Abstract group 691200.g: $S_4\times S_5\wr C_2$

• Label: 691200.g
• Order: \( 2^{10} \cdot 3^{3} \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $14$
• Trans deg.: $40$
• Rank: $3$

Group information

Description:	$S_4\times S_5\wr C_2$
Order:	\(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$S_4\times S_5\wr C_2$, of order \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \)
Composition factors:	$C_2$ x 6, $C_3$, $A_5$ x 2
Derived length:	$3$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	30	60
Elements	1	7959	3968	92136	624	111840	57600	46416	195360	13632	51264	28800	58560	23040	691200
Conjugacy classes	1	20	5	31	2	42	4	13	31	4	9	1	9	3	175
Divisions	1	20	5	31	2	42	4	13	31	4	9	1	9	3	175
Autjugacy classes	1	20	5	31	2	42	4	13	31	4	9	1	9	3	175

Dimension	1	2	3	4	6	8	10	12	16	20	24	25	30	32	36	40	48	50	60	64	72	75	80	96	100	108	120	144	150	180
Irr. complex chars.	8	6	8	1	2	8	8	4	12	4	10	8	8	6	8	8	12	6	4	1	2	8	4	4	1	4	10	4	2	4	175
Irr. rational chars.	8	6	8	1	2	8	8	4	12	4	10	8	8	6	8	8	12	6	4	1	2	8	4	4	1	4	10	4	2	4	175

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$40$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $14$ $\langle(1,3,7,6,4,2,5,8)(9,10)(11,13,12), (2,6,8)(5,9,7)(11,14,12,13), (1,2,4,6,7,8,9,10,5,3)(11,12,13)\rangle$
Transitive group:	40T146640				more information
Direct product:	$S_4$ $\, \times\, $ $(S_5\wr C_2)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(S_5\wr C_2)$ . $S_4$	$(S_4\times S_5^2)$ . $C_2$	$(S_5^2.S_4)$ . $C_2$	$S_5^2$ . $(C_2\times S_4)$	all 33
Aut. group:	$\Aut(A_5^2:C_2^3)$	$\Aut(A_4\times A_5^2)$	$\Aut(A_5^2:C_4\times C_2^2)$	$\Aut(S_4\times A_5^2)$	all 20

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 47481646 subgroups in 34387 conjugacy classes, 35 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_4\times S_5\wr C_2$
Commutator:	$G' \simeq$ $A_4\times \PSOPlus(4,5)$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_4\times S_5\wr C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2$	$G/\operatorname{Fit} \simeq$ $S_5^2:D_6$
Radical:	$R \simeq$ $S_4$	$G/R \simeq$ $A_5^2.D_4$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times A_5^2$	$G/\operatorname{soc} \simeq$ $S_3\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

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 225 (order at least 3072) are shown, as well as all normal subgroups of any index.

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

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 230400: $A_5^2.D_4^2$

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 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 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 69120: $S_5\times S_4^2$

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 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 34560: $A_4\times S_4\times S_5$ x 2, $A_5:S_4^2$ x 2, $A_5\times S_4^2$, $A_5:S_4^2$, $D_6\times S_4\times S_5$, $S_5\times \PSOPlus(4,3)$

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: $S_4^3:C_2$

Order 23040: $D_4\times S_4\times S_5$ x 2

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

Order 17280: $S_3\times S_4\times S_5$ x 9, $A_4^2:S_5$ x 2, $A_4\times S_4\times A_5$ x 2, $C_6\times S_4\times S_5$, $A_4^2:S_5$, $S_5\times A_4^2$, $D_6\times A_4:S_5$, $A_5\times \PSOPlus(4,3)$, $A_4\times D_6\times S_5$, $C_6:(S_4\times S_5)$, $C_6:S_4\times S_5$, $C_6:S_5\times S_4$, $D_6\times S_4\times A_5$

