Abstract group 192036096000.f: $C_2.A_7\wr S_3$

• Label: 192036096000.f
• Order: \( 2^{11} \cdot 3^{7} \cdot 5^{3} \cdot 7^{3} \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{7} \cdot 5^{3} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $23$
• Trans deg.: $42$
• Rank: $2$

Group information

Description:	$C_2.A_7\wr S_3$
Order:	\(192036096000\)\(\medspace = 2^{11} \cdot 3^{7} \cdot 5^{3} \cdot 7^{3} \)
Exponent:	\(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Automorphism group:	Group of order \(384072192000\)\(\medspace = 2^{12} \cdot 3^{7} \cdot 5^{3} \cdot 7^{3} \)
Composition factors:	$C_2$ x 2, $C_3$, $A_7$ x 3
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	12	14	15	18	20	21	24	28	30	35	40	42	56	60	70	84	105	140	210
Elements	1	3984751	55944350	1972993680	128787624	6757999010	374805360	7010841600	4445280000	4978905624	27308836800	9752985360	6854198400	9779616000	8165344320	9955008000	5334336000	12252643200	15043674240	1334672640	4800902400	21850214400	6858432000	3734035200	12765392640	5334336000	762048000	2743372800	1676505600	192036096000
Conjugacy classes	1	11	10	20	3	82	9	6	2	17	54	43	8	4	12	18	6	26	34	7	2	72	4	8	19	16	4	4	8	510
Divisions	1	11	10	20	3	82	5	6	2	17	54	23	8	4	12	10	6	14	34	4	2	38	2	8	10	8	2	2	4	402
Autjugacy classes	1	9	10	16	3	69	5	3	2	13	43	18	8	4	10	10	3	12	29	4	1	33	1	7	8	7	2	2	4	337

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$42$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $23$ $\langle(1,2,3,5,9,15)(4,7,13)(6,11,17)(8,12,10,16,20,21)(14,18,19)(22,23), (2,4,8,14,19,15,16,20,9,11,13,3,6,12)(5,10,17,18,21)(22,23)\rangle$
Transitive group:	42T7689				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$A_7^3$ . $D_6$	$(A_7\wr S_3)$ . $C_2$ (2)	$C_2$ . $(A_7\wr S_3)$	$(A_7\wr C_3)$ . $C_2^2$	all 6

Elements of the group are displayed as permutations of degree 23.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 9 normal subgroups (7 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	a subgroup isomorphic to $A_7\wr C_3$
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 36750 (order at least 5225472) are shown, as well as all normal subgroups of any index.

Order 192036096000: $C_2.A_7\wr S_3$

Order 96018048000: $A_7\wr S_3$ x 2, $C_2.A_7\wr C_3$

Order 64012032000: $C_2.A_7.A_7^2.C_2$

Order 48009024000: $A_7\wr C_3$

Order 32006016000: $A_7.A_7^2.C_2$ x 2, $C_2.A_7.A_7.A_7$

Order 16003008000: $A_7^3$

Order 9144576000: $C_2.A_6.A_7^2.C_2$

Order 4572288000: $A_6.A_7^2.C_2$ x 2, $C_2.A_7.A_7.A_6$

Order 4267468800: $(C_2\times \PSL(2,7)).A_7^2.C_2$ x 2

Order 3048192000: $(C_2\times A_5).A_7^2.C_2^2$

Order 2286144000: $A_7^2\times A_6$

Order 2133734400: $\PSL(2,7).A_7^2.C_2$ x 4, $(C_2\times \PSL(2,7)).A_7.A_7$ x 2

Order 1828915200: $(C_2\times C_3:S_4).A_7^2.C_2$

Order 1524096000: $A_5.A_7^2.C_2^2$ x 4, $(C_2\times A_5).A_7^2.C_2$ x 3, $C_2.A_7.A_5.A_7.C_2$

Order 1306368000: $C_2.A_7.A_6^2.C_2$

Order 1066867200: $\PSL(2,7)\times A_7^2$ x 2

Order 914457600: $C_3:S_4.A_7^2.C_2$ x 4, $(C_6\times A_4).A_7^2.C_2$ x 2, $(C_2\times C_3:S_4).A_7.A_7$, $(C_2\times C_3:S_3.C_2).A_7^2.C_2$

