Abstract group 101606400.g: $S_7\times A_8$

• Label: 101606400.g
• Order: \( 2^{10} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $15$
• Trans deg.: $56$
• Rank: $2$

Group information

Description:	$S_7\times A_8$
Order:	\(101606400\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$S_7\times S_8$, of order \(203212800\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \)
Composition factors:	$C_2$, $A_7$, $A_8$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	10	12	14	15	20	21	28	30	35	42	60	70	84	105
Elements	1	73311	432782	4317600	679224	13012482	4153680	1305864	16502640	1557360	3389568	4939200	2903040	7560000	13603968	3870720	12096000	3951360	2903040	2419200	1935360	101606400
Conjugacy classes	1	11	8	18	3	43	5	9	27	8	12	6	6	6	23	3	8	7	2	2	2	210
Divisions	1	11	8	18	3	43	3	9	27	5	8	6	4	4	16	2	5	4	1	1	1	180
Autjugacy classes	1	11	8	18	3	43	3	9	27	5	8	6	4	4	16	2	5	4	1	1	1	180

Dimension	1	6	7	14	15	20	21	28	35	42	45	56	64	70	84	90	98	105	120	126	140	147	168	196	210	245	252	270	280	294	300	315	336	384	392	400	420	441	490	525	540	560	588	630	675	700	735	784	840	882	896	900	945	960	980	1050	1120	1176	1225	1260	1280	1344	1350	1400	1470	1575	1800	1890	1960	2240	2450	3150
Irr. complex chars.	2	2	2	6	2	3	8	2	4	2	4	2	2	2	2	0	4	2	2	6	1	2	2	4	4	2	0	4	5	14	2	6	2	2	4	1	9	6	6	2	0	1	2	8	4	3	8	4	2	0	4	2	4	2	6	2	1	2	2	0	1	2	0	1	2	4	0	0	2	2	2	0	210
Irr. rational chars.	2	2	2	6	2	3	4	2	4	4	0	2	2	2	2	2	4	2	2	2	1	2	2	4	4	2	2	0	5	6	2	2	2	2	4	1	7	2	6	2	2	1	6	2	0	3	4	4	3	2	4	0	0	2	6	2	1	2	2	4	1	2	2	1	4	0	1	2	2	2	2	2	180

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$56$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	42	42	42
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $15$ $\langle(3,6)(5,7)(11,15,13,14,12), (1,2), (3,6,4,7,5)(9,10)(11,13,15,14), (3,6,5,7,4) \!\cdots\! \rangle$
Direct product:	$S_7$ $\, \times\, $ $A_8$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$A_8$ . $S_7$	$S_7$ . $A_8$	$(A_7\times A_8)$ . $C_2$	$A_7$ . $(C_2\times A_8)$	more information
Aut. group:	$\Aut(C_2^4\times A_7)$

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 6 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $A_7\times A_8$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 4800 (order at least 21168) are shown, as well as all normal subgroups of any index.

Order 101606400: $S_7\times A_8$

Order 50803200: $A_7\times A_8$

Order 14515200: $S_6\times A_8$

Order 12700800: $A_7\times S_7$

Order 7257600: $A_6\times A_8$

Order 6773760: $S_7\times C_2^3:\GL(3,2)$ x 2

Order 6350400: $A_7^2$

Order 4838400: $C_2\times A_8\times S_5$

Order 3628800: $S_6\times S_7$

Order 3386880: $A_7\times C_2^3.\PSL(2,7)$ x 2, $A_8\times \PSL(2,7)$

Order 2903040: $S_3\times S_4\times A_8$, $S_7\times \POPlus(4,3)$

Order 2419200: $S_5\times A_8$ x 3, $C_2\times A_8\times A_5$

Order 1814400: $S_6\times A_7$ x 2, $C_3:S_5\times S_7$, $A_6\times S_7$, $A_6:S_7$

Order 1451520: $A_4^2.(C_2\times S_7)$ x 2, $S_7\times A_4\wr C_2$ x 2, $S_3\wr C_2\times A_8$, $C_3:S_4\times A_8$, $A_8\times S_3\times A_4$, $C_3\times S_4\times A_8$, $\PSOPlus(4,3).S_7$, $A_4^2.C_2^2\times A_7$, $S_7\times \PSOPlus(4,3)$