Order 14400: $\POPlus(4,5)$ x 8, $S_5^2$ x 8, $A_5^2:C_2^2$ x 5, $A_5^2:C_2^2$ x 5, $A_5^2:C_2^2$ x 5, $A_5^2:C_4$ x 4, $C_2^2\times A_5^2$ x 2, $C_5:(S_4\times S_5)$, $C_5:S_4\times S_5$, $A_4\times A_5:F_5$, $D_5\times S_4\times A_5$, $A_5^2:C_4$, $C_4\times A_5^2$, $A_5^2:C_4$, $S_4\times C_5:S_5$, $A_4\times F_5\times A_5$, $A_4:F_5\times A_5$, $D_5\times A_4\times S_5$, $D_5\times A_4:S_5$, $(A_4\times A_5):F_5$, $C_5\times S_4\times S_5$, $S_3\times F_5\times S_5$, $A_5^2:C_4$

Order 13824: $S_4^3$, $A_4^3.C_2^3$, $A_4^2:C_4\times S_4$, $A_4^2:\GL(2,\mathbb{Z}/4)$, $A_4^3:D_4$, $S_4^2:S_4$, $A_4^3:D_4$

Order 11520: $S_5\times \GL(2,\mathbb{Z}/4)$ x 4, $S_4\times C_2^2:S_5$ x 4, $\GL(2,\mathbb{Z}/4):S_5$ x 4, $C_2^2\times S_4\times S_5$ x 4, $C_4:(S_4\times S_5)$ x 2, $S_4\times C_4:S_5$ x 2, $C_4:S_4\times S_5$ x 2, $C_4\times S_4\times S_5$ x 2, $D_4\times S_4\times A_5$ x 2, $D_4\times A_4\times S_5$ x 2, $D_4\times A_4:S_5$ x 2, $F_5\times S_4^2$, $D_4\times D_6\times S_5$, $C_2^2:S_4\times S_5$

Order 10800: $A_5\times \GL(2,4)$

Order 9600: $C_2^2\times F_5\times S_5$ x 2, $D_{10}:(C_4\times S_5)$ x 2, $F_5\times C_2^2:S_5$ x 2, $D_{10}:C_4\times S_5$ x 2, $S_4\times F_5^2$, $C_{20}:(C_4\times S_5)$, $F_5\times C_4:S_5$, $C_{20}:C_4\times S_5$, $C_4\times F_5\times S_5$, $D_4\times D_5\times S_5$, $D_4\times F_5\times A_5$, $D_4\times A_5:F_5$, $A_4\times F_5\wr C_2$, $F_5^2:S_4$, $D_5^2.\GL(2,\mathbb{Z}/4)$, $D_{10}^2.D_{12}$, $C_5^2:\OD_{16}\times S_4$, $D_5^2.C_2^2\times S_4$

Order 9216: $A_4^2:D_4^2$

Order 8640: $S_3\times A_4\times S_5$ x 5, $S_3\times S_4\times A_5$ x 5, $C_3:S_4\times S_5$ x 5, $S_4\times C_3:S_5$ x 5, $C_3:(S_4\times S_5)$ x 5, $C_3\times S_4\times S_5$ x 5, $S_3\times A_4:S_5$ x 5, $A_5\times A_4^2$, $A_4\times C_6:S_5$, $C_2\times \GL(2,4):S_4$, $C_6\times A_4\times S_5$, $C_6\times A_4:S_5$, $C_6\times S_4\times A_5$, $C_6:S_4\times A_5$, $A_4\times D_6\times A_5$, $A_5:D_6^2$

Order 7680: $D_4^2\times S_5$

Order 7200: $\PSOPlus(4,5)$ x 3, $A_5\times S_5$ x 3, $\SOPlus(4,4)$ x 3, $C_2\times A_5^2$ x 2, $D_5\times A_4\times A_5$, $C_5\times S_4\times A_5$, $A_4\times C_5:S_5$, $(C_5\times A_4):S_5$, $C_5\times A_4\times S_5$, $C_5\times A_4:S_5$, $A_5\times C_5:S_4$, $S_3\times D_5\times S_5$, $C_{15}:(C_4\times S_5)$, $S_3\times A_5:F_5$, $F_5\times C_3:S_5$, $C_{15}:C_4\times S_5$, $S_3\times F_5\times A_5$, $C_3\times F_5\times S_5$