Order 762048000: $A_5.A_7^2.C_2$ x 6, $A_7^2\times S_5$ x 2, $C_2.A_7.A_7.A_5$, $C_2.A_7.A_5.A_7$

Order 653184000: $A_7.A_6^2.C_2$ x 2, $C_2.A_6.A_7.A_6$

Order 609638400: $(C_2\times S_4).A_7^2.C_2$ x 4, $(C_2\times C_3:D_4).A_7^2.C_2$

Order 559872000: $C_2\times A_6\wr S_3$

Order 533433600: $(C_2\times C_7:C_3).A_7^2.C_2$

Order 508032000: $(C_2\times F_5).A_7^2.C_2$

Order 457228800: $C_3:S_3.C_2.A_7^2.C_2$ x 4, $(C_3\times A_4).A_7^2.C_2$ x 4, $C_3:S_4\times A_7^2$ x 2, $(C_2\times C_3:S_3).A_7^2.C_2$ x 2, $(C_6\times A_4).A_7.A_7$, $(C_2\times C_3:S_3.C_2).A_7.A_7$

Order 381024000: $A_7^2\times A_5$ x 2

Order 326592000: $A_6^2\times A_7$

Order 304819200: $S_4.A_7^2.C_2$ x 16, $C_3:D_4.A_7^2.C_2$ x 8, $(C_2\times A_4).A_7^2.C_2$ x 8, $(C_2\times S_4).A_7.A_7$ x 4, $C_2.A_7.\PSL(2,7).A_6$ x 2, $(C_2^2\times S_3).A_7^2.C_2$ x 2, $(C_2^2\times C_6).A_7^2.C_2$ x 2, $(C_2\times C_3:C_4).A_7^2.C_2$ x 2, $(C_2\times C_3:D_4).A_7.A_7$

Order 284497920: $C_2.A_7.\PSL(2,7)^2.C_2$ x 2

Order 279936000: $A_6\wr S_3$ x 2, $C_2\times A_6\wr C_3$

Order 266716800: $C_7:C_3.A_7^2.C_2$ x 2, $(C_2\times C_7:C_3).A_7.A_7$

Order 254016000: $F_5.A_7^2.C_2$ x 4, $D_{10}.A_7^2.C_2$ x 2, $(C_2\times F_5).A_7.A_7$

Order 228614400: $C_3:S_3.A_7^2.C_2$ x 4, $A_7^2\times C_3:S_3.C_2$ x 2, $(C_3\times C_6).A_7^2.C_2$ x 2, $A_7^2.C_3^2:C_4$ x 2, $(C_3\times A_4).A_7.A_7$, $(C_2\times C_3:S_3).A_7.A_7$

Order 217728000: $(C_2\times A_5).A_7.A_6.C_2$

Order 203212800: $(C_2\times D_4).A_7^2.C_2$

Order 186624000: $C_2.A_6.A_6^2.C_2$

Order 177811200: $C_{14}.A_7^2.C_2$

Order 152409600: $A_4.A_7^2.C_2$ x 16, $D_6.A_7^2.C_2$ x 13, $(C_2\times C_6).A_7^2.C_2$ x 12, $C_3:C_4.A_7^2.C_2$ x 8, $S_4.A_7.A_7$ x 6, $C_3:D_4.A_7.A_7$ x 4, $(C_2\times A_4).A_7.A_7$ x 4, $\PSL(2,7)\times A_6\times A_7$ x 2, $S_4\times A_7^2$, $A_7^2\times S_4$, $(C_2^2\times S_3).A_7.A_7$, $(C_2^2\times C_6).A_7.A_7$, $(C_2\times C_3:C_4).A_7.A_7$

Order 145152000: $C_2.A_7.A_5^2.D_4$

Order 142248960: $A_7.\PSL(2,7)^2.C_2$ x 4, $(C_2\times \PSL(2,7)).A_7.\PSL(2,7)$ x 2, $(C_2\times \PSL(2,7)).\PSL(2,7).A_7$

