Abstract group 2239488000.h: $A_6^3.S_4.C_2$

• Label: 2239488000.h
• Order: \( 2^{13} \cdot 3^{7} \cdot 5^{3} \)
• Exponent: \( 2^{4} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{7} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $30$
• Trans deg.: $30$
• Rank: $2$

Group information

Description:	$A_6^3.S_4.C_2$
Order:	\(2239488000\)\(\medspace = 2^{13} \cdot 3^{7} \cdot 5^{3} \)
Exponent:	\(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \)
Automorphism group:	Group of order \(4478976000\)\(\medspace = 2^{14} \cdot 3^{7} \cdot 5^{3} \)
Composition factors:	$C_2$ x 4, $C_3$, $A_6$ x 3
Derived length:	$3$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	15	16	18	20	24	30	40	60	80	120
Elements	1	467343	1568240	58201200	3048624	168218640	131258880	82944000	90742032	303091200	157109760	223948800	248832000	214228800	143078400	134991360	203109120	6220800	55987200	12441600	2239488000
Conjugacy classes	1	11	5	22	5	25	22	2	23	16	5	6	2	16	10	9	20	1	2	2	205
Divisions	1	11	5	22	5	25	22	2	23	16	5	6	2	16	10	9	11	1	1	1	194
Autjugacy classes	1	9	4	21	5	16	20	1	22	14	5	5	1	16	7	7	19	1	2	2	178

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$30$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	27	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $30$ $\langle(1,12,8,15,2,16,10,20,3,19)(4,11,9,18,7,17,6,14,5,13)(21,23,28)(22,30,26,27,24,25) \!\cdots\! \rangle$
Transitive group:	30T5027	36T97835			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:	$(A_6^3.S_4)$ . $C_2$	$(A_6^3.S_4)$ . $C_2$	$(A_6^3.A_4)$ . $C_2^2$	$(A_6^3.A_4.C_2)$ . $C_2$	all 8

Elements of the group are displayed as permutations of degree 30.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 10 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 $A_6^3.A_4$
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 8192 (order at least 273375) are shown, as well as all normal subgroups of any index.

Order 2239488000: $A_6^3.S_4.C_2$

Order 1119744000: $A_6^3.S_4$, $A_6^3.A_4.C_2$, $A_6^3.S_4$

Order 746496000: $A_6.A_6^2.D_4.C_2$

Order 559872000: $A_6^3.A_4$, $A_6^3.D_6$

Order 373248000: $A_6.A_6^2.D_4$ x 4, $A_6.A_6^2.C_4.C_2$, $A_6.A_6^2.C_2^3$, $A_6.A_6.A_6.C_2^3$

Order 279936000: $A_6\wr S_3$, $A_6^3.S_3$, $A_6^3.C_6$

Order 186624000: $A_6.A_6^2.C_2^2$ x 4, $A_6.A_6.A_6.C_2^2$ x 3, $A_6.A_6^2.C_4$ x 2

Order 139968000: $A_6\wr C_3$

Order 93312000: $A_6.A_6.A_6.C_2$ x 3, $A_6.A_6^2.C_2$ x 2

Order 74649600: $C_3:S_3.C_2.A_6^2.C_2^3.C_2$

Order 62208000: $A_5.A_6^2.C_2^3$

Order 46656000: $A_6.A_6.A_6$

Order 37324800: $C_3:S_3.C_2.A_6^2.D_4$ x 4, $C_3:S_3.C_2.A_6^2.C_4.C_2$, $C_3:S_3.C_2.A_6^2.C_2^3$, $C_3:S_3.C_2.A_6.A_6.C_2^3$

Order 31104000: $A_5.A_6^2.C_2^2$ x 6, $A_6.A_5.A_6.C_2^2$

Order 24883200: $S_4.A_6^2.C_2^3$

Order 20736000: $D_5.A_6^2.C_2^3.C_2$

Order 18662400: $C_3:S_3.C_2.A_6.A_6.C_2^2$ x 5, $C_3:S_3.C_2.A_6^2.C_2^2$ x 4, $C_3:S_3.A_6^2.D_4$ x 4, $C_3:S_3.A_6^2.C_2^3$ x 4, $C_3:S_3.C_2.A_6^2.C_4$ x 2

Order 16588800: $D_4.A_6^2.D_4.C_2$

Order 15552000: $A_5.A_6^2.C_2$ x 4, $A_5.A_6.A_6.C_2$ x 2, $A_6.A_5.A_6.C_2$

Order 12441600: $A_4.A_6^2.C_2^3$ x 8, $S_4.A_6^2.C_2^2$ x 6, $S_4.A_6.A_6.C_2^2$