Order 6912: $A_4:S_4^2$ x 2, $A_4\times S_4^2$ x 2, $A_4:S_4^2$, $D_6\times S_4^2$, $D_6\wr C_2\times S_4$, $A_4^3:C_2^2$, $A_4^3:C_4$, $A_4^3:C_2^2$, $A_4^3:C_2^2$, $A_4^3:C_4$, $A_4^3:C_2^2$, $S_4^2:D_6$

Order 6400: $D_5^2.D_4^2$

Order 5760: $C_2\times S_4\times S_5$ x 24, $S_3\times D_4\times S_5$ x 17, $A_5:\GL(2,\mathbb{Z}/4)$ x 4, $A_4\times C_2^2:S_5$ x 4, $(C_2\times S_4):S_5$ x 4, $A_5:\GL(2,\mathbb{Z}/4)$ x 4, $A_5:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^2\times A_4:S_5$ x 4, $C_2^2\times A_4\times S_5$ x 4, $C_2^2\times S_4\times A_5$ x 4, $A_5\times \GL(2,\mathbb{Z}/4)$ x 4, $D_5.S_4^2$ x 2, $S_3\times C_2^3:S_5$ x 2, $D_4\times A_4\times A_5$ x 2, $A_5\times C_4:S_4$ x 2, $C_4\times S_4\times A_5$ x 2, $C_6:(D_4\times S_5)$ x 2, $(C_4\times A_4):S_5$ x 2, $A_4\times C_4:S_5$ x 2, $C_4\times A_4\times S_5$ x 2, $C_4\times A_4:S_5$ x 2, $C_6:D_4\times S_5$ x 2, $\GL(2,4):C_2^5$ x 2, $A_4:(C_4\times S_5)$ x 2, $A_4:C_4\times S_5$ x 2, $A_4\times F_5\times S_4$ x 2, $A_5:C_4\times S_4$ x 2, $F_5\times \PSOPlus(4,3)$, $D_5\times S_4^2$, $C_2^2:A_4\times S_5$, $D_6\times F_5\times S_4$, $D_5.S_4^2$, $C_4:S_5\times D_6$, $C_6\times D_4\times S_5$, $C_2\times D_{12}:S_5$, $C_6:S_5\times D_4$, $C_4\times D_6\times S_5$, $C_2\times D_{12}\times S_5$, $D_4\times D_6\times A_5$, $(C_2^2\times A_5):S_4$, $A_5\times C_2^2:S_4$

Order 4800: $C_2\times F_5\times S_5$ x 12, $(C_{10}\times S_5):C_4$ x 4, $C_5:D_4\times S_5$ x 2, $C_5:(D_4\times S_5)$ x 2, $A_5\times D_{10}:C_4$ x 2, $C_2\times D_{10}\times S_5$ x 2, $D_5\times C_2^2:S_5$ x 2, $C_{10}.(C_4\times S_5)$ x 2, $A_5:C_4\times F_5$ x 2, $C_2^2\times F_5\times A_5$ x 2, $C_2\times D_{10}.S_5$ x 2, $(D_{10}\times A_5):C_4$ x 2, $D_{10}^2.D_6$ x 2, $D_{10}^2.D_6$ x 2, $D_{10}^2:D_6$ x 2, $D_{10}^2.D_6$ x 2, $D_{10}^2.D_6$, $D_{20}:S_5$, $C_4\times D_5\times S_5$, $D_{20}\times S_5$, $D_5\times C_4:S_5$, $D_4\times C_5:S_5$, $C_5\times D_4\times S_5$, $A_5\times C_{20}:C_4$, $D_4\times D_5\times A_5$, $C_4\times F_5\times A_5$, $A_4\times F_5^2$, $A_4:F_5^2$, $A_4:F_5^2$, $C_4\times A_5:F_5$, $(C_{20}\times A_5):C_4$, $(A_4\times C_5^2):\OD_{16}$, $(A_4\times C_5^2):\OD_{16}$, $D_{10}^2.C_{12}$, $C_5^2:Q_8:S_4$, $A_4\times D_5^2.C_2^2$, $C_5^2:Q_8\times S_4$, $C_5^2:C_8\times S_4$, $F_5^2:D_6$