Order 1209600: $A_5\times A_8$ x 2

Order 967680: $C_2^3.\PSL(2,7)\times S_6$ x 2, $(C_2\times S_4).A_8$ x 2, $S_7\times C_2^3:S_4$ x 2, $S_3\times A_8\times D_4$, $S_7\times C_2^3:S_4$

Order 907200: $A_6\times A_7$ x 2, $C_3:S_5\times A_7$, $C_3.A_7.S_5$, $S_7\times \GL(2,4)$

Order 846720: $S_7\times \GL(3,2)$ x 3, $F_8:C_3\times S_7$ x 2, $F_7\times A_8$

Order 806400: $C_2\times A_8\times F_5$

Order 725760: $S_3^2\times A_8$ x 2, $A_4\wr C_2\times A_7$ x 2, $A_4^2:S_7$ x 2, $A_7\times \PSOPlus(4,3)$, $C_3^2:C_4\times A_8$, $C_3\times A_4\times A_8$, $A_4^2:S_7$, $S_7\times A_4^2$

Order 604800: $S_5\times S_7$ x 2, $C_2\times S_5\times A_7$

Order 518400: $S_6^2$

Order 483840: $S_4\times A_8$ x 5, $S_7\times C_2^2:S_4$ x 4, $C_2^2.(S_4\times S_7)$ x 4, $(C_2^3\times A_7).S_4$ x 2, $C_2^3.(A_4\times S_7)$ x 2, $C_2^3:S_4\times A_7$ x 2, $A_6\times C_2^3.\PSL(2,7)$ x 2, $C_2\times A_4\times A_8$ x 2, $C_2^2\times S_3\times A_8$ x 2, $A_8\times C_3:D_4$ x 2, $Q_8.(A_4\times S_7)$, $C_2^3:S_4\times A_7$, $S_7\times \GL(2,\mathbb{Z}/4)$, $Q_8:A_4:S_7$, $C_4\times S_3\times A_8$, $C_3\times A_8\times D_4$, $A_8\times D_{12}$

Order 453600: $A_7\times \GL(2,4)$

Order 423360: $A_7\times \GL(3,2)$ x 4, $F_8:C_3\times A_7$ x 2, $C_7:C_3\times A_8$

Order 414720: $S_6\times \POPlus(4,3)$

Order 403200: $F_5\times A_8$ x 2, $D_{10}\times A_8$

Order 362880: $C_3\times S_3\times A_8$ x 2, $C_3:S_4\times S_7$, $S_3\times S_4\times A_7$, $A_4^2\times A_7$, $C_3:S_3\times A_8$, $S_7\times \SOPlus(4,2)$

Order 322560: $(C_2^4\times S_5).\GL(3,2)$ x 2, $C_2\times A_8\times D_4$, $S_7\times C_2\wr C_2^2$

Order 302400: $S_5\times A_7$ x 5, $A_5:S_7$ x 2, $A_5\times S_7$ x 2, $C_{15}:C_4\times S_7$, $C_2\times A_5\times A_7$

Order 282240: $F_8\times S_7$ x 2, $D_7\times A_8$

Order 259200: $A_6\times S_6$ x 2, $C_3:S_5\times S_6$, $A_6:S_6$

Order 241920: $D_6\times A_8$ x 6, $(C_2^2\times A_7):S_4$ x 4, $C_2^2:S_4\times A_7$ x 4, $C_2\times S_4\times S_7$ x 4, $A_4\times A_8$ x 3, $C_2\times C_6\times A_8$ x 2, $S_7\times C_2^4:C_3$ x 2, $C_2^3:A_4\times A_7$ x 2, $(C_2^2\times S_7):A_4$ x 2, $C_3:C_4\times A_8$, $C_{12}\times A_8$, $A_7:\GL(2,\mathbb{Z}/4)$, $C_2^2\times A_4\times S_7$, $Q_8:A_4\times A_7$, $A_4:C_4\times S_7$, $A_7\times \GL(2,\mathbb{Z}/4)$, $A_7:\GL(2,\mathbb{Z}/4)$, $A_7:\GL(2,\mathbb{Z}/4)$

