Abstract group 87091200.a: $S_4\times S_{10}$

• Label: 87091200.a
• Order: \( 2^{11} \cdot 3^{5} \cdot 5^{2} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{5} \cdot 5^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $14$
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$S_4\times S_{10}$
Order:	\(87091200\)\(\medspace = 2^{11} \cdot 3^{5} \cdot 5^{2} \cdot 7 \)
Exponent:	\(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Automorphism group:	$S_4\times S_{10}$, of order \(87091200\)\(\medspace = 2^{11} \cdot 3^{5} \cdot 5^{2} \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$, $A_{10}$
Derived length:	$3$

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	36	42	60	84
Elements	1	94959	279368	3403536	78624	10878840	86400	7257600	3628800	5848416	15044160	3369600	1717632	3628800	6459264	2246400	3628800	2073600	7378560	2419200	3628800	2903040	1036800	87091200
Conjugacy classes	1	17	7	30	2	47	1	8	2	13	38	5	4	2	9	3	2	2	9	1	3	3	1	210
Divisions	1	17	7	30	2	47	1	8	2	13	38	5	4	2	9	3	2	2	9	1	3	3	1	210
Autjugacy classes	1	17	7	30	2	47	1	8	2	13	38	5	4	2	9	3	2	2	9	1	3	3	1	210

Dimension	1	2	3	9	18	27	35	36	42	70	72	75	84	90	105	108	126	150	160	168	180	210	225	252	270	288	300	315	320	350	378	420	448	450	480	504	525	567	576	600	630	675	700	756	768	864	896	900	945	1050	1134	1344	1350	1536	1575	1701	2304
Irr. complex chars.	4	2	4	4	2	4	4	4	4	2	2	4	6	4	4	4	8	2	4	2	2	4	8	10	4	4	4	4	2	4	4	2	2	6	4	2	4	4	2	2	6	4	2	4	2	4	1	6	4	6	2	2	4	1	4	4	2	210
Irr. rational chars.	4	2	4	4	2	4	4	4	4	2	2	4	6	4	4	4	8	2	4	2	2	4	8	10	4	4	4	4	2	4	4	2	2	6	4	2	4	4	2	2	6	4	2	4	2	4	1	6	4	6	2	2	4	1	4	4	2	210

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$40$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Permutation group:	Degree $14$ $\langle(1,2,3,4,5,7,9,6)(8,10)(13,14), (5,6,8)(7,9)(11,12,13)\rangle$
Transitive group:	40T192725				more information
Direct product:	$S_4$ $\, \times\, $ $S_{10}$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$S_{10}$ . $S_4$	$S_4$ . $S_{10}$	$(S_4\times A_{10})$ . $C_2$	$(A_4\times S_{10})$ . $C_2$	all 11
Aut. group:	$\Aut(C_2^2\times S_{10})$	$\Aut(A_4\times A_{10})$	$\Aut(S_4\times A_{10})$	$\Aut(A_4\times S_{10})$	all 5

Elements of the group are displayed as permutations of degree 14.

Homology

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

Subgroups

There are 13 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 $A_4\times A_{10}$
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 5600 (order at least 15552) are shown, as well as all normal subgroups of any index.

Order 87091200: $S_4\times S_{10}$

Order 43545600: $A_4.S_{10}$, $A_4\times S_{10}$, $S_4\times A_{10}$

Order 29030400: $S_{10}\times D_4$

Order 21772800: $A_4\times A_{10}$, $S_3\times S_{10}$

Order 14515200: $C_2^2\times S_{10}$ x 2, $C_2^2.S_{10}$ x 2, $C_4.S_{10}$, $C_4\times S_{10}$, $A_{10}\times D_4$

Order 10886400: $S_3\times A_{10}$, $C_3.S_{10}$, $C_3\times S_{10}$

Order 8709120: $S_4\times S_9$

Order 7257600: $C_2\times S_{10}$ x 5, $C_2^2\times A_{10}$ x 2, $C_2.S_{10}$, $C_4\times A_{10}$

