2</sub><sup>&nbsp;</sup>
  | 6 ''C''<sub>2</sub>'<sup>&nbsp;</sup> || 6 ''σ<sub>d</sub><sup>&nbsp;</sup>''
  | colspan="2"  | ''θ''=3<sup>1/2</sup>
  |-
  | A<sub>1</sub>  ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1
  | &nbsp; || ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub>  ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 || &minus;1 || &minus;1
  | ''R<sub>z</sub>'' || &nbsp;
  |-
  | B<sub>1</sub>  ||  1 || &minus;1 ||  1 || &minus;1 ||  1 || &minus;1 ||  1 ||  1 || &minus;1
  | &nbsp; || &nbsp;
  |-
  | B<sub>2</sub>  ||  1 || &minus;1 ||  1 || &minus;1 ||  1 || &minus;1 ||  1 || &minus;1 ||  1
  | ''z''  || &nbsp;
  |-
  | E<sub>1</sub>  ||  2 || ''θ'' || 1 || 0 ||  &minus;1
  | &minus;''θ'' || &minus;2 || 0 || 0 || (''x'', ''y'') || &nbsp;
  |-
  | E<sub>2</sub>  ||  2 ||  1 || &minus;1 ||  &minus;2 || &minus;1 ||  1 || 2 || 0 || 0 || &nbsp;
  |  (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
  | E<sub>3</sub>  ||  2 || 0 || &minus;2 || 0 || 2 || 0 || &minus;2 || 0 || 0
  | &nbsp; || &nbsp;
  |-
  | E<sub>4</sub>  ||  2 || &minus;1 || &minus;1 || 2 || &minus;1 || &minus;1 || 2 || 0 || 0
  | &nbsp; || &nbsp;
  |-
  | E<sub>5</sub>  ||  2 || &minus;''θ'' || 1 || 0 ||  &minus;1
  | ''θ'' || &minus;2 || 0 ||  0
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'') || (''xz'', ''yz'')
  |-
  |}
|-
|}
 
=== [[Polyhedral group|Polyhedral]] symmetries ===
These symmetries are characterized by having more than one proper rotation axis of order greater than 2.
 
==== Cubic groups ====
These polyhedral groups are characterized by not having a ''C''<sub>5</sub> proper rotation axis.
 
