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

• Label: 2239488000.f
• 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	405135	1568240	42960240	3048624	168840720	329702400	82944000	75252240	287539200	157109760	223948800	248832000	105053760	118195200	149921280	144633600	6220800	55987200	37324800	2239488000
Conjugacy classes	1	10	5	21	3	25	24	2	12	16	5	6	2	10	10	9	8	1	2	2	174
Divisions	1	10	5	21	3	25	24	2	12	16	5	6	2	10	10	9	7	1	1	1	171
Autjugacy classes	1	8	4	20	3	16	22	1	10	14	4	5	1	10	7	6	7	1	1	1	142

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 $30$ $\langle(1,25,14,3,27,12,5,24,15,4,23,13,10,21,18,2,29,19)(6,30,16,7,28,11,9,26,20) \!\cdots\! \rangle$
Transitive group:	30T5025	36T97836			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:	$C_{2}^{2}$
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: $A_6^3.S_4$, $A_6^3.S_4$, $A_6^3.A_4.C_2$

Order 746496000: $A_6.A_6^2.C_2^3.C_2$

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, $C_3:S_3.C_2.A_6^2.C_2^3$, $C_3:S_3.C_2.A_6^2.C_2^2.C_2$, $C_3:S_3.C_2.A_6.A_6.C_2^3$

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

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

Order 20736000: $D_5.A_6^2.D_4.C_2$

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: $D_4.A_6^2.C_2^3.C_2$

Order 15552000: $A_5.A_6^2.C_2$ x 3, $A_5.A_6.A_6.C_2$ x 3, $A_6^2.C_2\times A_5$

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: $A_6^2\times A_5$

Order 7464960: $C_3^4.C_4:Q_8.C_2.A_6.C_2^2$

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, $D_4.A_6.A_6.C_2^2$ x 5, $C_2.A_6^2.C_2^3.C_2$ x 5, $D_4.A_6^2.C_2^2$ x 4, $C_4.A_6^2.C_4.C_2$ x 3, $C_2.A_6^2.D_4.C_2$ x 3, $D_4.A_6^2.C_4$ x 2, $C_4.A_6^2.C_2^2.C_2$, $A_6^2.(C_2^2\times D_4)$

Order 3732480: $C_3^4.C_4:Q_8.A_6.C_2^2$ x 4, $C_3^4.C_4:Q_8.C_2.A_6.C_2$ 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, $S_3.A_6.A_6.C_2^2$, $C_3:S_3.C_2.A_5.A_6.C_2^2$, $A_6^2\times S_4$

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: $C_3^6.(C_4^3.(C_2\times S_4))$

Order 2073600: $C_4.A_6.A_6.C_2^2$ x 18, $C_2^2.A_6^2.C_2^2$ x 17, $C_2.A_6^2.C_2^3$ 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, $C_2\times A_6^2.D_4$ 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, $A_6^2.(C_2\times D_4)$ x 2, $C_2.A_6^2.C_2^3$ x 2, $A_6^2.(C_2\times D_4)$ x 2, $C_2^2\times A_6^2.C_2^2$, $A_6^2.C_2^2:C_4$, $A_6^2.(C_2^2\times C_4)$

Order 1866240: $C_3^4.C_4:Q_8.A_6.C_2$ x 6, $C_3^4:C_2^2.C_2^3.A_6.C_2$ x 2, $C_3^4.\SD_{16}:C_2.A_6.C_2$ 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, $C_3^4.(\OD_{16}.C_2).A_6.C_2$ x 2, $C_3^4.(C_2\times Q_8).A_6.C_2^2$ x 2, $C_3^4.C_4:Q_8.C_2.A_6$

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, $C_3\times A_6^2:C_2^2$ x 2, $A_6^2:D_6$ x 2, $C_3:S_3.C_2.A_6.S_5$, $C_3:S_3.C_2.A_6.A_5.C_2$, $C_3:S_3.C_2.A_5.A_6.C_2$, $A_4.A_6.A_6$, $A_6^2.D_6$

Order 1296000: $C_5.A_6.A_6.C_2$ x 4, $C_5.A_6^2.C_2$ x 2, $A_5\wr S_3$ x 2, $A_5^3:C_6$ x 2, $A_5^3:S_3$ x 2, $D_5.A_6.A_6$, $A_6.A_5.A_5$

Order 1244160: $(A_4\times C_3:S_3).C_2^2.A_6.C_2^2$

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

Order 1119744: $(C_3:S_3)^3.C_2^3:S_4$ x 2, $C_3^5:D_6.C_2\wr S_4$ x 2, $C_3^6:(C_4^3.(C_2\times A_4))$, $C_3^6:(C_4^3.S_4)$, $C_3^6:(C_4^3.S_4)$

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 10, $C_2\times A_6^2.C_2^2$ x 8, $A_6^2:C_2^3$ 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, $C_2.A_6^2.C_2^2$ x 4, $C_2.A_6^2.C_2^2$ x 4, $A_6^2.D_4$ x 4, $A_6^2.D_4$ x 4, $C_2.A_6^2.C_4$ x 2, $C_2.A_6^2.C_2^2$ x 2, $A_6^2:D_4$ x 2, $A_6^2.(C_2\times C_4)$ x 2, $C_2\times A_6^2.C_4$ x 2, $D_4.A_6.A_6$, $C_3^2.C_4:F_5.A_6.C_2^2$, $A_6.A_6.C_2^3$, $A_6^2:(C_2\times C_4)$, $A_6^2.(C_2\times C_4)$, $S_4\times A_5:S_6$