Order 5443200: $C_3\times A_{10}$

Order 4354560: $A_4.S_9$, $A_4\times S_9$, $S_4\times A_9$

Order 3628800: $S_{10}$ x 3, $C_2\times A_{10}$ x 2

Order 2903040: $S_9\times D_4$

Order 2177280: $A_4\times A_9$, $S_3\times S_9$

Order 1935360: $C_2\times S_4\times S_8$

Order 1814400: $A_{10}$

Order 1451520: $C_2^2\times S_9$ x 2, $C_2^2.S_9$ x 2, $C_4\times S_9$, $C_4.A_9.C_2$, $A_9\times D_4$

Order 1088640: $S_3\times A_9$, $C_3:S_9$, $C_3\times S_9$

Order 967680: $S_4\times S_8$ x 4, $(C_2\times A_4).S_8$, $C_2\times A_4\times S_8$, $(C_2\times S_4).A_8$

Order 725760: $C_2\times S_9$ x 5, $C_2^2\times A_9$ x 2, $S_3\times S_4\times S_7$, $C_4\times A_9$, $A_9:C_4$

Order 691200: $S_4\times S_5\wr C_2$

Order 645120: $C_2\times S_8\times D_4$

Order 544320: $C_3\times A_9$

Order 483840: $A_4:S_8$ x 2, $A_4\times S_8$ x 2, $S_4\times A_8$ x 2, $D_6.S_8$, $C_2\times A_4\times A_8$

Order 414720: $S_6\times S_4^2$

Order 362880: $S_9$ x 3, $C_2\times A_9$ x 2, $C_3:S_4\times S_7$, $S_4\times C_3:S_7$, $S_3\times A_4\times S_7$, $C_3:(S_4\times S_7)$, $C_3\times S_4\times S_7$, $S_3\times S_4\times A_7$, $S_3\times A_4:S_7$

Order 345600: $S_4\times S_5^2$, $A_5^2.\GL(2,\mathbb{Z}/4)$, $(A_4\times A_5^2).D_4$, $A_4\times S_5\wr C_2$, $S_5^2.S_4$, $S_4\times A_5^2:C_4$, $S_4\times \POPlus(4,5)$

Order 322560: $D_4\times S_8$ x 8, $C_2^3.S_8$ x 2, $C_2^3\times S_8$ x 2, $C_2\times A_8\times D_4$, $C_2\times C_4:S_8$, $C_2\times C_4\times S_8$

Order 241920: $S_3\times S_8$ x 4, $D_6\times A_8$, $A_4\times A_8$, $C_6:S_8$, $C_6\times S_8$, $C_4.(D_6\times S_7)$, $C_2\times S_4\times S_7$

Order 230400: $A_5^2.D_4^2$

Order 207360: $A_6.S_4^2$ x 2, $A_4\times S_6\times S_4$ x 2, $A_6.S_4^2$, $A_6.S_4^2$, $S_6\times \PSOPlus(4,3)$

Order 181440: $(C_3\times A_4):S_7$, $C_3\times A_4\times S_7$, $C_3:S_4\times A_7$, $C_3\times S_4\times A_7$, $C_3\times A_4:S_7$, $S_3\times A_4\times A_7$, $A_4\times C_3:S_7$, $S_3^2\times S_7$, $A_9$

Order 172800: $A_4\times S_5^2$, $A_4:S_5^2$, $S_4\times \PSOPlus(4,5)$, $S_4\times A_5\times S_5$, $A_4:S_5^2$, $(A_4\times A_5^2):C_4$, $A_4\times A_5^2:C_4$, $\SOPlus(4,4):S_4$, $A_5^2:(C_2\times S_4)$, $A_4\times \POPlus(4,5)$, $S_4\times \SOPlus(4,4)$, $S_5^2:D_6$

Order 161280: $C_2^2\times S_8$ x 15, $C_2^2.A_8.C_2$ x 8, $A_8\times D_4$ x 4, $C_4:S_8$ x 4, $C_4\times S_8$ x 4, $C_2^3\times A_8$ x 2, $C_2\times C_4\times A_8$, $C_2^2.S_8$