{| class="wikitable"  style="text-align:center"
! Point<br>Group !! Canonical<br>group !! Order !! Character Table
|-
|  ''[[Tetrahedral group#Chiral tetrahedral symmetry|T]]'' || A<sub>4</sub> || 12
|  align="left" |
{|  style="text-align:center"
  | &nbsp; || E || 4 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | 4 ''C''<sub>3</sub><sup>2</sup>
  | 3 ''C''<sub>2</sub><sup>&nbsp;</sup>
  | colspan="2" | ''θ''=e<sup>2π ''i''/3</sup>
  |-
  | A || 1 ||  1 ||  1 ||  1 || &nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>
  |-
  | E || 1 <br> 1 || ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  | ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  | 1 <br> 1 || &nbsp;
  | (2 ''z''<sup>2</sup> &minus; ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, <br> ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>)
  |-
  | T || 3 || 0 || 0 || &minus;1
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'', ''R<sub>z</sub>''),<br>(''x'', ''y'', ''z'')
  | (''xy'', ''xz'', ''yz'')
  |-
|}
|-
|  ''[[Tetrahedral group#Achiral tetrahedral symmetry|T<sub>d</sub>]]'' || S<sub>4</sub> || 24
|  align="left" |
{|  style="text-align:center"
  | &nbsp; || E || 8 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | 3 ''C''<sub>2</sub><sup>&nbsp;</sup> || 6 ''S''<sub>4</sub><sup>&nbsp;</sup>
  | 6 ''σ<sub>d</sub><sup>&nbsp;</sup>''
  | colspan="2"  | &nbsp;
  |-
  | A<sub>1</sub> || 1 || 1 || 1 || 1 || 1 || &nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>
  |-
  | A<sub>2</sub> || 1 || 1 || 1 || &minus;1 || &minus;1 || &nbsp; || &nbsp;
  |-
  |  E  || 2 || &minus;1 || 2 || 0 || 0 || &nbsp;
  | (2 ''z''<sup>2</sup> &minus; ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, <br> ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>)
  |-
  | T<sub>1</sub> || 3 || 0 || &minus;1 || 1 || &minus;1
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'', ''R<sub>z</sub>'') || &nbsp;
  |-
  | T<sub>2</sub> || 3 || 0 || &minus;1 || &minus;1 || 1
  | (''x'', ''y'', ''z'') || (''xy'', ''xz'', ''yz'')
  |-
|}
|-
|  ''[[Tetrahedral group#Pyritohedral symmetry|T<sub>h</sub>]]'' || Z<sub>2</sub>&times;A<sub>4</sub> || 24
|  align="left" |
{| style="text-align:center"
  | &nbsp; || E || 4 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | 4 ''C''<sub>3</sub><sup>2</sup>
  | 3 ''C''<sub>2</sub><sup>&nbsp;</sup> || ''i''
  | 4 ''S''<sub>6</sub><sup>&nbsp;</sup> || 4 ''S''<sub>6</sub><sup>5</sup>
  | 3 ''σ<sub>h</sub><sup>&nbsp;</sup>''
  | colspan="2" | ''θ''=e<sup>2π ''i''/3</sup>
  |-
  | A<sub>g</sub>  || 1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1 ||  1
  | &nbsp; || ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>
  |-
  | A<sub>u</sub>  || 1 ||  1 ||  1 ||  1 ||  &minus;1 ||  &minus;1 ||  &minus;1 ||  &minus;1
  | &nbsp; || &nbsp;
  |-
  |  E<sub>g</sub> || 1 <br> 1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup> 
  |  1 <br> 1  ||  1 <br> 1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup>
  |  1 <br> 1
  |  &nbsp; 
  | (2 ''z''<sup>2</sup> &minus; ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, <br> ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>)
  |-
  |  E<sub>u</sub> || 1 <br> 1
  |  ''θ''<sup>&nbsp;</sup> <br> ''θ''<sup>C</sup>
  |  ''θ''<sup>C</sup> <br> ''θ''<sup>&nbsp;</sup> 
  |  1 <br> 1  ||  &minus;1 <br> &minus;1
  |  &minus;''θ''<sup>&nbsp;</sup> <br> &minus;''θ''<sup>C</sup>
  |  &minus;''θ''<sup>C</sup> <br> &minus;''θ''<sup>&nbsp;</sup>
  |  &minus;1 <br> &minus;1
  |  &nbsp; || &nbsp;
