Group structure of A4

The largest polytope that is generated by the action of the group is A120. Each element of the group is represented by a vertex of the polytope. The 3-dimensional boundaries that constitute to A120 are determined by the subgroup structure of the coxeter A4 group:

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	...	...	...	106	107	108	109	110	111	112	113	114	115	116	117	118	119	120
2	1	18	19	20	27	101	61	67	28	102	55	68	29	103	...	...	...	89	90	95	96	22	83	84	97	98	24	85	86	91	92	26
3	18	1	39	40	101	27	90	96	49	79	83	17	51	81	...	...	...	112	61	117	67	21	55	106	118	119	74	56	108	113	114	76
4	19	40	1	39	47	78	12	95	50	80	8	98	103	29	...	...	...	111	112	67	116	72	106	107	119	68	73	109	56	62	113	25
5	20	39	40	1	48	77	89	16	102	28	84	97	52	82	...	...	...	61	111	116	117	71	107	55	68	118	23	108	109	114	62	75
6	32	110	63	69	1	18	23	25	30	65	47	108	31	59	...	...	...	17	44	13	46	3	77	95	81	26	100	78	89	79	24	94
7	110	32	93	99	18	1	44	46	53	15	77	86	54	11	...	...	...	68	23	62	25	2	47	117	103	76	120	48	112	102	74	115
8	55	85	12	97	23	46	1	44	57	100	4	87	106	89	...	...	...	14	25	64	18	51	103	120	104	71	65	105	59	48	20	112
9	56	83	91	16	25	44	46	1	108	95	81	30	58	94	...	...	...	70	18	10	23	49	104	65	47	19	117	102	115	105	72	59
10	35	70	57	115	34	66	50	114	1	20	21	25	33	53	...	...	...	79	98	26	82	99	16	42	45	9	5	22	77	83	80	87
11	115	100	88	35	59	15	79	91	20	1	42	45	60	7	...	...	...	50	118	75	103	120	67	21	25	56	2	71	101	107	49	110
12	61	96	8	92	63	99	4	94	21	45	1	42	111	83	...	...	...	103	120	74	105	66	14	25	20	58	52	18	49	53	104	107
13	62	17	86	89	114	98	82	34	25	42	45	1	64	87	...	...	...	105	66	19	50	118	69	20	21	6	48	73	104	110	101	53
14	38	64	120	58	37	60	119	52	36	54	116	51	1	19	...	...	...	24	80	81	92	93	22	78	85	82	88	12	41	43	8	4
15	120	94	38	87	65	11	97	81	66	7	96	82	19	1	...	...	...	73	102	52	113	115	72	101	109	51	110	61	21	23	55	2
...
106	84	108	112	68	24	76	20	73	88	70	105	110	8	61	...	...	...	37	45	93	39	81	82	99	3	22	36	4	34	27	1	12
107	83	56	111	118	44	25	18	23	87	120	104	57	84	112	...	...	...	36	46	35	1	29	81	100	4	41	15	5	11	78	40	89
108	86	106	62	117	26	74	75	19	9	67	52	6	87	64	...	...	...	99	40	34	43	79	5	37	27	1	16	80	93	3	22	33
109	85	55	113	116	46	23	25	18	86	117	103	53	88	115	...	...	...	38	1	33	44	28	4	15	77	39	95	79	94	5	42	11
110	7	6	60	66	22	21	71	72	83	118	49	106	85	113	...	...	...	95	41	89	42	27	28	17	79	24	98	29	13	81	26	92
111	90	67	107	114	93	69	104	115	22	75	18	72	12	55	...	...	...	81	100	24	5	37	36	46	40	88	82	1	28	30	4	8
112	89	117	106	62	94	120	105	63	42	25	20	21	90	107	...	...	...	82	99	43	4	15	37	45	1	32	29	39	80	7	3	83
113	92	119	109	61	91	118	103	59	45	21	25	20	93	110	...	...	...	4	15	40	79	98	38	1	42	31	27	44	3	87	77	7
114	91	118	56	111	13	68	51	10	26	71	76	19	94	58	...	...	...	3	36	1	28	17	100	39	41	30	77	24	5	88	78	31
115	11	65	54	10	89	117	48	111	24	23	74	73	92	109	...	...	...	27	16	22	77	95	98	43	44	83	28	26	82	9	29	85
116	96	61	119	109	99	63	120	105	16	56	14	48	22	73	...	...	...	26	4	79	94	34	1	29	31	5	9	33	44	39	87	80
117	95	112	68	108	100	115	69	104	96	109	66	101	41	23	...	...	...	45	5	80	93	11	40	82	7	3	85	34	43	1	32	28
118	98	114	67	107	97	113	65	102	99	110	70	105	43	21	...	...	...	39	81	5	11	92	46	3	88	78	7	35	1	41	30	27
119	97	113	116	55	17	62	14	49	100	57	120	104	24	72	...	...	...	1	29	3	33	13	26	4	87	77	30	94	40	42	31	78
120	15	59	14	53	95	112	116	47	98	107	119	50	26	25	...	...	...	22	78	27	12	89	24	80	8	28	83	92	45	46	85	29

What are the largest geometric groups of rank 3?

rank 4: besides the full group coxA₄ ≅S₄ , there is a 60 dimensional sub group, which is the alternating group of S₅

rank 3:

d=24: 5 coxA₃ ≅ S₄,
d=12: 10 direct products of coxA₂×coxA₁

_{The largest representation of coxA4 is the truncated octahedron and the largest representation of}coxA₂×coxA_{1 is the hexagonal prism.}

_{Therefore the A120 has to be built of truncated octahedra and hexagonal prisms. At each vertex there meet two of each polyhedra. Consequently, the 120 vertices are distributed among 10
truncated octahedra and 20 hexagonal prisms.}

_{All other polytopes of the group can be obtained by shrinking the A120 and identifying vertices.}

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