Order 225792: $C_2^3:\GL(3,2)^2$ x 2

Order 207360: $A_4^2:(C_2\times S_6)$ x 2, $A_4\wr C_2\times S_6$ x 2, $S_6\times \PSOPlus(4,3)$, $A_4^2:(C_2\times S_6)$, $A_6\times \POPlus(4,3)$

Order 201600: $D_5\times A_8$ x 2, $C_{10}\times A_8$

Order 193536: $C_2^5.(S_3^2\times \GL(3,2))$ x 2

Order 181440: $C_3:S_4\times A_7$ x 2, $S_3^2\times S_7$ x 2, $A_7\times \SOPlus(4,2)$ x 2, $S_3^2:S_7$ x 2, $(C_3\times A_4):S_7$, $C_3\times A_4\times S_7$, $C_3\times S_4\times A_7$, $S_3\times A_4\times A_7$, $C_3^2\times A_8$, $A_7:\SOPlus(4,2)$, $C_3^2:C_4\times S_7$

Order 172800: $C_2\times S_5\times S_6$

Order 161280: $S_5\times C_2^3:\GL(3,2)$ x 6, $A_8\times D_4$ x 4, $S_7\times C_2^2\wr C_2$ x 3, $C_2^2.D_4.A_7.C_2$ x 3, $C_2^2\wr C_2.A_7.C_2$ x 3, $C_2^3:C_4\times S_7$ x 3, $A_5\times C_2^4:\GL(3,2)$ x 2, $C_2^3\times A_8$ x 2, $Q_8:C_2^2.A_7.C_2$, $D_4:C_2^2:S_7$, $C_2\wr C_2^2\times A_7$, $C_2\times C_4\times A_8$

Order 151200: $A_5\times A_7$ x 4, $(C_{15}\times A_7):C_4$, $C_3\times D_5\times S_7$, $C_{15}:C_4\times A_7$

Order 141120: $F_8\times A_7$ x 2, $C_7\times A_8$

Order 138240: $C_2^3.(S_4\times S_6)$ x 2, $C_2^3.(S_4\times S_6)$, $A_4^2.(C_2^3\times S_5)$

Order 129600: $A_6\times C_3:S_5$, $\GL(2,4):S_6$, $S_6\times \GL(2,4)$, $A_6^2$

Order 120960: $S_4\times S_7$ x 16, $C_2\times S_4\times A_7$ x 6, $C_2\times A_4\times S_7$ x 5, $S_3\times A_8$ x 5, $S_6\times \GL(3,2)$ x 4, $C_2\times A_4:S_7$ x 4, $C_6\times A_8$ x 3, $C_2^2:A_4\times A_7$ x 2, $F_8:C_3\times S_6$ x 2, $S_7\times C_3:D_4$, $S_7\times \SL(2,3)$, $C_2^2\times A_4\times A_7$, $A_4:C_4\times A_7$, $(A_4\times A_7):C_4$, $S_3\times D_4\times A_7$

Order 105840: $C_7:C_3\times S_7$, $F_7\times A_7$

Order 103680: $A_4\wr C_2\times A_6$ x 2, $A_4^2:S_6$ x 2, $A_6\times \PSOPlus(4,3)$, $A_4^2\times S_6$, $S_3\times S_4\times S_6$, $A_4^2:S_6$

Order 100800: $C_2\times F_5\times A_7$, $C_5\times A_8$, $F_5\times S_7$

Order 96768: $S_3\times A_4\times C_2^3:\GL(3,2)$ x 2, $(C_2^2\times C_6):(S_4\times \GL(3,2))$ x 2, $D_6^2:C_2^2:\GL(3,2)$ x 2, $C_3\times S_4\times C_2^3:\GL(3,2)$ x 2, $\GL(3,2)\times \POPlus(4,3)$

Order 90720: $S_3^2\times A_7$ x 4, $C_3\times A_4\times A_7$ x 2, $C_3\times S_3\times S_7$ x 2, $A_7:S_3^2$ x 2, $C_3^2:C_4\times A_7$ x 2, $A_7:S_3^2$ x 2, $C_3:S_3\times S_7$, $(C_3^2\times A_7):C_4$

Order 86400: $S_5\times S_6$ x 7, $C_2\times A_5\times S_6$, $C_2\times S_5\times A_6$, $C_2\times A_5:S_6$, $C_6:S_5^2$