Order 138240: $S_4\times S_6\times D_4$ x 2

Order 120960: $S_4\times S_7$ x 5, $S_3\times C_2^2:S_7$ x 2, $C_3:(D_4\times S_7)$ x 2, $S_7\times C_3:D_4$ x 2, $C_2\times D_6\times S_7$ x 2, $S_3\times A_8$ x 2, $C_3:S_8$ x 2, $C_3\times S_8$ x 2, $C_6\times A_8$, $C_3\times S_7\times D_4$, $S_7\times D_{12}$, $C_2\times A_4:S_7$, $C_2\times S_4\times A_7$, $C_2\times A_4\times S_7$, $D_4\times C_3:S_7$, $S_3\times D_4\times A_7$, $C_4\times S_3\times S_7$, $S_3\times C_4:S_7$, $D_{12}.A_7.C_2$

Order 115200: $C_2^2.A_5^2.D_4$ x 2, $S_5^2:D_4$ x 2, $C_2^2\times S_5\wr C_2$ x 2, $(C_2^2\times A_5^2):D_4$ x 2, $S_5^2\times D_4$, $C_4.A_5^2.D_4$, $C_4\times S_5\wr C_2$, $C_4.A_5^2.D_4$, $C_4.A_5^2.D_4$, $D_4\times \POPlus(4,5)$, $D_4\times A_5^2:C_4$

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

Order 92160: $C_2\wr S_5.S_4$

Order 90720: $C_3\times S_3\times S_7$ x 2, $A_7:S_3^2$ x 2, $C_3\times A_4\times A_7$, $C_3:S_3\times S_7$, $S_3^2\times A_7$, $A_7:S_3^2$

Order 86400: $A_5^2:S_4$, $A_4\times \PSOPlus(4,5)$, $A_5^2:S_4$, $A_4\times A_5\times S_5$, $S_4\times A_5^2$, $S_3\times S_5^2$, $A_5^2:S_4$, $A_4\times \SOPlus(4,4)$, $(A_5\times \GL(2,4)):D_4$, $A_5^2:D_{12}$, $S_5^2:C_6$, $A_5^2:C_4\times S_3$, $S_5^2:S_3$, $S_3\times \POPlus(4,5)$

Order 80640: $C_2\times S_8$ x 21, $C_2^2\times A_8$ x 7, $C_4\times A_8$ x 2, $A_8:C_4$ x 2, $C_2\times D_4\times S_7$

Order 69120: $C_2^2\times S_4\times S_6$ x 4, $\GL(2,\mathbb{Z}/4):S_6$ x 4, $S_4\times C_2^2:S_6$ x 4, $S_6\times \GL(2,\mathbb{Z}/4)$ x 4, $S_5\times S_4^2$ x 2, $D_4\times S_4\times A_6$ x 2, $D_4\times A_4:S_6$ x 2, $(D_4\times A_4).S_6$ x 2, $A_6:(D_4\times S_4)$ x 2, $S_4\times C_4:S_6$ x 2, $C_4:S_4\times S_6$ x 2, $C_4\times S_4\times S_6$ x 2, $C_2^2:S_4\times S_6$

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

Order 60480: $D_6\times S_7$ x 12, $S_4\times A_7$ x 3, $A_4:S_7$ x 3, $A_4\times S_7$ x 3, $C_3:D_4\times A_7$ x 2, $C_2\times C_6\times S_7$ x 2, $(C_6\times S_7):C_2$ x 2, $D_6:S_7$ x 2, $C_3\times C_2^2:S_7$ x 2, $C_2\times C_6:S_7$ x 2, $C_2\times D_6\times A_7$ x 2, $(C_2\times C_6):S_7$ x 2, $D_6:S_7$ x 2, $C_{12}:S_7$, $C_3\times D_4\times A_7$, $C_{12}\times S_7$, $C_3:(C_4\times S_7)$, $C_{12}:S_7$, $D_{12}\times A_7$, $C_4\times S_3\times A_7$, $C_2\times A_4\times A_7$, $C_3\times A_8$, $C_3:C_4\times S_7$, $C_{12}:S_7$, $D_6.S_7$