Order 139968000: $A_6\wr C_3$

Order 133358400: $C_7:C_3\times A_7^2$

Order 130636800: $(C_2\times C_3:S_4).A_7.A_6$

Order 127008000: $D_5.A_7^2.C_2$ x 6, $F_5\times A_7^2$ x 2, $C_{10}.A_7^2.C_2$ x 2, $D_{10}.A_7.A_7$

Order 114307200: $C_3^2.A_7^2.C_2$ x 4, $C_3:S_3.A_7\times A_7$ x 2, $(C_3\times C_6).A_7.A_7$

Order 108864000: $A_6\times A_7\times S_5$ x 2, $(C_2\times A_5).A_7.A_6$ x 2

Order 101606400: $D_4.A_7^2.C_2$ x 8, $C_2^3.A_7^2.C_2$ x 4, $(C_2\times \PSL(2,7)).A_5.A_7.C_2$ x 2, $(C_2\times C_4).A_7^2.C_2$ x 2, $(C_2\times D_4).A_7.A_7$

Order 93312000: $A_6.A_6^2.C_2$ x 2, $C_2.A_6.A_6.A_6$

Order 88905600: $C_7.A_7^2.C_2$ x 2, $C_{14}.A_7.A_7$

Order 87091200: $(C_2\times \PSL(2,7)).A_6^2.C_2$ x 2

Order 76204800: $C_6.A_7^2.C_2$ x 17, $A_7^2.D_6$ x 12, $C_2\times A_7^2\times S_3$ x 3, $A_4.A_7.A_7$ x 3, $(C_2\times C_6).A_7.A_7$ x 3, $D_6.A_7.A_7$ x 2, $C_3:C_4.A_7.A_7$ x 2, $A_4.A_7\times A_7$

Order 72576000: $A_7.A_5^2.D_4$ x 4, $C_2.A_7.A_5^2.C_4$, $C_2.A_7.A_5^2.C_2^2$, $(C_2\times A_5).A_5.A_7.C_2^2$

Order 71124480: $\PSL(2,7)^2\times A_7$ x 2, $A_7\times \PSL(2,7)^2$

Order 65318400: $C_3:S_4.A_7.A_6$ x 2, $(C_6\times A_4).A_7.A_6$, $(C_2\times C_3:S_3.C_2).A_7.A_6$

Order 63504000: $C_5.A_7^2.C_2$ x 4, $D_5.A_7.A_7$ x 2, $C_{10}.A_7.A_7$

Order 62208000: $(C_2\times A_5).A_6^2.C_2^2$

Order 60963840: $(C_2\times C_3:S_4).\PSL(2,7).A_7$ x 2

Order 57153600: $C_3^2.A_7.A_7$

Order 56899584: $C_2\times \PSL(2,7)\wr S_3$ x 2

Order 54432000: $A_7\times A_6\times A_5$, $A_5\times A_7\times A_6$

Order 52254720: $A_4^2.C_3:S_3.C_2^3.A_7$

Order 50803200: $C_2^2.A_7^2.C_2$ x 12, $C_4.A_7^2.C_2$ x 8, $A_7^2.D_4$ x 8, $S_5\times \PSL(2,7)\times A_7$ x 4, $D_4.A_7.A_7$ x 4, $C_2^3.A_7.A_7$ x 2, $C_2.A_7.\PSL(2,7).A_5$ x 2, $(C_2\times \PSL(2,7)).A_5.A_7$ x 2, $(C_2\times C_4).A_7.A_7$

Order 46656000: $A_6^3$

Order 44452800: $C_7.A_7.A_7$

Order 43545600: $\PSL(2,7).A_6^2.C_2$ x 4, $(C_2\times S_4).A_7.A_6$ x 4, $C_2.A_6.\PSL(2,7).A_6$ x 2, $(C_2\times C_3:S_4).A_5.A_7.C_2$, $(C_2\times C_3:D_4).A_7.A_6$