  |-
  | T<sub>g</sub> || 3 || 0 || 0 || &minus;1 || 3 || 0 || 0 || &minus;1
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'', ''R<sub>z</sub>'')
  | (''xy'', ''xz'', ''yz'')
  |-
  | T<sub>u</sub> || 3 || 0 || 0 || &minus;1 || &minus;3 || 0 || 0 || 1
  | (''x'', ''y'', ''z'') || &nbsp;
  |-
|}
|-
|  ''[[Octahedral symmetry#Chiral octahedral symmetry|O]]'' || S<sub>4</sub> || 24
|  align="left" |
{| style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | 6 ''C''<sub>4</sub><sup>&nbsp;</sup>
  | 3 ''C''<sub>2</sub><sup>&nbsp;</sup> (''C''<sub>4</sub><sup>2</sup>)
  | 8 ''C''<sub>3</sub><sup>&nbsp;</sup> || 6 ''C''<sub>2</sub><sup>&nbsp;</sup>
  | colspan="2"  | &nbsp;
  |-
  | A<sub>1</sub>  ||  1 ||  1 ||  1 ||  1 ||  1 ||  &nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>
  |-
  | A<sub>2</sub>  ||  1 || &minus;1 ||  1 || 1 || &minus;1 ||  &nbsp;  || &nbsp;
  |-
  |  E || 2 || 0 || 2 || &minus;1 ||  0 || &nbsp;
  | (2 ''z''<sup>2</sup> &minus; ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, <br> ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>)
  |-
  | T<sub>1</sub> || 3 || 1 || &minus;1 || 0 || &minus;1
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'', ''R<sub>z</sub>''), <br> (''x'', ''y'', ''z'')
  |  &nbsp; 
  |-
  | T<sub>2</sub> || 3  || &minus;1 || &minus;1 || 0 || 1
  |  &nbsp;  ||  (''xy'', ''xz'', ''yz'')
  |-
|}
|-
|  ''[[Octahedral symmetry#Achiral octahedral symmetry|O<sub>h</sub>]]''
| Z<sub>2</sub>&times;S<sub>4</sub> || 48
| align="left" |
{| style="text-align:center"
  | &nbsp; || E<sub>&nbsp;</sub><sup>&nbsp;</sup>
  | 8 ''C''<sub>3</sub><sup>&nbsp;</sup> || 6 ''C''<sub>2</sub><sup>&nbsp;</sup>
  | 6 ''C''<sub>4</sub><sup>&nbsp;</sup>
  | 3 ''C''<sub>2</sub><sup>&nbsp;</sup> (''C''<sub>4</sub><sup>2</sup>)
  |  ''i'' || 6 ''S''<sub>4</sub><sup>&nbsp;</sup>
  | 8 ''S''<sub>6</sub><sup>&nbsp;</sup> || 3 ''σ<sub>h</sub><sup>&nbsp;</sup>''
  | 6 ''σ<sub>d</sub><sup>&nbsp;</sup>''
  | colspan="2"  | &nbsp;
  |-
  | A<sub>1g</sub>  || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
  | &nbsp;  || ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>
  |-
  | A<sub>2g</sub>  || 1 || 1 || &minus;1 || &minus;1 || 1 || 1 || &minus;1 || 1 || 1 || &minus;1
  | &nbsp;  || &nbsp;
  |-
  |  E<sub>g</sub>  || 2 || &minus;1 || 0 || 0 || 2 || 2 || 0 || &minus;1 || 2 || 0
  |  &nbsp; 
  |  (2 ''z''<sup>2</sup> &minus; ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, <br> ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>)
  |-
  | T<sub>1g</sub> || 3 || 0 || &minus;1 || 1 || &minus;1 || 3 || 1 || 0 || &minus;1 || &minus;1
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'', ''R<sub>z</sub>'')
  |  &nbsp;
  |-
  | T<sub>2g</sub> || 3 || 0 || 1 || &minus;1 || &minus;1 || 3 || &minus;1 || 0 || &minus;1 || 1
  | &nbsp;  ||  (''xy'', ''xz'', ''yz'')
  |-
  | A<sub>1u</sub>  || 1 || 1 || 1 || 1 || 1
  | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1
  | &nbsp; || &nbsp;
  |-
  | A<sub>2u</sub>  || 1 || 1 || &minus;1 || &minus;1 || 1
  | &minus;1 || 1 || &minus;1 || &minus;1 || 1
  | &nbsp;  || &nbsp;
  |-
  |  E<sub>u</sub>  || 2 || &minus;1 || 0 || 0 || 2 || &minus;2 || 0 || 1 || &minus;2 || 0
  |  &nbsp;  ||  &nbsp;
  |-
  | T<sub>1u</sub> || 3 || 0 || &minus;1 || 1 || &minus;1 
  | &minus;3 || &minus;1 || 0 || 1 || 1
  | (''x'', ''y'', ''z'')  ||  &nbsp;
  |-
  | T<sub>2u</sub> || 3 || 0 || 1 || &minus;1 || &minus;1
  | &minus;3 || 1 || 0 || 1 || &minus;1
  | &nbsp;  ||  &nbsp;
|}
|-
|}
 
