Abstract group 384072192000.h: $A_7\wr C_3.C_2^3$

• Label: 384072192000.h
• Order: \( 2^{12} \cdot 3^{7} \cdot 5^{3} \cdot 7^{3} \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{7} \cdot 5^{3} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $23$
• Trans deg.: $42$
• Rank: $3$

Group information

Description:	$A_7\wr C_3.C_2^3$
Order:	\(384072192000\)\(\medspace = 2^{12} \cdot 3^{7} \cdot 5^{3} \cdot 7^{3} \)
Exponent:	\(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Automorphism group:	Group of order \(768144384000\)\(\medspace = 2^{13} \cdot 3^{7} \cdot 5^{3} \cdot 7^{3} \)
Composition factors:	$C_2$ x 3, $C_3$, $A_7$ x 3
Derived length:	$2$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	12	14	15	18	20	21	24	28	30	35	36	40	42	56	60	70	84	105	140	210
Elements	1	9890623	55944350	2581467840	128787624	22633363970	374805360	10211443200	4445280000	10283521752	56042068320	11124671760	6854198400	41785632000	11179244160	9955008000	21337344000	14538787200	47796497280	1334672640	10668672000	9601804800	35567078400	6858432000	15362887680	18252138240	9906624000	762048000	2743372800	1676505600	384072192000
Conjugacy classes	1	23	10	40	3	138	5	12	2	37	144	27	8	8	24	10	12	16	62	4	2	4	42	2	30	12	10	2	2	4	696
Divisions	1	23	10	40	3	138	5	12	2	37	144	27	8	8	24	10	12	16	62	4	2	4	42	2	30	12	10	2	2	4	696
Autjugacy classes	1	15	10	26	3	97	5	6	2	23	88	20	8	6	16	10	6	13	43	4	1	2	35	1	18	9	8	2	2	4	484

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$42$
Rank:	$3$
Inequivalent generating triples:	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 $23$ $\langle(1,2,5)(3,8,19)(4,11,15,6,14,12,21,9,13,20,18,16)(7,17,10)(22,23), (1,3,9,11,14,7,18) \!\cdots\! \rangle$
Transitive group:	42T7888				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:	$(A_7^3.D_6)$ . $C_2$ (4)	$A_7^3$ . $(C_2\times D_6)$	$C_2$ . $(A_7^3.D_6)$	$(A_7\wr S_3)$ . $C_2^2$ (2)	all 13

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 23 normal subgroups (11 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 2450 (order at least 156764160) are shown, as well as all normal subgroups of any index.

Order 384072192000: $A_7\wr C_3.C_2^3$

Order 192036096000: $A_7^3.D_6$ x 4, $C_2.A_7\wr S_3$, $A_7\wr C_3.C_2^2$, $A_7\wr C_3.C_2^2$

Order 128024064000: $C_2.A_7.A_7^2.C_2^2$

Order 96018048000: $A_7\wr S_3$ x 2, $A_7^3.S_3$ x 2, $A_7^3.C_6$ x 2, $C_2.A_7\wr C_3$

Order 64012032000: $A_7.A_7^2.C_2^2$ x 4, $C_2.A_7.A_7^2.C_2$ x 2, $C_2.A_7.A_7.A_7.C_2$

Order 48009024000: $A_7\wr C_3$

Order 32006016000: $A_7.A_7^2.C_2$ x 4, $A_7.A_7.A_7.C_2$ x 2, $C_2.A_7.A_7.A_7$

Order 18289152000: $C_2.A_6.A_7^2.C_2^2$

Order 16003008000: $A_7^3$

Order 9144576000: $A_6.A_7^2.C_2^2$ x 4, $C_2.A_6.A_7^2.C_2$ x 2, $C_2.A_7.A_7.A_6.C_2$

Order 6096384000: $(C_2\times A_5).A_7^2.C_2^3$

Order 4572288000: $A_6.A_7^2.C_2$ x 4, $A_7.A_7.A_6.C_2$ x 2, $C_2.A_7.A_7.A_6$

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

Order 3657830400: $(C_2\times C_3:S_4).A_7^2.C_2^2$

Order 3048192000: $A_5.A_7^2.C_2^3$ x 8, $(C_2\times A_5).A_7^2.C_2^2$ x 7, $C_2.A_7.A_5.A_7.C_2^2$

Order 2612736000: $C_2.A_7.A_6^2.C_2^2$

Order 2286144000: $A_7^2\times A_6$

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

Order 1828915200: $C_3:S_4.A_7^2.C_2^2$ x 8, $(C_6\times A_4).A_7^2.C_2^2$ x 4, $(C_2\times C_3:S_4).A_7^2.C_2$ x 2, $(C_2\times C_3:S_4).A_7.A_7.C_2$, $(C_2\times C_3:S_3.C_2).A_7^2.C_2^2$