Order 10368000: $D_5.A_6^2.D_4$ x 4, $D_5.A_6^2.C_4.C_2$, $D_5.A_6^2.C_2^3$, $D_5.A_6.A_6.C_2^3$

Order 9331200: $C_3:S_3.A_6^2.C_2^2$ x 12, $C_3:S_3.C_2.A_6.A_6.C_2$ x 5, $C_3^2.A_6^2.C_2^3$ x 4, $C_3:S_3.A_6.A_6.C_2^2$ x 4, $C_3:S_3.C_2.A_6^2.C_2$ x 2, $C_3:S_3.A_6^2.C_4$ x 2, $A_6^2.\SOPlus(4,2)$ x 2

Order 8294400: $C_4.A_6^2.D_4.C_2$ x 8, $D_4.A_6^2.D_4$ x 4, $D_4.A_6^2.C_4.C_2$, $D_4.A_6^2.C_2^3$, $D_4.A_6.A_6.C_2^3$

Order 7776000: $A_5\times A_6^2$

Order 7464960: $C_3^4.(C_4\times C_8).C_2.A_6.C_2^2$

Order 6220800: $A_4.A_6^2.C_2^2$ x 24, $S_4.A_6^2.C_2$ x 4, $A_4.A_6.A_6.C_2^2$ x 4, $S_4.A_6.A_6.C_2$ x 3, $S_3.A_6^2.C_2^3$

Order 5184000: $D_5.A_6.A_6.C_2^2$ x 5, $D_5.A_6^2.C_2^2$ x 4, $C_5.A_6^2.D_4$ x 4, $D_5.A_6^2.C_4$ x 2, $A_6.A_5^2.C_2^2$

Order 4665600: $C_3^2.A_6^2.C_2^2$ x 13, $C_3:S_3.A_6.A_6.C_2$ x 5, $C_3^2.A_6.A_6.C_2^2$ x 2, $A_6^2.S_3^2$ x 2, $C_3^2.A_6^2.C_4$, $C_3:S_3.C_2\times A_6^2$, $C_3:S_3.A_6^2.C_2$, $C_3^2.A_6^2:C_4$

Order 4147200: $C_4.A_6^2.D_4$ x 16, $C_2^2.A_6^2.C_2^3$ x 7, $C_4.A_6^2.C_2^3$ x 6, $C_4.A_6.A_6.C_2^3$ x 6, $C_2.A_6^2.D_4.C_2$ x 6, $D_4.A_6.A_6.C_2^2$ x 5, $D_4.A_6^2.C_2^2$ x 4, $C_4.A_6^2.C_4.C_2$ x 3, $D_4.A_6^2.C_4$ x 2, $C_2.A_6^2.C_2^3.C_2$ x 2, $C_4.A_6^2.C_2^2.C_2$, $A_6^2.(C_2^2\times D_4)$

Order 3732480: $C_3^4.C_4^2.C_2.A_6.C_2^2$ x 4, $C_3^4.C_4^2.C_2^2.A_6.C_2$ x 3

Order 3110400: $C_3.A_6^2.C_2^3$ x 8, $A_4.A_6^2.C_2$ x 8, $A_4.A_6.A_6.C_2$ x 6, $S_3.A_6^2.C_2^2$ x 4, $A_6^2.(C_2\times D_6)$ x 2, $S_3.A_6.A_6.C_2^2$, $C_3:S_3.C_2.A_6.S_5.C_2$, $A_6^2\times S_4$

Order 2592000: $D_5.A_6.A_6.C_2$ x 5, $C_5.A_6^2.C_4$ x 3, $C_5.A_6^2.C_2^2$ x 3, $C_5.A_6.A_6.C_2^2$ x 3, $D_5.A_6^2.C_2$ x 2, $A_6.A_5^2.C_2$ x 2, $A_5^3:D_6$ x 2, $A_6.A_5.S_5$

Order 2332800: $C_3^2.A_6.A_6.C_2$ x 4, $C_3^2.A_6^2.C_2$ x 2, $(C_3\times A_6)\wr C_2$ x 2, $C_3:S_3.A_6.A_6$