Order 57600: $S_5^2:C_2^2$ x 13, $S_5^2:C_4$ x 3, $\POPlus(4,5):C_4$ x 3, $\SOPlus(4,4):D_4$ x 2, $(C_2^2\times A_5^2):C_4$ x 2, $A_5^2:(C_2\times D_4)$ x 2, $(C_2^2\times A_5).S_5.C_2$ x 2, $(C_2^2\times A_5).S_5.C_2$ x 2, $C_2^2\times S_5^2$ x 2, $A_5^2:C_2^4$ x 2, $A_5^2:C_4\times C_2^2$ x 2, $D_4\times \SOPlus(4,4)$, $C_4\times S_5^2$, $A_5:(D_4\times S_5)$, $C_4:S_5\times S_5$, $D_4\times A_5\times S_5$, $F_5\times S_4\times S_5$, $D_4\times \PSOPlus(4,5)$, $\SOPlus(4,4):D_4$, $C_4\times \POPlus(4,5)$, $C_4:\POPlus(4,5)$, $A_5^2:C_4^2$, $(C_4\times A_5^2):C_4$, $(C_2\times A_5^2).D_4$

Order 55296: $S_4^3.C_2^2$

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

Order 46080: $S_4\times C_2^4:S_5$ x 4, $S_6\times D_4^2$, $C_2^5.(S_4\times A_5)$, $C_2\wr S_5\times A_4$, $A_4.C_2\wr S_5$

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

Order 43200: $C_3:S_5^2$, $A_5^2:D_6$, $A_5^2:D_6$, $(A_5\times \GL(2,4)):C_4$, $A_5^2:D_6$, $A_5^2:D_6$, $A_5^2:C_{12}$, $C_3\times \POPlus(4,5)$, $C_3\times S_5^2$, $A_4\times A_5^2$, $C_3:S_5^2$, $A_5^2:D_6$

Order 41472: $S_4^2.\SOPlus(4,2)$

Order 40320: $D_4\times S_7$ x 8, $S_8$ x 6, $C_2\times A_8$ x 5, $C_2^3:S_7$ x 2, $C_2^3\times S_7$ x 2, $C_2\times D_4\times A_7$, $C_2\times C_4\times S_7$, $C_2\times C_4:S_7$

Order 36288: $S_4\times {}^2G(2,3)$

Order 34560: $C_2\times S_4\times S_6$ x 22, $C_2^2\times A_4\times S_6$ x 4, $C_2^2\times A_4:S_6$ x 4, $A_4\times S_4\times S_5$ x 4, $A_4\times C_2^2:S_6$ x 4, $A_6:\GL(2,\mathbb{Z}/4)$ x 4, $A_6\times \GL(2,\mathbb{Z}/4)$ x 4, $A_5:S_4^2$ x 4, $(C_2\times S_4):S_6$ x 4, $A_6:\GL(2,\mathbb{Z}/4)$ x 4, $A_6:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^2\times S_4\times A_6$ x 4, $D_4\times A_4\times A_6$ x 2, $(C_4\times A_4):S_6$ x 2, $C_4\times A_4\times S_6$ x 2, $A_4\times C_4:S_6$ x 2, $C_4\times S_4\times A_6$ x 2, $C_4\times A_4:S_6$ x 2, $A_5\times S_4^2$ x 2, $A_5:S_4^2$ x 2, $S_5\times \PSOPlus(4,3)$ x 2, $S_3\times D_4\times S_6$ x 2, $C_4:S_4\times A_6$ x 2, $A_4:(C_4\times S_6)$ x 2, $A_6:C_4\times S_4$ x 2, $A_4:C_4\times S_6$ x 2, $S_4\times S_6:C_2$, $C_2^2:S_4\times A_6$, $(C_2^2\times A_6):S_4$, $C_2^2:A_4\times S_6$, $D_6\times S_4\times S_5$