Order 4608: $D_4\times S_4^2$ x 2, $S_4^2:C_2^3$ x 2, $C_2^6:\SOPlus(4,2)$ x 2, $C_2^6:\SOPlus(4,2)$ x 2, $S_4^2:D_4$ x 2, $C_2^3:S_4^2$, $(C_4\times A_4^2).D_4$, $C_4\times S_4\wr C_2$, $(C_4\times A_4^2):D_4$, $S_4^2:D_4$, $A_4^2:C_4\times D_4$, $C_2^5.D_6^2$

Order 4320: $S_3^2\times S_5$ x 8, $\GL(2,4):S_4$ x 3, $\GL(2,4):S_4$ x 3, $\GL(2,4):S_4$ x 3, $S_4\times \GL(2,4)$ x 3, $S_3\times A_4\times A_5$ x 3, $A_4\times C_3:S_5$ x 3, $C_3\times A_4\times S_5$ x 3, $S_3\times C_6:S_5$ x 2, $C_6\times S_3\times S_5$ x 2, $C_6\times A_4\times A_5$, $S_3\times D_6\times A_5$, $C_2\times A_5:S_3^2$, $C_6:S_3\times S_5$

Order 3840: $C_4:D_4\times S_5$ x 4, $A_5.D_4^2$ x 4, $A_5:D_4^2$ x 4, $A_5:D_4^2$ x 4, $C_2^2\times D_4\times S_5$ x 4, $S_5\times C_2^2\wr C_2$ x 4, $C_4\times D_4\times S_5$ x 2, $A_5:D_4^2$ x 2, $D_4\times F_5\times S_4$ x 2, $C_4:D_4\times S_5$, $A_5:D_4^2$, $A_5\times D_4^2$

Order 3600: $C_5\times A_4\times A_5$, $D_{15}:S_5$, $C_5\times S_3\times S_5$, $\GL(2,4):D_{10}$, $\GL(2,4):F_5$, $F_5\times \GL(2,4)$, $\GL(2,4):D_{10}$, $\GL(2,4):D_{10}$, $\GL(2,4):F_5$, $\GL(2,4):F_5$, $D_{15}\times S_5$, $\GL(2,4):D_{10}$, $A_5^2$

Order 3456: $S_3\times S_4^2$ x 9, $S_4\times A_4^2$ x 2, $A_4^2:S_4$ x 2, $C_2\times A_4^2.D_6$ x 2, $A_4\times D_6\times S_4$ x 2, $S_3^2:\GL(2,\mathbb{Z}/4)$ x 2, $S_3^2:C_4\times S_4$ x 2, $D_6^2:D_{12}$ x 2, $C_2\times D_6^2:D_6$ x 2, $S_4\times D_6^2$, $D_6\times \PSOPlus(4,3)$, $C_6\times S_4^2$, $C_6:S_4^2$, $A_4^2:S_4$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $(A_4\times C_6^2):D_4$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $C_2^4:S_3^3$, $C_6^2:C_4\times S_4$, $A_4\times D_6\wr C_2$, $A_4^2:S_4$, $A_4^3:C_2$, $A_4^2:C_4\times S_3$, $C_2^4:S_3^3$, $S_4^2:C_6$, $(C_3\times A_4^2):D_4$, $S_4^2:S_3$, $A_4^2:D_{12}$

Order 3200: $F_5^2:D_4$ x 2, $F_5^2:C_2^3$ x 2, $D_{10}^2.D_4$ x 2, $D_{10}^2.D_4$ x 2, $D_4\times F_5^2$, $C_4:F_5\wr C_2$, $C_4:F_5\wr C_2$, $C_4\times F_5\wr C_2$, $F_5^2:D_4$, $D_4\times C_5^2:\OD_{16}$, $D_{10}^2.C_2^3$

Order 3072: $C_4^2.(D_4\times S_4)$

Order 1024: $C_4^2:D_4^2$

Order 27: $C_3^3$

Order 25: $C_5^2$

Order 24: $S_4$

Order 12: $A_4$

Order 4: $C_2^2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 225 (order at least 3072) are shown, as well as all normal subgroups of any index.