Order 2239488: $C_3^6.(C_4\times C_8^2):D_6$

Order 2073600: $C_4.A_6.A_6.C_2^2$ x 18, $C_2^2.A_6^2.C_2^2$ x 18, $C_2.A_6^2.C_2^3$ x 17, $C_4.A_6^2.C_2^2$ x 14, $C_2.A_6^2.D_4$ x 12, $A_6^2.C_2^4$ x 8, $D_4.A_6.A_6.C_2$ x 5, $C_4.A_6^2.C_4$ x 4, $C_2^2.A_6.A_6.C_2^2$ x 4, $C_2.A_6.A_6.C_2^3$ x 4, $A_6^2.(C_2\times D_4)$ x 4, $C_2.A_6^2.C_4.C_2$ x 3, $D_4.A_6^2.C_2$ x 2, $(C_2\times A_6^2).D_4$ x 2, $C_2\times \PGL(2,9)\wr C_2$ x 2, $A_6^2.(C_2\times D_4)$ x 2, $A_6^2.(C_2\times D_4)$ x 2, $A_6^2.(C_2\times D_4)$ x 2, $A_6^2.(C_2\times D_4)$ x 2, $C_2.A_6^2.C_2^3$ x 2, $A_6^2.(C_2\times D_4)$ x 2, $\PSOPlus(4,9).D_4$, $A_6^2.(C_2^2\times C_4)$

Order 1866240: $C_3^4.C_4^2.C_2.A_6.C_2$ x 6, $C_3^4.\OD_{16}:C_2.A_6.C_2$ x 2, $C_3^4.D_8:C_2.A_6.C_2$ x 2, $C_3^4.C_4^2.A_6.C_2^2$ x 2, $C_3^4.C_4\wr C_2.A_6.C_2$ x 2, $C_3^4.(C_8.C_4).A_6.C_2$ x 2, $C_3^4.(C_2\times C_8).A_6.C_2^2$ x 2, $C_3^4.C_4^2.C_2^2.A_6$

Order 1555200: $C_3.A_6^2.C_2^2$ x 16, $C_3.A_6.A_6.C_2^2$ x 4, $A_6^2:D_6$ x 4, $A_6^2.D_6$ x 4, $S_3.A_6.A_6.C_2$ x 2, $C_3:S_3.C_2.A_5.A_6.C_2$ x 2, $C_3\times A_6^2:C_2^2$ x 2, $A_6^2:D_6$ x 2, $C_3:S_3.C_2.A_6.A_5.C_2$, $A_4.A_6.A_6$, $A_6^2.D_6$

Order 1296000: $C_5.A_6.A_6.C_2$ x 3, $C_5.A_6^2.C_2$ x 2, $A_5\wr S_3$ x 2, $A_5^3:C_6$ x 2, $A_5^3:S_3$ x 2, $D_5.A_6.A_6$, $C_5\times A_6.A_6.C_2$, $A_6.A_5.A_5$

Order 1244160: $C_6^2.(C_4\times S_3).A_6.C_2^2$

Order 1166400: $C_3^2.A_6.A_6$

Order 1119744: $C_3^6:(C_4^2.\GL(2,\mathbb{Z}/4))$ x 2, $C_3^6.(C_2\times C_8^2:D_6)$ x 2, $C_3^6:(C_8^2:D_{12})$, $C_3^6:((C_4\times C_8^2):C_6)$, $C_3^6:(C_4^3.S_4)$

Order 1036800: $C_2.A_6.A_6.C_2^2$ x 19, $A_6^2.C_2^3$ x 11, $C_4.A_6.A_6.C_2$ x 10, $C_2.A_6^2.C_2^2$ x 8, $A_6^2:C_2^3$ x 8, $C_2\times A_6^2.C_2^2$ x 7, $A_6^2.C_2^3$ x 7, $C_2^2.A_6.A_6.C_2$ x 6, $C_4.A_6^2.C_2$ x 4, $C_2^2.A_6^2.C_2$ x 4, $C_2.A_6^2.C_2^2$ x 4, $A_6^2:D_4$ x 4, $C_2.A_6^2.C_2^2$ x 4, $C_2.A_6^2.C_2^2$ x 4, $A_6^2.D_4$ x 4, $\PGL(2,9)\wr C_2$ x 4, $A_6^2:C_8$ x 2, $C_2.A_6^2.C_2^2$ x 2, $A_6^2:(C_2\times C_4)$ x 2, $C_2\times A_6^2:C_4$ x 2, $D_4.A_6.A_6$, $C_5.(C_2\times F_9).A_6.C_2^2$, $A_6.A_6.C_2^3$, $A_6^2.(C_2\times C_4)$, $A_6^2.(C_2\times C_4)$, $S_4\times A_5:S_6$

