Editing Coxeter group (section)

===Properties===
Some properties of the finite irreducible Coxeter groups are given in the following table. The order of a reducible group can be computed by the product of its irreducible subgroup orders.
{{sort-under}}
{| class="wikitable sortable sort-under"
! {{verth|Rank ''n''}} || {{verth|Group<br />symbol}} || {{verth|Alternate<br />symbol}} || [[Coxeter notation|Bracket<br />notation]]||Coxeter<br />graph || data-sort-type="number"|Reflections<br />{{math|1=''m'' = {{sfrac|1|2}}''nh''}}<ref>{{cite book|title=Regular Polytopes|last=Coxeter|first=H. S. M.|chapter=12.6. The number of reflections|date=January 1973 |publisher=Courier Corporation |language=en|isbn=0-486-61480-8}}</ref>||data-sort-type="number"|[[Coxeter element|Coxeter number]]<br />''h''||data-sort-type="number"| [[Order (group theory)|Order]] || Group structure<ref name="wilson">{{Citation|last1=Wilson|first1=Robert A.|author-link=Robert Arnott Wilson|title=The finite simple groups|publisher=[[Springer-Verlag]]|location=Berlin, New York|series=[[Graduate Texts in Mathematics]] 251|isbn=978-1-84800-987-5|doi=10.1007/978-1-84800-988-2|year=2009|chapter=Chapter 2|volume=251}}</ref> || Related [[Uniform polytope|polytopes]]
|- align=center
!1 ||''A''<sub>1</sub> 
| ''A''<sub>1</sub> || [ ]|| {{CDD|node}} || 1 ||2 || 2 || <math>S_2</math> || { }
|- align=center
!2 ||''A''<sub>2</sub> 
| ''A''<sub>2</sub> || [3]|| {{CDD|node|3|node}} || 3 ||3 || 6 || <math>S_3\cong D_6\cong \operatorname{GO}^-_2(2)\cong \operatorname{GO}^+_2(4)</math> || [[equilateral triangle|{3}]] 
|- align=center
!3 ||''A''<sub>3</sub> 
| ''A''<sub>3</sub> || [3,3]|| {{CDD|node|3|node|3|node}} || 6 ||4 || 24 || <math>S_4</math> || [[regular tetrahedron|{3,3}]] 
|- align=center
!4 ||''A''<sub>4</sub> 
| ''A''<sub>4</sub> || [3,3,3]|| {{CDD|node|3|node|3|node|3|node}} || 10 ||5 || 120 || <math>S_5</math> || [[5-cell|{3,3,3}]] 
|- align=center
!5 ||''A''<sub>5</sub> 
| ''A''<sub>5</sub> || [3,3,3,3]|| {{CDD|node|3|node|3|node|3|node|3|node}} || 15 ||6 || 720 || <math>S_6</math> || [[5-simplex|{3,3,3,3}]] 
|- align=center
!''n'' ||''A''<sub>''n''</sub> || ''A''<sub>''n''</sub> || [3<sup>''n''−1</sup>]|| {{CDD|node|3|node|3}}...{{CDD|3|node|3|node}} || ''n''(''n'' + 1)/2 ||''n'' + 1 || (''n'' + 1)! || <math>S_{n+1}</math> || [[simplex|''n''-simplex]] 
|- align=center
!2 ||''B''<sub>2</sub>
| ''C''<sub>2</sub> || [4]|| {{CDD|node|4|node}} || 4 ||4 || 8 || <math>C_{2}\wr S_{2} \cong D_8\cong \operatorname{GO}^-_2(3)\cong \operatorname{GO}^+_2(5)</math> || [[square|{4}]]
|- align=center
!3 ||''B''<sub>3</sub>
| ''C''<sub>3</sub> || [4,3]|| {{CDD|node|4|node|3|node}}|| 9 ||6 || 48 || <math>C_{2}\wr S_{3}\cong S_4\times 2</math> || [[cube|{4,3}]] / [[regular octahedron|{3,4}]] 
|- align=center
!4 ||''B''<sub>4</sub>
| ''C''<sub>4</sub> || [4,3,3]|| {{CDD|node|4|node|3|node|3|node}}|| 16 ||8 || 384 || <math>C_{2}\wr S_{4}</math> || [[tesseract|{4,3,3}]] / [[16-cell|{3,3,4}]] 
|- align=center
!5 ||''B''<sub>5</sub>
| ''C''<sub>5</sub> || [4,3,3,3]|| {{CDD|node|4|node|3|node|3|node|3|node}} || 25 || 10 || 3840 || <math>C_{2}\wr S_{5}</math> || [[5-cube|{4,3,3,3}]] / [[5-orthoplex|{3,3,3,4}]] 
|- align=center
!''n'' ||''B''<sub>''n''</sub>|| ''C''<sub>''n''</sub> || [4,3<sup>''n''−2</sup>]|| {{CDD|node|4|node|3}}...{{CDD|3|node|3|node}}|| ''n''<sup>2</sup> ||2''n'' || 2<sup>''n''</sup> ''n''! || <math>C_{2}\wr S_{n}</math> || ''n''-cube / [[orthoplex|''n''-orthoplex]] 