Order 1524096000: $A_5.A_7^2.C_2^2$ x 28, $(C_2\times A_5).A_7^2.C_2$ x 6, $A_7.A_5.A_7.C_2^2$ x 4, $C_2.A_7.A_5.A_7.C_2$ x 3, $C_2.A_7.A_7.S_5$

Order 1306368000: $A_7.A_6^2.C_2^2$ x 4, $C_2.A_7.A_6^2.C_2$ x 2, $C_2.A_6.A_7.A_6.C_2$

Order 1219276800: $(C_2\times S_4).A_7^2.C_2^2$ x 2, $(C_2\times C_3:D_4).A_7^2.C_2^2$

Order 1119744000: $A_6\wr C_3.C_2^3$

Order 1066867200: $A_7^2\times \PSL(2,7)$, $(C_2\times C_7:C_3).A_7^2.C_2^2$

Order 1016064000: $(C_2\times F_5).A_7^2.C_2^2$

Order 914457600: $(C_3\times A_4).A_7^2.C_2^2$ x 16, $C_3:S_4.A_7^2.C_2$ x 8, $C_3:S_3.C_2.A_7^2.C_2^2$ x 8, $C_3:S_4.A_7.A_7.C_2$ x 4, $(C_6\times A_4).A_7^2.C_2$ x 4, $(C_2\times C_3:S_3).A_7^2.C_2^2$ x 4, $(C_6\times A_4).A_7.A_7.C_2$ x 2, $(C_2\times C_3:S_3.C_2).A_7^2.C_2$ x 2, $(C_2\times C_3:S_4).A_7.A_7$, $(C_2\times C_3:S_3.C_2).A_7.A_7.C_2$

Order 762048000: $A_5.A_7^2.C_2$ x 12, $A_7.A_5.A_7.C_2$ x 4, $A_7.A_7.S_5$ x 2, $S_5\times A_7^2$, $C_2.A_7.A_7.A_5$, $C_2.A_7.A_5.A_7$, $A_7^2\times S_5$

Order 653184000: $A_7.A_6^2.C_2$ x 4, $A_6.A_7.A_6.C_2$ x 2, $C_2.A_6.A_7.A_6$

Order 609638400: $S_4.A_7^2.C_2^2$ x 16, $C_3:D_4.A_7^2.C_2^2$ x 16, $(C_2\times A_4).A_7^2.C_2^2$ x 8, $(C_2\times S_4).A_7^2.C_2$ x 5, $(C_2^2\times S_3).A_7^2.C_2^2$ x 4, $(C_2^2\times C_6).A_7^2.C_2^2$ x 4, $(C_2\times C_3:C_4).A_7^2.C_2^2$ x 4, $(C_2\times S_4).A_7.A_7.C_2$ x 2, $(C_2\times C_3:D_4).A_7^2.C_2$ x 2, $(C_2\times C_3:D_4).A_7.A_7.C_2$

Order 559872000: $A_6^3.D_6$ x 4, $C_2\times A_6\wr S_3$, $C_2\times A_6^3.S_3$, $C_2\times A_6^3.C_6$

Order 533433600: $C_7:C_3.A_7^2.C_2^2$ x 4, $(C_2\times C_7:C_3).A_7^2.C_2$ x 2, $(C_2\times C_7:C_3).A_7.A_7.C_2$

Order 508032000: $F_5.A_7^2.C_2^2$ x 8, $D_{10}.A_7^2.C_2^2$ x 4, $(C_2\times F_5).A_7^2.C_2$ x 2, $(C_2\times F_5).A_7.A_7.C_2$

Order 457228800: $C_3:S_3.A_7^2.C_2^2$ x 12, $(C_3\times A_4).A_7^2.C_2$ x 8, $A_7^2.\SOPlus(4,2)$ x 8, $C_3:S_3.C_2.A_7^2.C_2$ x 4, $C_3:S_3.C_2.A_7.A_7.C_2$ x 4, $A_7^2.C_2\times C_3:S_3.C_2$ x 4, $(C_3\times C_6).A_7^2.C_2^2$ x 4, $(C_3\times A_4).A_7.A_7.C_2$ x 4, $(C_2\times C_3:S_3).A_7^2.C_2$ x 4, $C_3:S_4\times A_7^2$ x 2, $(C_2\times C_3:S_3).A_7.A_7.C_2$ 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 435456000: $(C_2\times A_5).A_7.A_6.C_2^2$