Order 933120: $C_3^4.(C_2\times C_8).A_6.C_2$ x 9, $C_3^4.\OD_{16}.A_6.C_2$ x 6, $C_3^4.C_4^2.A_6.C_2$ x 5, $C_3^4:C_2^2.C_2.A_6.C_2^2$ x 4, $C_3^4.D_4:C_2.A_6.C_2$ x 4, $C_3^4.\SD_{16}.A_6.C_2$ x 2, $C_3^4.C_8.C_2.A_6.C_2$ x 2, $C_3^4.\OD_{16}:C_2.A_6$, $C_3^4.Q_{16}.A_6.C_2$, $C_3^4.D_8:C_2.A_6$, $C_3^4.D_8.A_6.C_2$, $C_3^4.C_4^2.C_2.A_6$, $C_3^4.C_4\wr C_2.A_6$, $C_3^4.(C_8.C_4).A_6$

Order 864000: $A_5^3:C_2^2$ x 2, $D_5.A_5.A_6.C_2^2$

Order 829440: $C_3^2.D_4:C_8.A_6.C_2^2$, $S_6\times S_4\wr C_2$

Order 777600: $C_3.A_6.A_6.C_2$ x 4, $A_6^2:C_6$ x 4, $A_6^2:S_3$ x 4, $C_3:S_3.A_5.A_6.C_2$ x 2, $C_3:S_3.C_2.A_5.A_6$, $A_6^2:S_3$, $A_6^2:C_6$, $S_3\times A_6^2$

Order 746496: $C_3^6.C_4.C_2.C_2^4.C_2^3$

Order 691200: $D_4.A_6.S_5.C_2$

Order 648000: $A_5\wr C_3$ x 2, $C_5.A_6.A_6$

Order 622080: $(A_4\times C_3:S_3.C_2).A_6.C_2^2$ x 4, $(A_4\times C_3:S_3).C_2^2.A_6.C_2$ x 2, $C_6^2.(C_4\times S_3).A_6.C_2$, $C_3^4.C_4^2.C_2^2.S_5$

Order 576000: $(C_2\times D_5\wr C_2).A_6.C_2^2$

Order 559872: $C_3^6:(C_2\times C_8^2:C_6)$ x 2, $C_3^6:(C_8^2:D_6)$ x 2, $C_3^6:(C_8^2:D_6)$ x 2, $C_3^6:(C_8^2:D_6)$ x 2, $C_3^6:((C_2\times C_4^2).S_4)$, $C_3^6:(C_8^2:D_6)$, $C_3^6:(C_8^2:C_{12})$, $C_3^6.C_4^3:D_6$

Order 518400: $A_6.A_6.C_2^2$ x 15, $C_2.A_6.A_6.C_2$ x 11, $\POPlus(4,9)$ x 8, $A_6^2:C_2^2$ x 6, $A_6^2.C_2^2$ x 6, $A_6^2:C_2^2$ x 6, $(D_5\times C_3:S_3.C_2).A_6.C_2^2$ x 4, $C_5.(C_2\times F_9).A_6.C_2$ x 3, $A_6^2.C_2^2$ x 3, $A_6^2:C_4$ x 2, $A_6^2:C_4$ x 2, $C_4.A_6.A_6$, $C_3:S_3.C_2.A_5^2.C_2^2$, $C_2^2.A_6.A_6$, $(A_4\times A_5):S_6$, $S_4\times A_5\times A_6$, $S_6^2$, $A_6.S_5\times A_4$

Order 466560: $C_3^4.C_8.A_6.C_2$ x 15, $C_3^4:C_2^2.C_2.A_6.C_2$ x 8, $C_3^4.(C_2\times C_4).A_6.C_2$ x 7, $C_3^4.D_4.A_6.C_2$ x 6, $C_3^4.Q_8.A_6.C_2$ x 4, $C_3^4.\OD_{16}.A_6$ x 2, $C_3^4.(C_2\times C_8).A_6$ x 2, $C_3^3:(C_3:C_4).A_6.C_2^2$ x 2, $(C_3^2\times C_3:S_3.C_2).A_6.C_2^2$ x 2, $C_3^4.\SD_{16}.A_6$, $C_3^4.Q_{16}.A_6$, $C_3^4.D_8.A_6$, $C_3^4.D_4:C_2.A_6$, $C_3^4.C_8.C_2.A_6$, $C_3^4.C_4^2.A_6$

