The Group structure of coxEter A3

Group table

The group coxA₃ consists of 24 elements. The full set of elements can be represented as all symmetry transformations of the 24 vertices of a truncated octahedron. The group is uniquely described by its group table. The elements are ordered with increasing order. The elements 2 to 10 are of order 2, which can immediately seen from the group table, since they square to unit. The elements 11 to 18 are of order three and the remaining ones are of order four.

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
2	1	8	11	23	12	21	3	14	16	4	6	19	9	20	10	22	24	13	15	7	17	5	18
3	8	1	19	7	20	5	2	22	24	13	15	11	17	12	18	14	16	4	6	23	9	21	10
4	14	24	1	10	13	19	15	11	5	9	21	6	2	8	22	23	20	7	18	12	16	17	3
5	21	7	10	1	22	3	23	20	4	18	14	16	12	17	13	15	11	24	9	2	6	8	19
6	16	22	15	20	1	9	13	7	12	23	10	8	24	4	2	19	21	17	5	18	3	11	14
7	23	5	24	3	9	1	21	6	19	16	17	18	15	14	11	12	13	10	22	8	20	2	4
8	3	2	13	21	15	23	1	17	18	19	20	4	22	6	24	9	10	11	12	5	14	7	16
9	11	20	14	22	7	6	18	1	17	2	19	21	4	24	23	10	8	12	3	13	5	16	15
10	12	19	5	4	16	24	17	18	1	20	2	22	21	23	6	8	9	3	11	14	13	15	7
11	9	18	2	16	19	13	20	4	23	14	7	12	1	3	17	5	15	21	24	6	10	22	8
12	10	17	20	15	2	14	19	21	6	5	16	3	18	11	1	13	7	22	23	24	8	4	9
13	22	16	8	18	4	11	6	19	21	17	5	15	3	1	14	7	12	23	10	20	24	9	2
14	4	15	9	17	21	12	24	2	22	1	13	7	11	18	5	16	3	6	8	19	23	10	20
15	24	14	6	12	8	17	4	23	20	7	18	1	16	13	3	11	5	9	21	10	2	19	22
16	6	13	23	11	10	18	22	24	2	15	1	17	7	5	12	3	14	8	4	9	19	20	21
17	19	12	22	14	23	15	10	8	9	3	11	5	13	16	7	18	1	20	2	4	21	24	6
18	20	11	21	13	24	16	9	10	8	12	3	14	5	7	15	1	17	2	19	22	4	6	23
19	17	10	3	24	11	4	12	13	7	22	23	20	8	2	9	21	6	5	16	15	18	14	1
20	18	9	12	6	3	22	11	5	15	21	24	2	10	19	8	4	23	14	7	16	1	13	17
21	5	23	18	8	14	2	7	12	13	10	22	24	20	9	4	6	19	16	17	3	15	1	11
22	13	6	17	9	5	20	16	3	14	8	4	23	19	10	21	24	2	15	1	11	7	18	12
23	7	21	16	2	17	8	5	15	11	24	9	10	6	22	19	20	4	18	14	1	12	3	13
24	15	4	7	19	18	10	14	16	3	6	8	9	23	21	20	2	22	1	13	17	11	12	5

Subgroup structure

Some subsets of elements form groups themselves. For instance {1,2} is a sub group of order two. All possible subgroups are listed below.

[1]

2: subgroups in 2 classes of conjugacy

C₂	C₂
[1,3]	[1,2]
[1,5]	[1,4]
[1,7]	[1,6]
	[1,8]
	[1,9]
	[1,10]

3: C₃

[1,11,14]
[1,12,16]
[1,13,15]
[1,17,18]

4: three classes of conjugacy

D₂	D₂	C₄
	[1,2,3,8]	[1,3,21,23]
[1,3,5,7]	[1,4,5,10]	[1,5,19,24]
	[1,6,7,9]	[1,7,20,22]

6: D₃≅coxA₂≅S₃

[1,2,4,9,11,14]
[1,2,6,10,12,16]
[1,4,6,8,13,15]
[1,8,9,10,17,18]

8: D₄≅coxB₂

[1,2,3,5,7,8,21,23]
[1,3,4,5,7,10,19,24]
[1,3,5,6,7,9,20,22]

12: A₄

[1,3,5,7,11,12,13,14,15,16,17,18]

24: coxA₃≅ S₄

[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]

Geometric Interpretation of the Subgroups

Truncated Octahedron

Each subgroup U provides a unique partition of the group in coset spaces gU:

D₂≅CoxI₂	D₃≅CoxI₃	D₃≅CoxI₃
[1,2,3,8]	[1,2,6,10,12,16]	[1,8,9,10,17,18]
[4,14,15,24]	[3,8,15,18,20,24]	[2,3,14,16,22,24]
[5,7,21,23]	[4,5,13,14,21,22]	[4,5,11,15,20,23]
[6,13,16,22]	[7,9,11,17,19,23]	[6,7,12,13,19,21]
[9,11,18,20]
[10,12,17,19]
It turns out, that these sets of four vertices are precisely realized as squares on the truncated octahedron. All of these squares transform simultaneously under the action of the D₂.	Similarly, these sets of six vertices are realized as two groups of hexagons on the truncated octahedron. Each of these sets transforms separately under the action of the corresponding D₃.

This link shows a geogebra file, where the structure of these vertices is visualized.

Constraints from the Subgroup structure

This shows, that there are tight constraints from the subgroup structure, telling what kind of polyhedra can be constructed. In order to have a vertex, where a certain number of faces meet, this requires the existence of the corresponding subgroups that have to be coxeter groups of rank two. Therefore, among all subgroups, the subgroups of coxeter type play a crucial role for the construction of polyhedra.

Cuboctahedron & truncated Tetrahedron

The vertices of the cuboctahedronand the truncated tetrahedron themselves constitute a subgroup of the full group, In comparison to the full group all reflexions are removed. There are 12 rotations left [1,3,5,7,11,12,13,14,15,16,17,18]. Correspondingly, only subgroups with these elements survive:

[1]