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

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 230400: $A_5^2.D_4^2$

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 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 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 69120: $S_5\times S_4^2$

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 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 34560: $A_4\times S_4\times S_5$ x 2, $A_5:S_4^2$ x 2, $A_5\times S_4^2$, $A_5:S_4^2$, $D_6\times S_4\times S_5$, $S_5\times \PSOPlus(4,3)$

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: $S_4^3:C_2$

Order 23040: $D_4\times S_4\times S_5$ x 2

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

Order 17280: $S_3\times S_4\times S_5$ x 9, $A_4^2:S_5$ x 2, $A_4\times S_4\times A_5$ x 2, $C_6\times S_4\times S_5$, $A_4^2:S_5$, $S_5\times A_4^2$, $D_6\times A_4:S_5$, $A_5\times \PSOPlus(4,3)$, $A_4\times D_6\times S_5$, $C_6:(S_4\times S_5)$, $C_6:S_4\times S_5$, $C_6:S_5\times S_4$, $D_6\times S_4\times A_5$

Order 14400: $\POPlus(4,5)$ x 8, $S_5^2$ x 8, $A_5^2:C_2^2$ x 5, $A_5^2:C_2^2$ x 5, $A_5^2:C_2^2$ x 5, $A_5^2:C_4$ x 4, $C_2^2\times A_5^2$ x 2, $C_5:(S_4\times S_5)$, $C_5:S_4\times S_5$, $A_4\times A_5:F_5$, $D_5\times S_4\times A_5$, $A_5^2:C_4$, $C_4\times A_5^2$, $A_5^2:C_4$, $S_4\times C_5:S_5$, $A_4\times F_5\times A_5$, $A_4:F_5\times A_5$, $D_5\times A_4\times S_5$, $D_5\times A_4:S_5$, $(A_4\times A_5):F_5$, $C_5\times S_4\times S_5$, $S_3\times F_5\times S_5$, $A_5^2:C_4$

Order 13824: $S_4^3$, $A_4^3.C_2^3$, $A_4^2:C_4\times S_4$, $A_4^2:\GL(2,\mathbb{Z}/4)$, $A_4^3:D_4$, $S_4^2:S_4$, $A_4^3:D_4$

Order 11520: $S_5\times \GL(2,\mathbb{Z}/4)$ x 4, $S_4\times C_2^2:S_5$ x 4, $\GL(2,\mathbb{Z}/4):S_5$ x 4, $C_2^2\times S_4\times S_5$ x 4, $C_4:(S_4\times S_5)$ x 2, $S_4\times C_4:S_5$ x 2, $C_4:S_4\times S_5$ x 2, $C_4\times S_4\times S_5$ x 2, $D_4\times S_4\times A_5$ x 2, $D_4\times A_4\times S_5$ x 2, $D_4\times A_4:S_5$ x 2, $F_5\times S_4^2$, $D_4\times D_6\times S_5$, $C_2^2:S_4\times S_5$

Order 10800: $A_5\times \GL(2,4)$

Order 9600: $C_2^2\times F_5\times S_5$ x 2, $D_{10}:(C_4\times S_5)$ x 2, $F_5\times C_2^2:S_5$ x 2, $D_{10}:C_4\times S_5$ x 2, $S_4\times F_5^2$, $C_{20}:(C_4\times S_5)$, $F_5\times C_4:S_5$, $C_{20}:C_4\times S_5$, $C_4\times F_5\times S_5$, $D_4\times D_5\times S_5$, $D_4\times F_5\times A_5$, $D_4\times A_5:F_5$, $A_4\times F_5\wr C_2$, $F_5^2:S_4$, $D_5^2.\GL(2,\mathbb{Z}/4)$, $D_{10}^2.D_{12}$, $C_5^2:\OD_{16}\times S_4$, $D_5^2.C_2^2\times S_4$