Order 40642560: $C_2.A_6.\PSL(2,7)^2.C_2$ x 2

Order 38102400: $A_7^2.C_6$ x 6, $S_3.A_7.A_7$ x 4, $C_6.A_7.A_7$ x 4, $A_7^2.S_3$ x 4, $(C_2\times C_7:C_3).A_7.A_6$

Order 37324800: $(C_2\times C_3:S_4).A_6^2.C_2$

Order 36288000: $A_7.A_5^2.C_2^2$ x 3, $C_2.A_7.A_5^2.C_2$ x 2, $A_7.A_5^2.C_4$ x 2, $A_5.A_5.C_2^2\times A_7$ x 2, $(C_2\times A_5).A_5.A_7.C_2$ x 2, $A_5.S_5.C_2\times A_7$, $(C_2\times F_5).A_7.A_6$, $(C_2\times A_5).A_7.A_5.C_2$

Order 32659200: $C_3:S_3.C_2\times A_7\times A_6$ x 2, $(C_3\times A_4).A_7.A_6$, $(C_2\times C_3:S_3).A_7.A_6$

Order 31752000: $C_5.A_7.A_7$

Order 31104000: $A_5.A_6^2.C_2^2$ x 4, $(C_2\times A_5).A_6^2.C_2$ x 3, $C_2.A_6.A_6.S_5$

Order 30481920: $C_3:S_4.\PSL(2,7).A_7$ x 4, $(C_6\times A_4).\PSL(2,7).A_7$ x 2, $(C_2\times C_3:S_3.C_2).\PSL(2,7).A_7$ x 2

Order 28449792: $\PSL(2,7)\wr S_3$ x 4, $C_2\times \PSL(2,7)\wr C_3$ x 2

Order 26127360: $A_4^2.S_3\wr C_2.A_7$ x 4, $C_2^4.C_3^4.C_2^3.A_7$ x 2, $A_4^2.C_3:S_3.C_2^2.A_7$

Order 25401600: $A_7^2.C_2^2$ x 12, $C_2^2.A_7.A_7$ x 6, $C_4.A_7.A_7$ x 2, $A_7^2.C_4$ x 2, $\PSL(2,7)\times A_7\times A_5$, $\PSL(2,7)\times A_5\times A_7$, $C_2.A_7.A_7.C_2$, $A_5\times \PSL(2,7)\times A_7$, $A_5\times A_7\times \PSL(2,7)$

Order 21772800: $S_4.A_7.A_6$ x 4, $C_3:S_4.A_5.A_7.C_2$ x 4, $C_3:D_4.A_7.A_6$ x 4, $(C_2\times A_4).A_7.A_6$ x 4, $A_6\times S_4\times A_7$ x 3, $A_6^2\times \PSL(2,7)$ x 2, $(C_6\times A_4).A_5.A_7.C_2$ x 2, $S_4\times A_6\times A_7$, $C_3:S_3.(C_2^2\times A_5).A_7.C_2$, $(C_2^2\times S_3).A_7.A_6$, $(C_2^2\times C_6).A_7.A_6$, $(C_2\times C_3:S_4).A_7.A_5$, $(C_2\times C_3:S_4).A_5.A_7$, $(C_2\times C_3:C_4).A_7.A_6$

Order 20736000: $C_2.A_6.A_5^2.D_4$, $S_5^3.D_6$

Order 20321280: $A_6.\PSL(2,7)^2.C_2$ x 4, $(C_2\times S_4).\PSL(2,7).A_7$ x 4, $(C_2\times S_4).A_7.\PSL(2,7)$ x 4, $(C_2\times \PSL(2,7)).\PSL(2,7).A_6$ x 3, $(C_2\times C_3:D_4).\PSL(2,7).A_7$ x 2

Order 19051200: $C_3.A_7.A_7$ x 2, $C_7:C_3.A_7.A_6$

Order 18966528: $(C_2\times \PSL(2,7)).\PSL(2,7)^2.C_2$ x 4

Order 18662400: $C_3:S_4.A_6^2.C_2$ x 4, $(C_6\times A_4).A_6^2.C_2$ x 2, $(C_2\times C_3:S_4).A_6.A_6$, $(C_2\times C_3:S_3.C_2).A_6^2.C_2$