|- align=center
!4 ||''D''<sub>4</sub> 
| ''B''<sub>4</sub> || [3<sup>1,1,1</sup>]|| {{CDD|nodes|split2|node|3|node}}|| 12 ||6 || 192 || <math>C_{2}^3 S_{4}\cong 2^{1+4}\colon S_3</math> || [[16-cell|h{4,3,3}]] / [[16-cell|{3,3<sup>1,1</sup>}]]
|- align=center
!5 ||''D''<sub>5</sub> 
| ''B''<sub>5</sub> || [3<sup>2,1,1</sup>]|| {{CDD|nodes|split2|node|3|node|3|node}} || 20 ||8 || 1920 || <math>C_{2}^4 S_{5}</math>|| [[5-demicube|h{4,3,3,3}]] / [[5-orthoplex|{3,3,3<sup>1,1</sup>}]]
|- align=center
!''n'' ||''D''<sub>''n''</sub> || ''B''<sub>''n''</sub> || [3<sup>''n''−3,1,1</sup>]|| {{CDD|nodes|split2|node|3}}...{{CDD|3|node|3|node}}|| ''n''(''n'' − 1) ||2(''n'' − 1) || 2<sup>''n''&minus;1</sup> ''n''! || <math>C_{2}^{n-1} S_{n}</math> || [[demihypercube|''n''-demicube]] / ''n''-orthoplex
|- align=center
!6 ||[[E6 (mathematics)|''E''<sub>6</sub>]]
|''E''<sub>6</sub> || [3<sup>2,2,1</sup>]|| {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}} || 36 ||12 || 51840 
|
<math>\operatorname{GO}_6^{-}(2) \cong \operatorname{SO}_5(3) \cong \operatorname{PSp}_4(3) \colon 2 \cong \operatorname{PSU}_4(2) \colon 2</math>
|
[[2 21 polytope|2<sub>21</sub>]], [[1 22 polytope|1<sub>22</sub>]] 
|- align=center
!7 ||[[E7 (mathematics)|''E''<sub>7</sub>]]
|''E''<sub>7</sub> || [3<sup>3,2,1</sup>]|| {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}|| 63 ||18 || 2903040 || <math> \operatorname{GO}_7(2)\times 2 \cong \operatorname{Sp}_6(2)\times 2 </math>|| [[3 21 polytope|3<sub>21</sub>]], [[2 31 polytope|2<sub>31</sub>]], [[1 32 polytope|1<sub>32</sub>]] 
|- align=center
!8 ||[[E8 (mathematics)|''E''<sub>8</sub>]]
| ''E''<sub>8</sub> || [3<sup>4,2,1</sup>]|| {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}|| 120 ||30 || 696729600 || <math>2\cdot\operatorname{GO}_8^{+}(2)</math>||[[4 21 polytope|4<sub>21</sub>]], [[2 41 polytope|2<sub>41</sub>]], [[1 42 polytope|1<sub>42</sub>]] 
|- align=center
!4 ||[[F4 (mathematics)|''F''<sub>4</sub>]]
|''F''<sub>4</sub> || [3,4,3]|| {{CDD|node|3|node|4|node|3|node}} || 24 ||12 || 1152 ||<math>\operatorname{GO}^+_4(3)\cong 2^{1+4}\colon(S_3 \times S_3)</math>|| [[24-cell|{3,4,3}]] 
|- align=center
!2 ||[[G2 (mathematics)|''G''<sub>2</sub>]]
|| – (''D''{{supsub|6|2}}) || [6]|| {{CDD|node|6|node}} || 6 ||6 || 12 || <math>D_{12}\cong \operatorname{GO}^-_2(5)\cong \operatorname{GO}^+_2(7)</math>|| [[hexagon|{6}]] 
|- align=center
!2 ||''I''<sub>2</sub>(5) 
|| ''G''<sub>2</sub> || [5]||{{CDD|node|5|node}} || 5 || 5 || 10 || <math>D_{10}\cong \operatorname{GO}^-_2(4)</math>|| [[pentagon|{5}]] 
|- align=center
!3 ||''H''<sub>3</sub> 
| ''G''<sub>3</sub> 
|| [3,5]|| {{CDD|node|5|node|3|node}} || 15 ||10 || 120 || <math>2\times A_5</math>|| [[icosahedron|{3,5}]] / [[dodecahedron|{5,3}]] 
|- align=center
!4 ||''H''<sub>4</sub> 
| ''G''<sub>4</sub> || [3,3,5]|| {{CDD|node|5|node|3|node|3|node}} || 60 ||30 || 14400 || <math>2\cdot(A_5\times A_5)\colon 2</math>{{efn|an index 2 subgroup of <math>\operatorname{GO}^+_4(5)</math>}}|| [[120-cell|{5,3,3}]] / [[600-cell|{3,3,5}]] 
|- align=center
!2 ||''I''<sub>2</sub>(''n'') 
|| ''D''{{supsub|''n''|2}}|| [''n'']|| {{CDD|node|n|node}} || ''n'' ||''n'' || 2''n'' 
|
<math>D_{2n}</math>

<math>\cong \operatorname{GO}^-_2(n-1)</math> when ''n'' = ''p''<sup>''k''</sup> + 1, ''p'' prime
<math>\cong \operatorname{GO}^+_2(n+1)</math> when ''n'' = ''p''<sup>''k''</sup> − 1, ''p'' prime

| [[regular polygon|{''p''}]] 
|}