Order 406425600: $(C_2\times D_4).A_7^2.C_2^2$

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

Order 373248000: $C_2.A_6.A_6^2.C_2^2$

Order 355622400: $C_{14}.A_7^2.C_2^2$

Order 326592000: $A_7\times A_6^2$

Order 304819200: $(C_2\times C_6).A_7^2.C_2^2$ x 48, $D_6.A_7^2.C_2^2$ x 41, $C_3:C_4.A_7^2.C_2^2$ x 32, $A_4.A_7^2.C_2^2$ x 32, $S_4.A_7^2.C_2$ x 20, $C_3:D_4.A_7^2.C_2$ x 16, $(C_2\times A_4).A_7^2.C_2$ x 10, $S_4.A_7.A_7.C_2$ x 8, $C_3:D_4.A_7.A_7.C_2$ x 8, $(C_2^2\times S_3).A_7^2.C_2$ x 4, $(C_2^2\times C_6).A_7^2.C_2$ x 4, $(C_2\times C_3:C_4).A_7^2.C_2$ x 4, $(C_2\times A_4).A_7.A_7.C_2$ x 4, $(C_2\times S_4).A_7.A_7$ x 3, $(C_2^2\times S_3).A_7.A_7.C_2$ x 2, $(C_2^2\times C_6).A_7.A_7.C_2$ x 2, $(C_2\times C_3:C_4).A_7.A_7.C_2$ x 2, $C_2.A_7.\PSL(2,7).A_6$, $(C_2\times C_3:D_4).A_7.A_7$

Order 290304000: $C_2.A_7.A_5^2.D_4.C_2$

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

Order 279936000: $A_6\wr S_3$ x 2, $A_6^3.S_3$ x 2, $A_6^3.C_6$ x 2, $C_2\times A_6\wr C_3$

Order 266716800: $C_7:C_3.A_7^2.C_2$ x 4, $C_7:C_3.A_7.A_7.C_2$ x 2, $(C_2\times C_7:C_3).A_7.A_7$

Order 261273600: $(C_2\times C_3:S_4).A_7.A_6.C_2$

Order 254016000: $D_5.A_7^2.C_2^2$ x 20, $F_5.A_7^2.C_2$ x 8, $F_5.A_7.A_7.C_2$ x 4, $D_{10}.A_7^2.C_2$ x 4, $C_{10}.A_7^2.C_2^2$ x 4, $D_{10}.A_7.A_7.C_2$ x 2, $(C_2\times F_5).A_7.A_7$

Order 228614400: $C_3^2.A_7^2.C_2^2$ x 16, $C_3:S_3.A_7.A_7.C_2$ x 6, $A_7^2.S_3^2$ x 6, $C_3:S_3.A_7^2.C_2$ x 4, $(C_3\times C_6).A_7^2.C_2$ x 4, $C_3:S_3.C_2\times A_7^2$ x 2, $A_7^2.C_3^2:C_4$ x 2, $C_2\times C_3^2.A_7.A_7.C_2$, $(C_3\times C_6).A_7.A_7.C_2$, $(C_3\times A_4).A_7.A_7$, $(C_2\times C_3:S_3).A_7.A_7$

Order 217728000: $A_5.A_7.A_6.C_2^2$ x 4, $(C_2\times A_5).A_7.A_6.C_2$ x 4

Order 203212800: $D_4.A_7^2.C_2^2$ x 16, $C_2^3.A_7^2.C_2^2$ x 8, $(C_2\times C_4).A_7^2.C_2^2$ x 4, $(C_2\times D_4).A_7^2.C_2$ x 2, $(C_2\times D_4).A_7.A_7.C_2$

Order 186624000: $A_6.A_6^2.C_2^2$ x 4, $C_2.A_6.A_6^2.C_2$ x 2, $C_2.A_6.A_6.A_6.C_2$

Order 177811200: $C_7.A_7^2.C_2^2$ x 4, $C_{14}.A_7^2.C_2$ x 2, $C_{14}.A_7.A_7.C_2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 384072192000: $A_7\wr C_3.C_2^3$ ($C_1$)

Order 192036096000: $A_7^3.D_6$ ($C_2$) x 4, $C_2.A_7\wr S_3$ ($C_2$), $A_7\wr C_3.C_2^2$ ($C_2$), $A_7\wr C_3.C_2^2$ ($C_2$)

Order 96018048000: $A_7\wr S_3$ ($C_2^2$) x 2, $A_7^3.S_3$ ($C_2^2$) x 2, $A_7^3.C_6$ ($C_2^2$) x 2, $C_2.A_7\wr C_3$ ($C_2^2$)

Order 64012032000: $C_2.A_7.A_7.A_7.C_2$ ($S_3$)

Order 48009024000: $A_7\wr C_3$ ($C_2^3$)

Order 32006016000: $A_7.A_7.A_7.C_2$ ($D_6$) x 2, $C_2.A_7.A_7.A_7$ ($D_6$)

Order 16003008000: $A_7^3$ ($C_2\times D_6$)

Order 2: $C_2$ ($A_7^3.D_6$)

Order 1: $C_1$ ($A_7\wr C_3.C_2^3$)

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

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

The $696 \times 696$ rational character table is not available for this group.