Order 18144000: $A_7.A_5^2.C_2$ x 4, $S_5\times A_5\times A_7$ x 2, $A_5\times A_7\times S_5$ x 2, $A_5.S_5\times A_7$ x 2, $(C_2\times A_5).A_7.A_5$ x 2, $F_5\times A_7\times A_6$, $D_{10}.A_7.A_6$, $A_7\times F_5\times A_6$, $(C_2\times A_5).A_5.A_7$

Order 17781120: $(C_2\times C_7:C_3).A_7.\PSL(2,7)$ x 2

Order 16934400: $(C_2\times F_5).\PSL(2,7).A_7$ x 2

Order 16329600: $C_3:S_3.A_7.A_6$ x 2, $(C_3\times C_6).A_7.A_6$

Order 15552000: $A_5.A_6^2.C_2$ x 6, $A_6.A_6.S_5$ x 2, $C_2.A_6.A_6.A_5$, $(C_2\times A_5).A_6.A_6$

Order 15240960: $C_3:S_3.C_2\times A_7\times \PSL(2,7)$ x 2, $(C_3\times A_4).\PSL(2,7).A_7$ x 2, $(C_2\times C_3:S_3).\PSL(2,7).A_7$ x 2, $C_3:S_3.C_2\times \PSL(2,7)\times A_7$, $A_7\times \PSL(2,7)\times C_3:S_3.C_2$

Order 14515200: $(C_2\times S_4).A_5.A_7.C_2$ x 4, $(C_2\times A_5).\PSL(2,7).A_6.C_2$ x 2, $C_6.(C_2^3\times A_5).A_7.C_2$, $(C_2\times D_4).A_7.A_6$

Order 14224896: $\PSL(2,7)\wr C_3$ x 2

Order 13547520: $(C_2\times A_5).\PSL(2,7)^2.C_2^2$ x 2

Order 13063680: $(C_3\times A_4^2).D_6.A_7$ x 6, $A_4^2.C_3^2.C_4.A_7$ x 2, $C_3^4.(C_2\times C_4\wr C_2).A_7$, $C_2^4.C_3^4.C_2^2.A_7$, $A_4^2.S_3^2.A_7$, $A_4.(C_6\times S_3\times A_4).A_7$, $(C_3^3\times A_4).C_2^3.C_2.A_7$

Order 12700800: $C_2.A_7.A_7$ x 7, $A_7\wr C_2$ x 4, $A_7\times S_7$ x 2, $C_{14}.A_7.A_6$, $C_7.(C_6\times A_5).A_7.C_2$

Order 12441600: $(C_2\times S_4).A_6^2.C_2$ x 4, $(C_2\times C_3:D_4).A_6^2.C_2$

Order 12096000: $(C_2\times F_5).A_5.A_7.C_2$

Order 10886400: $S_5\times C_3:S_3.C_2\times A_7$ x 4, $A_4.A_7.A_6$ x 4, $(C_3\times A_4).A_5.A_7.C_2$ x 4, $D_6\times A_7\times A_6$ x 3, $(C_2\times C_6).A_7.A_6$ x 3, $D_6.A_7.A_6$ x 2, $C_3:S_4.A_7.A_5$ x 2, $C_3:S_4.A_5.A_7$ x 2, $C_3:C_4.A_7.A_6$ x 2, $(C_2\times C_3:S_3).A_5.A_7.C_2$ x 2, $C_3:S_3.(C_2^2\times A_5).A_7$, $(C_6\times A_4).A_7.A_5$, $(C_6\times A_4).A_5.A_7$, $(C_2\times C_7:C_3).A_6^2.C_2$, $(C_2\times C_3:S_3.C_2).A_7.A_5$

Order 10368000: $A_6.A_5^2.D_4$ x 4, $S_5^3.S_3$ x 4, $C_2.A_6.A_5^2.C_4$, $C_2.A_6.A_5^2.C_2^2$, $C_2.A_6.A_5.S_5.C_2$, $(C_2\times F_5).A_6^2.C_2$, $A_5^3.(C_2\times S_4)$, $A_5^3.(C_2\times S_4)$, $S_5^3.C_6$