Order 82944: $C_2^6.S_3^4$

Order 80640: $(C_2^2\times S_7):C_4$ x 6, $C_2^2\times A_8$ x 5, $C_2\times D_4\times S_7$ x 4, $(C_2\times D_4):S_7$ x 4, $(C_2\times S_7):D_4$ x 4, $C_2^2:C_4\times S_7$ x 4, $A_5\times C_2^3:\GL(3,2)$ x 4, $(C_2\times D_4):S_7$ x 3, $(C_2^3\times A_7):C_4$ x 3, $C_2^4:S_7$ x 3, $C_2^2\wr C_2\times A_7$ x 3, $A_7\times C_2^3:C_4$ x 3, $C_4\times A_8$ x 2, $A_7\times D_4:C_2^2$, $D_4:C_2.A_7.C_2$, $D_4:C_2\times S_7$, $C_2^4\times S_7$

Order 75600: $C_{15}\times S_7$, $C_{15}:S_7$, $C_3\times D_5\times A_7$

Order 69120: $S_5\times \POPlus(4,3)$ x 9, $C_2^2:S_4\times S_6$ x 4, $C_2^2:S_4:S_6$ x 4, $A_4^2:(C_2^2\times S_5)$ x 2, $A_4^2:C_2^2\times S_5$ x 2, $A_6\times C_2^2:S_4:C_2$ x 2, $C_2^3:A_4:S_6$ x 2, $C_2^3:A_4\times S_6$ x 2, $(C_2^3\times A_6):S_4$, $C_2^3:S_4\times A_6$, $A_4^2:C_2^3\times A_5$, $A_4^2.(C_2^2\times S_5)$, $S_6\times \GL(2,\mathbb{Z}/4)$, $A_4^2:(C_2^2\times S_5)$, $Q_8:A_4\times S_6$

Order 64800: $A_6\times \GL(2,4)$

Order 64512: $(C_2^4\times S_4).\GL(3,2)$ x 4, $C_2^5.(D_6\times \GL(3,2))$ x 2

Order 60480: $S_4\times A_7$ x 14, $A_4\times S_7$ x 10, $A_4:S_7$ x 9, $C_2\times A_4\times A_7$ x 6, $A_6\times \GL(3,2)$ x 4, $C_3:D_4\times A_7$ x 3, $F_8:C_3\times A_6$ x 2, $C_2\times D_6\times A_7$ x 2, $D_6\times S_7$ x 2, $C_3\times A_8$ x 2, $C_3\times D_4\times A_7$, $C_2\times C_6\times S_7$, $(C_6\times S_7):C_2$, $D_{12}\times A_7$, $C_4\times S_3\times A_7$, $A_7\times \SL(2,3)$, $C_3:S_5\times \GL(3,2)$, $C_3:C_4\times S_7$, $(C_2\times C_6):S_7$, $D_6:S_7$

Order 56448: $F_7\times C_2^3:\GL(3,2)$ x 2

Order 53760: $C_2^4:\GL(3,2)\times F_5$ x 2

Order 52920: $C_7:C_3\times A_7$ x 2

Order 51840: $S_6\times \SOPlus(4,2)$ x 2, $C_3:S_4\times S_6$ x 2, $C_3\times S_4\times S_6$, $S_4\times A_5:S_3^2$, $A_4^2\times A_6$, $C_3:(S_4\times S_6)$, $S_3\times S_4\times A_6$, $S_3\times A_4:S_6$, $S_4\times C_3:S_6$, $S_3\times A_4\times S_6$

Order 50400: $F_5\times A_7$ x 3, $D_{10}\times A_7$, $A_7:F_5$, $D_5\times S_7$

Order 48384: $S_3^2\times C_2^3:\GL(3,2)$ x 4, $A_4\wr C_2\times \GL(3,2)$ x 2, $(C_2\times C_6^2):(C_4\times \GL(3,2))$ x 2, $C_2^5:(C_3^2\times \GL(3,2))$ x 2, $\GL(3,2)\times \PSOPlus(4,3)$

Order 46080: $(C_2^4\times S_5).S_4$ x 2, $S_6\times C_2\wr C_2^2$, $C_2^4.(S_4\times S_5)$

Order 45360: $C_3\times S_3\times A_7$ x 4, $C_3:S_3\times A_7$ x 2, $C_3^2:S_7$ x 2, $C_3^2\times S_7$, $C_3^2:S_7$