==== Icosahedral groups ====
{{see also|Icosahedral symmetry}}
 
These polyhedral groups are characterized by having a ''C''<sub>5</sub> proper rotation axis.
 
{| class="wikitable"  style="text-align:center"
! Point<br>Group !! Canonical<br>group !!Order !! Character Table
|-
|  ''I'' || A<sub>5</sub> || 60
|  align="left" |
{|  style="text-align:center"
  | &nbsp; || E || 12 ''C''<sub>5</sub><sup>&nbsp;</sup>
  | 12 ''C''<sub>5</sub><sup>2</sup>
  | 20 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | 15 ''C''<sub>2</sub><sup>&nbsp;</sup>
  | colspan="2" | ''θ''=π/5</sup>
  |-
  | A || 1 ||  1 ||  1 ||  1 ||  1 ||&nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>
  |-
  | T<sub>1</sub> || 3 || 2 cos(''θ'') || 2 cos(3''θ'') || 0 || &minus;1
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'', ''R<sub>z</sub>''),<br>(''x'', ''y'', ''z'') || &nbsp; 
  |-
  | T<sub>2</sub> || 3 || 2 cos(3''θ'') || 2 cos(''θ'') || 0 || &minus;1
  | &nbsp;  || &nbsp;
  |-
  | G || 4 || &minus;1 || &minus;1 || 1 || 0 || &nbsp;  || &nbsp;
  |-
  | H || 5 || 0 || 0 || &minus;1 || 1 || &nbsp; 
  | (2 ''z''<sup>2</sup> &minus; ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, <br> ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, <br>''xy'', ''xz'', ''yz'')
  |-
|}
|-
|  ''I<sub>h</sub>'' || Z<sub>2</sub>&times;A<sub>5</sub> || 120
|  align="left" |
{|  style="text-align:center"
  | &nbsp; || E || 12 ''C''<sub>5</sub><sup>&nbsp;</sup>
  | 12 ''C''<sub>5</sub><sup>2</sup>
  | 20 ''C''<sub>3</sub><sup>&nbsp;</sup>
  | 15 ''C''<sub>2</sub><sup>&nbsp;</sup> || ''i'' 
  | 12 ''S''<sub>10</sub><sup>&nbsp;</sup>
  | 12 ''S''<sub>10</sub><sup>3</sup>
  | 20 ''S''<sub>6</sub><sup>&nbsp;</sup>
  | 15 ''σ''
  | colspan="2" | ''θ''=π/5</sup>
  |-
  | A<sub>g</sub> || 1 ||  1 ||  1 ||  1 ||  1 || 1 ||  1 ||  1 ||  1 ||  1 ||&nbsp;
  | ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>
  |-
  | T<sub>1g</sub> || 3 || 2 cos(''θ'') || 2 cos(3''θ'') || 0 || &minus;1
  | 3 || 2 cos(3''θ'') || 2 cos(''θ'') || 0 || &minus;1
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'', ''R<sub>z</sub>'') || &nbsp; 
  |-
  | T<sub>2g</sub> || 3 || 2 cos(3''θ'') || 2 cos(''θ'') || 0 || &minus;1
  | 3 || 2 cos(''θ'') || 2 cos(3''θ'') || 0 || &minus;1 || &nbsp;  || &nbsp;
  |-
  | G<sub>g</sub> || 4 || &minus;1 || &minus;1 || 1 || 0 || 4 || &minus;1 || &minus;1 || 1 || 0
  | &nbsp;  || &nbsp;
  |-
  | H<sub>g</sub> || 5 || 0 || 0 || &minus;1 || 1 || 5 || 0 || 0 || &minus;1 || 1 || &nbsp; 
  | (2 ''z''<sup>2</sup> &minus; ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, <br> ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, <br>''xy'', ''xz'', ''yz'')
  |-
  | A<sub>u</sub> || 1 ||  1 ||  1 ||  1 ||  1
  | &minus;1 ||  &minus;1 ||  &minus;1 ||  &minus;1 ||  &minus;1
  | &nbsp; || &nbsp;
  |-
  | T<sub>1u</sub> || 3 || 2 cos(''θ'') || 2 cos(3''θ'') || 0 || &minus;1
  | &minus;3 || &minus;2 cos(3''θ'') || &minus;2 cos(''θ'') || 0 || 1
  | (''x'', ''y'', ''z'') || &nbsp; 
  |-
  | T<sub>2u</sub> || 3 || 2 cos(3''θ'') || 2 cos(''θ'') || 0 || &minus;1
  | &minus;3 || &minus;2 cos(''θ'') || &minus;2 cos(3''θ'') || 0 || 1
  | &nbsp;  || &nbsp;
  |-
  | G<sub>u</sub> || 4 || &minus;1 || &minus;1 || 1 || 0
  | &minus;4 || 1 || 1 || &minus;1 || 0
  | &nbsp;  || &nbsp;
  |-
  | H<sub>u</sub> || 5 || 0 || 0 || &minus;1 || 1 || &minus;5 || 0 || 0 || 1 || &minus;1
  | &nbsp;  || &nbsp;
  |-
|}
|-
|}
 
=== Linear (cylindrical) groups ===
These groups are characterized by having a proper rotation axis ''C''<sub>∞</sub> around which the symmetry is invariant to ''any'' rotation.
 