Order 32256: $S_4\times C_2^3:\GL(3,2)$ x 2, $A_4\times C_2^4:\GL(3,2)$

Order 31104: $S_3\wr S_3\times S_4$

Order 30720: $C_2^5.(D_4\times S_5)$

Order 30240: $S_3\times S_7$ x 16, $C_6:S_7$ x 6, $D_6\times A_7$ x 6, $C_6\times S_7$ x 6, $C_2\times C_6\times A_7$ x 2, $A_4\times A_7$ x 2, $C_6.S_7$, $C_{12}\times A_7$, $A_7:C_{12}$, $C_3:C_4\times A_7$

Order 28800: $S_5\wr C_2$ x 13, $A_5^2:C_2^3$ x 8, $C_2\times S_5^2$ x 8, $C_2\times A_5^2:C_4$ x 6, $A_5^2:D_4$ x 2, $A_5^2:D_4$ x 2, $A_5^2:D_4$ x 2, $A_5^2:C_2^3$ x 2, $A_5^2:C_2^3$ x 2, $A_5^2:D_4$ x 2, $A_5^2:D_4$ x 2, $A_5^2:C_2^3$ x 2, $\SOPlus(4,4):C_4$, $A_5^2:(C_2\times C_4)$, $C_2.S_5^2$, $C_4\times A_5\times S_5$, $A_5^2:D_4$, $C_2.S_5^2$, $D_5\times S_4\times S_5$, $A_5^2:D_4$, $D_4\times A_5^2$, $F_5\times S_4\times A_5$, $C_4\times \PSOPlus(4,5)$, $A_5:F_5\times S_4$, $A_4\times F_5\times S_5$, $A_5^2:D_4$, $A_4:F_5\times S_5$, $F_5\times A_4:S_5$, $A_4:(F_5\times S_5)$, $C_4\times \SOPlus(4,4)$, $A_5^2:D_4$, $A_5^2:D_4$

Order 27648: $S_4^3:C_2$ x 8, $C_2\times S_4^3$ x 2, $C_2\times A_4^2:\GL(2,\mathbb{Z}/4)$, $C_2\times A_4^3:D_4$, $C_2\times S_4^2:S_4$, $A_4^3.C_2^4$, $A_4^3.(C_2^2\times C_4)$, $C_2\times A_4^3:D_4$

Order 25920: $C_3:S_4\times A_6$ x 2, $A_4\times C_3:S_6$ x 2, $(C_3\times A_4):S_6$ x 2, $C_3\times S_4\times A_6$ x 2, $C_3\times A_4\times S_6$ x 2, $S_3\times A_4\times A_6$ x 2, $C_3\times A_4:S_6$ x 2, $S_3^2\times S_6$

Order 24192: $S_3\times S_4\times \GL(3,2)$

Order 23040: $A_6:D_4^2$ x 4, $C_2^2\times S_6\times D_4$ x 4, $A_6:D_4^2$ x 4, $A_6:D_4^2$ x 4, $S_6\times C_2^2\wr C_2$ x 4, $C_4:D_4\times S_6$ x 4, $D_4\times S_4\times S_5$ x 4, $A_6:D_4^2$ x 2, $C_4\times D_4\times S_6$ x 2, $(A_4\times C_2^4):S_5$ x 2, $S_4\times C_2^4:A_5$ x 2, $A_4\times C_2^4:S_5$ x 2, $C_2\wr S_5\times S_3$, $D_4^2\times A_6$, $C_4:D_4\times S_6$, $A_6:D_4^2$, $C_2^7:\GL(2,4)$

Order 22680: $C_3^2\times A_7$

Order 21600: $S_5\times \GL(2,4)$, $S_3\times A_5^2$, $\GL(2,4):S_5$, $\GL(2,4):S_5$, $A_5^2:S_3$, $\GL(2,4):S_5$, $A_5^2:C_6$