Order 43200: $C_3:S_5^2$ x 5, $A_5:S_6$ x 5, $A_5\times S_6$ x 5, $S_5\times A_6$ x 5, $C_{15}:C_4\times S_6$, $A_5^2:D_6$, $C_2\times \GL(2,4):S_5$, $C_6\times A_5\times S_5$, $C_2\times A_5\times A_6$

Order 41472: $S_4\times A_4^2:D_6$ x 2, $A_4^2.(D_6\times S_4)$ x 2, $A_4^2.(D_6\times S_4)$ x 2, $S_3\times A_4^3:C_2^2$ x 2, $A_4^3.(C_2^2\times C_6)$, $S_4\times A_4^2:D_6$, $A_4^2.(D_6\times S_4)$, $S_3\times A_4^3:C_2^2$, $A_4^3:C_2^2\times S_3$, $A_4^2:(C_6^2:D_4)$, $A_4^3.(C_2\times D_6)$, $(S_3\times A_4).S_4^2$

Order 40320: $D_4\times S_7$ x 13, $C_2^3:S_7$ x 8, $C_2^3\times S_7$ x 7, $C_2\times D_4\times A_7$ x 5, $(C_2\times C_4):S_7$ x 5, $C_2^3.S_7$ x 4, $C_2\times C_4\times S_7$ x 4, $C_2^2:C_4\times A_7$ x 4, $C_2\times C_4:S_7$ x 4, $(C_2\times C_4):S_7$ x 3, $D_4:S_7$ x 3, $C_2\times A_8$ x 3, $C_2\times S_5\times \GL(3,2)$ x 3, $F_8\times S_6$ x 2, $F_8:C_6\times S_5$ x 2, $D_4:C_2\times A_7$, $Q_8:S_7$, $C_2^4\times A_7$, $Q_8\times S_7$

Order 37800: $C_{15}\times A_7$

Order 35280: $C_7\times S_7$, $D_7\times A_7$

Order 34560: $A_4^2:(C_2\times S_5)$ x 10, $A_4\wr C_2\times S_5$ x 10, $C_2\times S_4\times S_6$ x 6, $A_4^2:(C_2\times S_5)$ x 5, $A_5\times \POPlus(4,3)$ x 5, $S_5\times \PSOPlus(4,3)$ x 5, $C_2^2:S_4\times A_6$ x 4, $(C_2^2\times A_6):S_4$ x 4, $C_2^3:A_4\times A_6$ x 2, $C_2^2:A_4\times S_6$ x 2, $(C_2^2\times S_6):A_4$ x 2, $C_2\times A_4^2:S_5$ x 2, $A_4^2:C_2^2\times A_5$ x 2, $C_2^2\times A_4\times S_6$, $A_4^2.C_2^2.A_5$, $A_6:\GL(2,\mathbb{Z}/4)$, $A_6\times \GL(2,\mathbb{Z}/4)$, $C_2\times S_5\times A_4^2$, $C_2\times A_4^2:S_5$, $Q_8:A_4\times A_6$, $S_3\times D_4\times S_6$, $A_6:\GL(2,\mathbb{Z}/4)$, $A_6:\GL(2,\mathbb{Z}/4)$, $A_4:C_4\times S_6$

Order 32256: $S_4\times C_2^3:\GL(3,2)$ x 10, $A_4\times C_2^4:\GL(3,2)$ x 4, $(D_6\times C_2^4).\GL(3,2)$ x 4, $C_2^5:(S_3\times \GL(3,2))$ x 4, $C_2^3:S_4\times \GL(3,2)$ x 2, $D_{12}\times C_2^3:\GL(3,2)$ x 2, $C_2^4.(D_6\times \GL(3,2))$ x 2, $C_3\times D_4\times C_2^3.\PSL(2,7)$ x 2, $C_2^3:S_4\times \PSL(2,7)$

Order 30240: $A_4\times A_7$ x 9, $D_6\times A_7$ x 8, $S_3\times S_7$ x 6, $C_2\times C_6\times A_7$ x 3, $C_6\times S_7$ x 3, $C_3:C_4\times A_7$ x 2, $C_6:S_7$ x 2, $C_6.S_7$, $C_{12}\times A_7$, $\GL(2,4)\times \PSL(2,7)$, $F_7\times S_6$