{| class="wikitable"  style="text-align:center"
! Point<br>Group !!  Character Table
|-
|  ''C<sub>∞v</sub>''
|  align="left" |
{|  style="text-align:center"
  | &nbsp; || E || 2 ''C''<sub>∞</sub><sup>Φ</sup>
  | ...
  | ∞ σ<sub>v</sub><sup>&nbsp;</sup>
  | colspan="2" | &nbsp;
  |-
  | A<sub>1</sub>=Σ<sup>+</sup> || 1 ||  1 || ... ||  1 || ''z''
  | ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | A<sub>2</sub>=Σ<sup>&minus;</sup> || 1 || 1 || ... || &minus;1 ||  ''R<sub>z</sub>''
  | &nbsp; 
  |-
  | E<sub>1</sub>=Π || 2 || 2 cos(Φ) || ... || 0 
  | (''x'', ''y''), (''R<sub>x</sub>'', ''R<sub>y</sub>'')  || (''xz'', ''yz'')
  |-
  | E<sub>2</sub>=Δ || 2 || 2 cos(2Φ) || ... || 0
  | &nbsp; || (''x''<sup>2</sup> - ''y''<sup>2</sup>, ''xy'')
  |-
  | E<sub>3</sub>=Φ || 2 || 2 cos(3Φ) || ... || 0
  |  &nbsp; || &nbsp;
  |-
  | ... || ... || ... || ... || ... || &nbsp; || &nbsp;
  |-
|}
|-
|  ''D<sub>∞h</sub>''
|  align="left" |
{|  style="text-align:center"
  | &nbsp; || E || 2 ''C''<sub>∞</sub><sup>Φ</sup> || ...
  | ∞ σ<sub>v</sub><sup>&nbsp;</sup> || ''i''
  | 2 ''S''<sub>∞</sub><sup>Φ</sup> || ... || ∞ ''C''<sub>2</sub><sup>&nbsp;</sup>
  | colspan="2" | &nbsp;
  |-
  | Σ<sub>g</sub><sup>+</sup> || 1 ||  1 || ... ||  1 || 1 || 1 || ... || 1
  | &nbsp; || ''x''<sup>2</sup> + ''y''<sup>2</sup>, ''z''<sup>2</sup>
  |-
  | Σ<sub>g</sub><sup>&minus;</sup> || 1 ||  1 || ...
  | &minus;1 || 1 || 1 || ... || &minus;1
  | ''R<sub>z</sub>'' || &nbsp;
  |-
  | Π<sub>g</sub> || 2 || 2 cos(Φ) || ... || 0 ||2 || &minus;2 cos(Φ) || .. || 0
  | (''R<sub>x</sub>'', ''R<sub>y</sub>'')  || (''xz'', ''yz'')
  |-
  | Δ<sub>g</sub> || 2 || 2 cos(2Φ) || ... || 0 || 2 || 2 cos(2Φ) || .. || 0
  | &nbsp; || (''x''<sup>2</sup> &minus; ''y''<sup>2</sup>, ''xy'')
  |-
  | ... || ... || ... || ... || ... || ... || ... || ... || ... || &nbsp; || &nbsp;
  |-
  | Σ<sub>u</sub><sup>+</sup> || 1 ||  1 || ...
  |  1 || &minus;1 || &minus;1 || ... || &minus;1
  | ''z'' || &nbsp;
  |-
  | Σ<sub>u</sub><sup>&minus;</sup> || 1 ||  1 || ...
  | &minus;1 || &minus;1 || &minus;1 || ... || 1
  | &nbsp; || &nbsp;
  |-
  | Π<sub>u</sub> || 2 || 2 cos(Φ) || ...
  | 0 || &minus;2 || 2 cos(Φ) || .. || 0
  | (''x'', ''y'')  || &nbsp;
  |-
  | Δ<sub>u</sub> || 2 || 2 cos(2Φ) || ...
  | 0 || &minus;2 || &minus;2 cos(2Φ) || .. || 0
  | &nbsp; || &nbsp;
  |-
  | ... || ... || ... || ... || ... || ... || ... || ... || ... || &nbsp; || &nbsp;
|}
|-
|}
 
== See also ==
*[[Linear combination of atomic orbitals|Linear combination of atomic orbitals molecular orbital method]]
*[[Raman spectroscopy]]
*[[Molecular vibration|Vibrational spectroscopy (molecular vibration)]]
*[[List of small groups]]
*[[Cubic harmonic]]s
 
== Notes ==
{{Reflist}}
 
==External links==
*[http://gernot-katzers-spice-pages.com/character_tables/ Character tables for many more point groups] (includes symmetry transformations of Cartesian products up to sixth order)
 
== Further reading ==
* {{cite book | last = Bunker | first = Philip | coauthors = Jensen, Per | title = Molecular Symmetry and Spectroscopy, Second edition | publisher = NRC Research Press | year = 2006 | location = [[Ottawa]] | isbn = 0-660-19628-X}}
 
[[Category:Theoretical chemistry]]
[[Category:Physical chemistry]]
[[Category:Group theory]]
[[Category:Finite groups]]

Latest revision as of 10:54, 11 December 2014

Hi there. Let me begin by introducing the writer, her name is Sophia. Doing ballet is some thing she would by no means give up. Distributing manufacturing is where my main income arrives from and psychic readings online psychic reading (conniecolin.com) it's some thing I truly enjoy. For many years he's been living in Alaska and he doesn't strategy on altering it.

Also visit my weblog free online tarot card readings

Retrieved from "https://en.formulasearchengine.com/index.php?title=Hausdorff_density&oldid=254123"