Order 10160640: $S_4.\PSL(2,7).A_7$ x 8, $C_3:D_4.\PSL(2,7).A_7$ x 8, $(C_2\times A_4).\PSL(2,7).A_7$ x 4, $(C_2\times A_4).A_7.\PSL(2,7)$ x 4, $A_7\times S_4\times \PSL(2,7)$ x 3, $\PSL(2,7)\times A_7\times S_4$ x 2, $S_4\times \PSL(2,7)\times A_7$ x 2, $D_6.(C_2\times \PSL(2,7)).A_7$ x 2, $A_6\times \PSL(2,7)^2$ x 2, $(C_2^2\times C_6).\PSL(2,7).A_7$ x 2, $(C_2\times C_3:C_4).\PSL(2,7).A_7$ x 2, $\PSL(2,7)^2\times A_6$, $A_7\times \PSL(2,7)\times S_4$

Order 9676800: $(C_2\times \PSL(2,7)).A_5^2.D_4$ x 2

Order 9483264: $\PSL(2,7).\PSL(2,7)^2.C_2$ x 8, $(C_2\times \PSL(2,7)).\PSL(2,7).\PSL(2,7)$ x 4

Order 9331200: $C_3:S_3.C_2.A_6^2.C_2$ x 4, $(C_3\times A_4).A_6^2.C_2$ x 4, $C_3:S_4.A_6.A_6$ x 2, $(C_2\times C_3:S_3).A_6^2.C_2$ x 2, $(C_6\times A_4).A_6.A_6$, $(C_2\times C_3:S_3.C_2).A_6.A_6$

Order 9072000: $A_5^2\times A_7$ x 3, $D_5\times A_7\times A_6$, $C_{10}.A_7.A_6$, $A_7\times D_5\times A_6$

Order 8890560: $C_7:C_3\times A_7\times \PSL(2,7)$, $A_7\times C_7:C_3\times \PSL(2,7)$

Order 8709120: $A_4^2.D_6.C_2.A_7$ x 3, $(C_3\times A_4^2).C_2^3.A_7$ x 3, $(C_2\times C_3:S_4).\PSL(2,7).A_6$ x 2, $C_2^4:C_3.(C_2\times S_3^2).A_7$, $A_4^2.C_6.C_2^2.A_7$, $A_4.D_6^2.C_2.A_7$

Order 8467200: $F_5\times \PSL(2,7)\times A_7$ x 2, $D_{10}.\PSL(2,7).A_7$ x 2, $A_7\times F_5\times \PSL(2,7)$ x 2

Order 8164800: $C_3^2.A_7.A_6$

Order 8128512: $(C_2\times C_3:S_4).\PSL(2,7)^2.C_2$ x 2

Order 7776000: $A_6^2\times A_5$, $A_5\times A_6^2$

Order 7620480: $C_3:S_3.\PSL(2,7).A_7$ x 4, $(C_3\times C_6).\PSL(2,7).A_7$ x 2, $(C_{21}\times A_4).C_6.C_2.A_7$

Order 7464960: $C_2^4.C_3^4.C_2^3.C_2.A_6$

Order 7257600: $S_4\times S_5\times A_7$ x 8, $S_4.A_5.A_7.C_2$ x 8, $C_3:D_4.A_5.A_7.C_2$ x 8, $(C_2\times A_4).A_5.A_7.C_2$ x 8, $D_4.A_7.A_6$ x 4, $(C_2\times S_4).A_7.A_5$ x 4, $(C_2\times S_4).A_5.A_7$ x 4, $(C_2\times A_5).\PSL(2,7).A_6$ x 4, $\PSL(2,7)\times S_5\times A_6$ x 3, $D_6.(C_2\times A_5).A_7.C_2$ x 2, $C_6.(C_2^2\times A_5).A_7.C_2$ x 2, $C_2^3.A_7.A_6$ x 2, $(C_2^2\times C_6).A_5.A_7.C_2$ x 2, $S_5\times A_6\times \PSL(2,7)$, $D_6.(C_2^2\times A_5).A_7$, $(C_5\times A_4).D_6.C_2^2.A_7$, $(C_2\times C_4).A_7.A_6$, $(C_2\times C_3:D_4).A_7.A_5$