Order 28800: $C_2\times S_5^2$ x 2, $C_2\times F_5\times S_6$

Order 28224: $\GL(3,2)^2$ x 3, $F_8:C_3\times \GL(3,2)$ x 2, $(C_2^2\times C_{14}):(C_3\times \GL(3,2))$ x 2

Order 27648: $(C_2^2\times D_6).S_4^2$ x 2, $A_4^3.C_2^4$ x 2, $(C_{12}\times A_4^2).C_2^4$, $(C_2^2\times D_6).S_4^2$

Order 26880: $F_5\times C_2^3:\GL(3,2)$ x 4, $C_2^3:\GL(3,2)\times D_{10}$ x 2

Order 25920: $S_3^2\times S_6$ x 4, $S_3^2:S_6$ x 4, $A_6:\SOPlus(4,2)$ x 2, $C_3^2:C_4\times S_6$ x 2, $A_6\times \SOPlus(4,2)$ x 2, $C_3:S_4\times A_6$ x 2, $(C_3\times A_4):S_6$ x 2, $C_3\times A_4\times S_6$ x 2, $C_3^3:(D_4\times S_5)$, $S_4\times C_3^2:S_5$, $C_3^2:(S_4\times S_5)$, $C_3^2:(S_4\times S_5)$, $C_3^2:S_5\times S_4$, $A_4\times C_3:S_6$, $C_3\times S_4\times A_6$, $S_3\times A_4\times A_6$, $C_3\times A_4:S_6$, $S_3\times \GL(2,4):S_4$, $A_4\times A_5:S_3^2$, $S_3\times S_4\times \GL(2,4)$

Order 25200: $D_5\times A_7$ x 3, $C_{10}\times A_7$, $C_5:S_7$, $C_5\times S_7$

Order 24192: $C_6^2:C_2^2:\GL(3,2)$ x 4, $S_3\times S_4\times \GL(3,2)$ x 3, $F_8:C_6^2:D_6$ x 2, $C_6^2:C_2^2:\GL(3,2)$ x 2, $A_4^2\times \GL(3,2)$, $F_7\times \POPlus(4,3)$

Order 23040: $C_2^3:S_4\times S_5$ x 18, $C_2^3:S_4\times S_5$ x 5, $C_2^3:S_4\times S_5$ x 4, $(C_2^3\times S_5):S_4$ x 4, $A_6:C_2\wr C_2^2$ x 3, $C_2^4:(C_2\times S_6)$ x 3, $C_2^3:C_4\times S_6$ x 3, $S_6\times C_2^2\wr C_2$ x 3, $C_2^4:S_4\times A_5$ x 2, $C_2^4:A_4\times S_5$ x 2, $(C_2^4\times A_5):S_4$ x 2, $(C_2^4\times A_5):S_4$, $C_4.(C_2^3\times S_6)$, $(C_2^3\times A_6):D_4$, $C_2^4.(A_4\times S_5)$, $(C_2\times S_5).\GL(2,\mathbb{Z}/4)$, $C_2\wr C_2^2\times A_6$, $C_2^4:S_4\times A_5$, $A_4^2:C_2^3\times F_5$

Order 22680: $C_3^2\times A_7$ x 2

Order 21600: $A_5\times A_6$ x 4, $S_5\times \GL(2,4)$ x 3, $\GL(2,4):S_5$ x 3, $\GL(2,4):S_5$ x 3, $(C_{15}\times A_6):C_4$, $C_3\times D_5\times S_6$, $C_{15}:C_4\times A_6$, $C_6\times A_5^2$

Order 21504: $(D_4\times C_2^4).\GL(3,2)$ x 2

Order 20160: $A_8$

Order 5040: $S_7$

Order 2520: $A_7$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 101606400: $S_7\times A_8$ ($C_1$)

Order 50803200: $A_7\times A_8$ ($C_2$)

Order 20160: $A_8$ ($S_7$)

Order 5040: $S_7$ ($A_8$)

Order 2520: $A_7$ ($C_2\times A_8$)

Order 1: $C_1$ ($S_7\times A_8$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $210 \times 210$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $180 \times 180$ rational character table (warning: may be slow to load).