Order 432000: $D_5.A_6.S_5$ x 2, $A_5^3:C_2$ x 2, $A_5^2:S_5$ x 2, $A_5^2:S_5$ x 2, $D_5.A_5.A_6.C_2$

Order 414720: $C_4:F_9.A_6.C_2^2$ x 4, $(D_4\times C_3:S_3.C_2).A_6.C_2^2$ x 4, $(C_4\times F_9).A_6.C_2^2$ x 4, $C_3^2.D_4:C_8.A_6.C_2$ x 3, $S_4^2.S_6$, $S_6\times S_4^2$, $S_6\times \POPlus(4,3)$, $A_4^2.(C_4\times S_6)$, $(A_4^2\times A_6).D_4$, $(A_4^2\times A_6).D_4$, $A_6\times A_4^2.D_4$

Order 388800: $C_3^2.A_6.S_5$, $C_3:S_3.A_6.A_5$, $C_3\times A_6^2$

Order 373248: $C_3^6.C_4^2.C_2^4.C_2$ x 4, $C_3^6.(C_2\times C_4\times C_8).C_2^3$ x 4, $C_3^6.C_2.C_4^3.C_2^2$ x 3, $C_3^6.C_8.C_4^2.C_2^2$, $C_3^6.C_2^3.C_2^4.C_2^2$, $C_3^6.C_2^3.C_2.C_2^4.C_2$, $C_3^6.C_2.C_2^4.C_2^4$

Order 368640: $D_4^2.C_2^2.A_6.C_2^2$

Order 345600: $C_4.A_6.S_5.C_2$ x 3, $S_4\times \POPlus(4,5)$ x 2, $D_4.A_6.A_5.C_2$, $C_4.A_6.A_5.C_2^2$, $(D_5\times S_4).A_6.C_2^2$, $(D_4\times A_5).A_6.C_2$, $(D_4\times A_5).S_6$

Order 311040: $(A_4\times C_3:S_3.C_2).A_6.C_2$ x 6, $C_3^4.C_4\wr C_2.S_5$ x 4, $C_3^4.\OD_{16}:C_2.S_5$ x 2, $C_3^4.D_8:C_2.S_5$ x 2, $C_3^4.C_8.C_2^2.S_5$ x 2, $C_3^4.C_4^2.C_2.S_5$ x 2, $C_3^4.(C_8.C_4).S_5$ x 2, $(C_3^2\times A_4):C_4.A_6.C_2$ x 2, $(C_3:S_3\times S_4).A_6.C_2$ x 2, $C_3^4.C_4^2.C_2^2.A_5$, $(S_3\times C_3:S_3.C_2).A_6.C_2^2$, $(A_4\times C_3:S_3).C_2^2.A_6$

Order 288000: $(C_2\times D_5^2).A_6.C_2^2$ x 8, $(C_2\times C_5:F_5).A_6.C_2^2$ x 4, $(C_2\times D_5\wr C_2).A_6.C_2$ x 3

Order 279936: $C_3^5:D_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_3^6:C_2^4.S_4$ x 2, $C_3^6:(C_8^2:S_3)$ x 2, $C_3^6:(C_8^2:C_6)$ x 2, $C_3^6:(C_4^2:D_{12})$, $C_3^6:C_4\wr S_3$, $C_3^6:(C_4^3:C_6)$, $C_3^6:(C_8^2:S_3)$, $C_3^6:(C_8^2:S_3)$, $C_3^6:(C_8^2:C_6)$

Order 276480: $(D_4\times S_4).A_6.C_2^2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 2239488000: $A_6^3.S_4.C_2$ ($C_1$)

Order 1119744000: $A_6^3.S_4$ ($C_2$), $A_6^3.A_4.C_2$ ($C_2$), $A_6^3.S_4$ ($C_2$)

Order 559872000: $A_6^3.A_4$ ($C_2^2$)

Order 373248000: $A_6.A_6.A_6.C_2^3$ ($S_3$)

Order 186624000: $A_6.A_6.A_6.C_2^2$ ($D_6$)

Order 93312000: $A_6.A_6.A_6.C_2$ ($S_4$)

Order 46656000: $A_6.A_6.A_6$ ($C_2\times S_4$)

Order 1: $C_1$ ($A_6^3.S_4.C_2$)

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 3 larger groups in the database.

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

Character theory

Complex character table

The $205 \times 205$ character table is not available for this group.

Rational character table

The $194 \times 194$ rational character table is not available for this group.