Order 9216: $A_4^2:D_4^2$

Order 8640: $S_3\times A_4\times S_5$ x 5, $S_3\times S_4\times A_5$ x 5, $C_3:S_4\times S_5$ x 5, $S_4\times C_3:S_5$ x 5, $C_3:(S_4\times S_5)$ x 5, $C_3\times S_4\times S_5$ x 5, $S_3\times A_4:S_5$ x 5, $A_5\times A_4^2$, $A_4\times C_6:S_5$, $C_2\times \GL(2,4):S_4$, $C_6\times A_4\times S_5$, $C_6\times A_4:S_5$, $C_6\times S_4\times A_5$, $C_6:S_4\times A_5$, $A_4\times D_6\times A_5$, $A_5:D_6^2$

Order 7680: $D_4^2\times S_5$

Order 7200: $\PSOPlus(4,5)$ x 3, $A_5\times S_5$ x 3, $\SOPlus(4,4)$ x 3, $C_2\times A_5^2$ x 2, $D_5\times A_4\times A_5$, $C_5\times S_4\times A_5$, $A_4\times C_5:S_5$, $(C_5\times A_4):S_5$, $C_5\times A_4\times S_5$, $C_5\times A_4:S_5$, $A_5\times C_5:S_4$, $S_3\times D_5\times S_5$, $C_{15}:(C_4\times S_5)$, $S_3\times A_5:F_5$, $F_5\times C_3:S_5$, $C_{15}:C_4\times S_5$, $S_3\times F_5\times A_5$, $C_3\times F_5\times S_5$

Order 6912: $A_4:S_4^2$ x 2, $A_4\times S_4^2$ x 2, $A_4:S_4^2$, $D_6\times S_4^2$, $D_6\wr C_2\times S_4$, $A_4^3:C_2^2$, $A_4^3:C_4$, $A_4^3:C_2^2$, $A_4^3:C_2^2$, $A_4^3:C_4$, $A_4^3:C_2^2$, $S_4^2:D_6$

Order 6400: $D_5^2.D_4^2$

Order 5760: $C_2\times S_4\times S_5$ x 24, $S_3\times D_4\times S_5$ x 17, $A_5:\GL(2,\mathbb{Z}/4)$ x 4, $A_4\times C_2^2:S_5$ x 4, $(C_2\times S_4):S_5$ x 4, $A_5:\GL(2,\mathbb{Z}/4)$ x 4, $A_5:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^2\times A_4:S_5$ x 4, $C_2^2\times A_4\times S_5$ x 4, $C_2^2\times S_4\times A_5$ x 4, $A_5\times \GL(2,\mathbb{Z}/4)$ x 4, $D_5.S_4^2$ x 2, $S_3\times C_2^3:S_5$ x 2, $D_4\times A_4\times A_5$ x 2, $A_5\times C_4:S_4$ x 2, $C_4\times S_4\times A_5$ x 2, $C_6:(D_4\times S_5)$ x 2, $(C_4\times A_4):S_5$ x 2, $A_4\times C_4:S_5$ x 2, $C_4\times A_4\times S_5$ x 2, $C_4\times A_4:S_5$ x 2, $C_6:D_4\times S_5$ x 2, $\GL(2,4):C_2^5$ x 2, $A_4:(C_4\times S_5)$ x 2, $A_4:C_4\times S_5$ x 2, $A_4\times F_5\times S_4$ x 2, $A_5:C_4\times S_4$ x 2, $F_5\times \PSOPlus(4,3)$, $D_5\times S_4^2$, $C_2^2:A_4\times S_5$, $D_6\times F_5\times S_4$, $D_5.S_4^2$, $C_4:S_5\times D_6$, $C_6\times D_4\times S_5$, $C_2\times D_{12}:S_5$, $C_6:S_5\times D_4$, $C_4\times D_6\times S_5$, $C_2\times D_{12}\times S_5$, $D_4\times D_6\times A_5$, $(C_2^2\times A_5):S_4$, $A_5\times C_2^2:S_4$