Order 933120: $C_3^4.(C_2\times Q_8).A_6.C_2$ x 9, $C_3^4.D_4:C_2.A_6.C_2$ 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, $C_3^4.\OD_{16}.A_6.C_2$ x 4, $C_3^4.Q_8.C_2.A_6.C_2$ x 4, $C_3^4.\SD_{16}.A_6.C_2$ x 2, $C_3^4.Q_{16}.A_6.C_2$ x 2, $C_3^4.C_4:C_4.A_6.C_2$ x 2, $C_3^4:C_2^2.C_2^3.A_6$, $C_3^4.\SD_{16}:C_2.A_6$, $C_3^4.C_4\wr C_2.A_6$, $C_3^4.C_4:Q_8.A_6$, $C_3^4.(\OD_{16}.C_2).A_6$

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

Order 829440: $(D_4\times C_3:S_3).C_2^2.A_6.C_2^2$, $S_6\times S_4\wr C_2$

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, $C_3:S_3.C_2.A_6.A_5$, $A_6^2:S_3$, $A_6^2:C_6$, $S_3\times A_6^2$

Order 746496: $C_3^6.C_2^3.C_2^5.C_2^2$

Order 691200: $(D_4\times A_5).A_6.C_2^2$

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 3, $C_3^4.C_4:Q_8.C_2.S_5$

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

Order 559872: $C_3^5:D_6.C_2\wr A_4$ x 2, $C_3^6:((C_2\times C_4^2).S_4)$ x 2, $C_3^6:((C_2\times C_4^2).S_4)$, $C_3^6:(C_4^3.A_4)$, $C_3^6:((C_2\times C_4^2).S_4)$, $C_3^6.C_4^3:D_6$

Order 518400: $A_6.A_6.C_2^2$ x 15, $C_2.A_6.A_6.C_2$ x 11, $A_6^2.C_2^2$ x 8, $A_6^2:C_2^2$ x 6, $A_6^2:C_2^2$ x 6, $\POPlus(4,9)$ x 6, $(D_5\times C_3:S_3.C_2).A_6.C_2^2$ x 4, $C_3^2.C_4:F_5.A_6.C_2$ 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: $C_3^4:C_2^2.C_2.A_6.C_2$ x 16, $C_3^4.Q_8.A_6.C_2$ x 13, $C_3^4.(C_2\times C_4).A_6.C_2$ x 7, $C_3^4.D_4.A_6.C_2$ x 6, $C_3^4.C_8.A_6.C_2$ x 4, $C_3^4.Q_8.C_2.A_6$ x 2, $C_3^4.D_4:C_2.A_6$ x 2, $C_3^4.(C_2\times Q_8).A_6$ x 2, $C_3^2:(C_3:S_3.C_2).A_6.C_2^2$ x 2, $(C_3^2\times C_3:S_3.C_2).A_6.C_2^2$ x 2, $C_3^4.\SD_{16}.A_6$, $C_3^4.\OD_{16}.A_6$, $C_3^4.Q_{16}.A_6$, $C_3^4.C_4^2.A_6$, $C_3^4.C_4:C_4.A_6$

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, $D_5.A_6.A_5.C_2$

Order 414720: $C_4:\PSU(3,2).A_6.C_2^2$ x 4, $(D_4\times C_3:S_3.C_2).A_6.C_2^2$ x 4, $(C_3\times C_{12}).Q_8.A_6.C_2^2$ x 4, $(C_4\times C_3:S_3).C_2^3.A_6.C_2$ 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: $C_3^6.C_2^3.C_2\wr C_2^2$ x 8, $C_3^6.C_4^3.C_2^3$ x 2, $C_3^6.C_2^3.C_2^5.C_2$ x 2, $C_3^6.C_2^3.C_2^4.C_2^2$ x 2, $C_3^4.C_6^2.C_2^3.C_2.C_2^3$

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

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

Order 311040: $(A_4\times C_3:S_3.C_2).A_6.C_2$ x 6, $C_3^4.\SD_{16}:C_2.S_5$ x 4, $C_3^4.C_4\wr C_2.S_5$ x 4, $C_3^4:C_2^2.C_2^3.S_5$ x 2, $C_3^4.C_4:Q_8.S_5$ x 2, $C_3^4.(\OD_{16}.C_2).S_5$ 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, $C_6^2.(C_4\times S_3).A_6$, $C_3^4.C_4:Q_8.C_2.A_5$, $(S_3\times C_3:S_3.C_2).A_6.C_2^2$

Order 288000: $D_5:F_5.A_6.C_2^2$ x 4, $D_5\wr C_2:C_2.A_6.C_2$ 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, $C_3^6:(C_4^2:D_{12})$, $C_3^6:C_4\wr S_3$, $C_3^6:(C_4^3:C_6)$, $C_3^5:D_6.(C_4\times S_4)$, $(C_3:S_3)^3.(C_2\times S_4)$, $C_3^6:((C_2\times C_4^2).A_4)$

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

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 $174 \times 174$ character table is not available for this group.

Rational character table

The $171 \times 171$ rational character table is not available for this group.