Order 6912000: $(C_2\times A_5).A_5^2.D_4.C_2$

Order 6773760: $A_5.\PSL(2,7)^2.C_2^2$ x 8, $(C_2\times A_5).\PSL(2,7)^2.C_2$ x 6, $(C_2\times \PSL(2,7)).A_5.\PSL(2,7).C_2$ x 3, $(C_2\times D_4).A_7.\PSL(2,7)$ x 2

Order 6531840: $C_2^4.C_3^4.C_2.A_7$ x 5, $C_3^4.C_4\wr C_2.A_7$ x 4, $A_4.(S_3\times C_3:S_3).C_2.A_7$ x 3, $C_3^4.C_2^3.C_2^2.A_7$ x 2, $(C_6^2\times A_4).C_6.A_7$ x 2, $C_6^2.C_6^2.C_2.A_7$, $C_3^4.(C_2\times \OD_{16}).A_7$, $A_4.(C_6\times C_3:S_3).C_2.A_7$, $(C_3^3\times A_4).C_2^3.A_7$, $(C_3\times C_6^2).D_6.C_2.A_7$

Order 6350400: $A_7^2$ x 3, $C_7:C_3\times A_7\times S_5$ x 2, $C_7.A_7.A_6$, $C_7.(C_6\times A_5).A_7$, $(C_2\times C_7:C_3).A_7.A_5$

Order 6220800: $S_4.A_6^2.C_2$ x 16, $C_3:D_4.A_6^2.C_2$ x 8, $(C_2\times A_4).A_6^2.C_2$ x 8, $(C_2\times S_4).A_6.A_6$ x 4, $(C_2^2\times S_3).A_6^2.C_2$ x 2, $(C_2^2\times C_6).A_6^2.C_2$ x 2, $(C_2\times C_3:C_4).A_6^2.C_2$ x 2, $(C_2\times C_3:S_4).A_6.S_5$, $(C_2\times C_3:D_4).A_6.A_6$

Order 6048000: $F_5.S_5\times A_7$ x 4, $D_{10}.A_5.A_7.C_2$ x 2, $(C_2\times F_5).A_7.A_5$, $(C_2\times F_5).A_5.A_7$

Order 5927040: $C_{14}.A_7.\PSL(2,7)$ x 2

Order 5806080: $A_4^2.C_2^3.C_2.A_7$ x 3, $C_6^2.C_2^5.C_2.A_7$, $C_2\times A_4^2.D_4.A_7$

Order 5443200: $C_3:S_3.A_5.A_7.C_2$ x 6, $S_3.A_7.A_6$ x 4, $C_6.A_7.A_6$ x 4, $C_7:C_3.A_6^2.C_2$ x 2, $C_3:S_3.C_2.A_7.A_5$ x 2, $C_3:S_3.C_2.A_5.A_7$ x 2, $(C_3\times C_6).A_5.A_7.C_2$ x 2, $C_3\times A_4.A_7.A_5$, $(C_3\times A_4).A_5.A_7$, $(C_2\times C_7:C_3).A_6.A_6$, $(C_2\times C_3:S_3).A_7.A_5$, $(C_2\times C_3:S_3).A_5.A_7$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 192036096000: $C_2.A_7\wr S_3$ ($C_1$)

Order 96018048000: $A_7\wr S_3$ ($C_2$) x 2, $C_2.A_7\wr C_3$ ($C_2$)

Order 48009024000: $A_7\wr C_3$ ($C_2^2$)

Order 32006016000: $C_2.A_7.A_7.A_7$ ($S_3$)

Order 16003008000: $A_7^3$ ($D_6$)

Order 2: $C_2$ ($A_7\wr S_3$)

Order 1: $C_1$ ($C_2.A_7\wr S_3$)

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 4 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $510 \times 510$ character table is not available for this group.

Rational character table

The $402 \times 402$ rational character table is not available for this group.