Order 4800: $C_2\times F_5\times S_5$ x 12, $(C_{10}\times S_5):C_4$ x 4, $C_5:D_4\times S_5$ x 2, $C_5:(D_4\times S_5)$ x 2, $A_5\times D_{10}:C_4$ x 2, $C_2\times D_{10}\times S_5$ x 2, $D_5\times C_2^2:S_5$ x 2, $C_{10}.(C_4\times S_5)$ x 2, $A_5:C_4\times F_5$ x 2, $C_2^2\times F_5\times A_5$ x 2, $C_2\times D_{10}.S_5$ x 2, $(D_{10}\times A_5):C_4$ x 2, $D_{10}^2.D_6$ x 2, $D_{10}^2.D_6$ x 2, $D_{10}^2:D_6$ x 2, $D_{10}^2.D_6$ x 2, $D_{10}^2.D_6$, $D_{20}:S_5$, $C_4\times D_5\times S_5$, $D_{20}\times S_5$, $D_5\times C_4:S_5$, $D_4\times C_5:S_5$, $C_5\times D_4\times S_5$, $A_5\times C_{20}:C_4$, $D_4\times D_5\times A_5$, $C_4\times F_5\times A_5$, $A_4\times F_5^2$, $A_4:F_5^2$, $A_4:F_5^2$, $C_4\times A_5:F_5$, $(C_{20}\times A_5):C_4$, $(A_4\times C_5^2):\OD_{16}$, $(A_4\times C_5^2):\OD_{16}$, $D_{10}^2.C_{12}$, $C_5^2:Q_8:S_4$, $A_4\times D_5^2.C_2^2$, $C_5^2:Q_8\times S_4$, $C_5^2:C_8\times S_4$, $F_5^2:D_6$

Order 4608: $D_4\times S_4^2$ x 2, $S_4^2:C_2^3$ x 2, $C_2^6:\SOPlus(4,2)$ x 2, $C_2^6:\SOPlus(4,2)$ x 2, $S_4^2:D_4$ x 2, $C_2^3:S_4^2$, $(C_4\times A_4^2).D_4$, $C_4\times S_4\wr C_2$, $(C_4\times A_4^2):D_4$, $S_4^2:D_4$, $A_4^2:C_4\times D_4$, $C_2^5.D_6^2$

Order 4320: $S_3^2\times S_5$ x 8, $\GL(2,4):S_4$ x 3, $\GL(2,4):S_4$ x 3, $\GL(2,4):S_4$ x 3, $S_4\times \GL(2,4)$ x 3, $S_3\times A_4\times A_5$ x 3, $A_4\times C_3:S_5$ x 3, $C_3\times A_4\times S_5$ x 3, $S_3\times C_6:S_5$ x 2, $C_6\times S_3\times S_5$ x 2, $C_6\times A_4\times A_5$, $S_3\times D_6\times A_5$, $C_2\times A_5:S_3^2$, $C_6:S_3\times S_5$

Order 3840: $C_4:D_4\times S_5$ x 4, $A_5.D_4^2$ x 4, $A_5:D_4^2$ x 4, $A_5:D_4^2$ x 4, $C_2^2\times D_4\times S_5$ x 4, $S_5\times C_2^2\wr C_2$ x 4, $C_4\times D_4\times S_5$ x 2, $A_5:D_4^2$ x 2, $D_4\times F_5\times S_4$ x 2, $C_4:D_4\times S_5$, $A_5:D_4^2$, $A_5\times D_4^2$

Order 3600: $C_5\times A_4\times A_5$, $D_{15}:S_5$, $C_5\times S_3\times S_5$, $\GL(2,4):D_{10}$, $\GL(2,4):F_5$, $F_5\times \GL(2,4)$, $\GL(2,4):D_{10}$, $\GL(2,4):D_{10}$, $\GL(2,4):F_5$, $\GL(2,4):F_5$, $D_{15}\times S_5$, $\GL(2,4):D_{10}$, $A_5^2$