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

Order 20736: $S_3^2:S_4^2$ x 4, $S_3^2.S_4^2$ x 2, $S_4\times C_6^2:D_{12}$ x 2, $S_3^2:S_4^2$ x 2, $D_6^2:C_6\times S_4$ x 2, $C_6^2:D_{12}:S_4$, $(A_4\times D_6^2):D_6$, $C_3^2:C_4\times S_4^2$

Order 20160: $C_2^2\times S_7$ x 15, $C_2^2:S_7$ x 8, $C_4:S_7$ x 4, $C_4\times S_7$ x 4, $D_4\times A_7$ x 4, $C_2^3\times A_7$ x 2, $C_2\times C_4\times A_7$, $C_2\times A_7:C_4$, $A_8$

Order 19200: $F_5\wr C_2\times S_4$, $D_4\times F_5\times S_5$

Order 18432: $C_2\times A_4^2.D_4^2$, $C_2^5.S_4^2$

Order 18144: $A_4\times {}^2G(2,3)$

Order 17280: $S_4\times S_6$ x 26, $S_3\times S_4\times S_5$ x 11, $C_2\times A_4:S_6$ x 10, $C_2\times S_4\times A_6$ x 10, $C_2\times A_4\times S_6$ x 10, $A_4^2:S_5$ x 4, $C_3:(D_4\times S_6)$ x 4, $C_3:D_4\times S_6$ x 4, $S_3\times C_2^2:S_6$ x 4, $A_4\times S_4\times A_5$ x 4, $C_2\times D_6\times S_6$ x 4, $C_2^2\times A_4\times A_6$ x 4, $C_4\times A_4\times A_6$ x 2, $A_4\times A_6:C_4$ x 2, $A_4:C_4\times A_6$ x 2, $A_4^2:S_5$ x 2, $S_5\times A_4^2$ x 2, $D_{12}:S_6$ x 2, $S_3\times C_4:S_6$ x 2, $A_5\times \PSOPlus(4,3)$ x 2, $C_3\times D_4\times S_6$ x 2, $S_3\times D_4\times A_6$ x 2, $D_4\times C_3:S_6$ x 2, $(A_4\times A_6):C_4$ x 2, $D_{12}\times S_6$ x 2, $C_4\times S_3\times S_6$ x 2, $C_6\times S_4\times S_5$, $\PGL(2,9):S_4$, $S_6:S_4$, $A_4\times S_6:C_2$, $A_6.(C_2\times S_4)$, $A_6.C_2\times S_4$, $S_4\times \PGL(2,9)$, $C_2^2:A_4\times A_6$, $D_6\times A_4:S_5$, $A_4\times D_6\times S_5$, $C_6:(S_4\times S_5)$, $C_6:S_4\times S_5$, $C_6:S_5\times S_4$, $D_6\times S_4\times A_5$

Order 16128: $C_2\times S_4\times \PGL(2,7)$, $S_3\times C_2^4:\GL(3,2)$, $A_4\times C_2^3:\GL(3,2)$

Order 15552: $C_3^3:S_4^2$, $C_3^3:S_4^2$, $C_3^3:S_4^2$, $C_3^3:S_4^2$, $S_3\wr C_3\times S_4$, $(A_4\times S_3^3):S_3$, $A_4\times S_3\wr S_3$

Order 24: $S_4$

Order 12: $A_4$

Order 4: $C_2^2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 87091200: $S_4\times S_{10}$ ($C_1$)

Order 43545600: $A_4.S_{10}$ ($C_2$), $A_4\times S_{10}$ ($C_2$), $S_4\times A_{10}$ ($C_2$)

Order 21772800: $A_4\times A_{10}$ ($C_2^2$)

Order 14515200: $C_2^2\times S_{10}$ ($S_3$)

Order 7257600: $C_2^2\times A_{10}$ ($D_6$)

Order 3628800: $S_{10}$ ($S_4$)

Order 1814400: $A_{10}$ ($C_2\times S_4$)

Order 24: $S_4$ ($S_{10}$)

Order 12: $A_4$ ($C_2\times S_{10}$)

Order 4: $C_2^2$ ($S_3\times S_{10}$)

Order 1: $C_1$ ($S_4\times S_{10}$)

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 0 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

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