Order 3456: $S_3\times S_4^2$ x 9, $S_4\times A_4^2$ x 2, $A_4^2:S_4$ x 2, $C_2\times A_4^2.D_6$ x 2, $A_4\times D_6\times S_4$ x 2, $S_3^2:\GL(2,\mathbb{Z}/4)$ x 2, $S_3^2:C_4\times S_4$ x 2, $D_6^2:D_{12}$ x 2, $C_2\times D_6^2:D_6$ x 2, $S_4\times D_6^2$, $D_6\times \PSOPlus(4,3)$, $C_6\times S_4^2$, $C_6:S_4^2$, $A_4^2:S_4$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $(A_4\times C_6^2):D_4$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $C_2^4:S_3^3$, $C_6^2:C_4\times S_4$, $A_4\times D_6\wr C_2$, $A_4^2:S_4$, $A_4^3:C_2$, $A_4^2:C_4\times S_3$, $C_2^4:S_3^3$, $S_4^2:C_6$, $(C_3\times A_4^2):D_4$, $S_4^2:S_3$, $A_4^2:D_{12}$

Order 3200: $F_5^2:D_4$ x 2, $F_5^2:C_2^3$ x 2, $D_{10}^2.D_4$ x 2, $D_{10}^2.D_4$ x 2, $D_4\times F_5^2$, $C_4:F_5\wr C_2$, $C_4:F_5\wr C_2$, $C_4\times F_5\wr C_2$, $F_5^2:D_4$, $D_4\times C_5^2:\OD_{16}$, $D_{10}^2.C_2^3$

Order 3072: $C_4^2.(D_4\times S_4)$

Order 1024: $C_4^2:D_4^2$

Order 27: $C_3^3$

Order 25: $C_5^2$

Order 24: $S_4$

Order 12: $A_4$

Order 4: $C_2^2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 691200: $S_4\times S_5\wr C_2$ ($C_1$)

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

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

Order 115200: $C_2^2\times S_5\wr C_2$ ($S_3$)

Order 86400: $A_5^2:S_4$ ($D_4$), $A_4\times \PSOPlus(4,5)$ ($C_2^3$), $S_4\times A_5^2$ ($D_4$)

Order 57600: $C_2^2\times S_5^2$ ($D_6$), $A_5^2:C_2^4$ ($D_6$), $A_5^2:C_4\times C_2^2$ ($D_6$)

Order 43200: $A_4\times A_5^2$ ($C_2\times D_4$)

Order 28800: $S_5\wr C_2$ ($S_4$), $A_5^2:C_2^3$ ($C_2\times D_6$)

Order 14400: $C_2^2\times A_5^2$ ($S_3\times D_4$), $\POPlus(4,5)$ ($C_2\times S_4$), $S_5^2$ ($C_2\times S_4$), $A_5^2:C_4$ ($C_2\times S_4$)

Order 7200: $\PSOPlus(4,5)$ ($C_2^2\times S_4$)

Order 3600: $A_5^2$ ($D_4\times S_4$)

Order 24: $S_4$ ($A_5^2.D_4$)

Order 12: $A_4$ ($S_5^2:C_2^2$)

Order 4: $C_2^2$ ($S_5^2:D_6$)

Order 1: $C_1$ ($S_4\times S_5\wr C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 691200: $S_4\times S_5\wr C_2$ ($C_1$)

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

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

Order 115200: $C_2^2\times S_5\wr C_2$ ($S_3$)

Order 86400: $A_5^2:S_4$ ($D_4$), $A_4\times \PSOPlus(4,5)$ ($C_2^3$), $S_4\times A_5^2$ ($D_4$)

Order 57600: $C_2^2\times S_5^2$ ($D_6$), $A_5^2:C_2^4$ ($D_6$), $A_5^2:C_4\times C_2^2$ ($D_6$)

Order 43200: $A_4\times A_5^2$ ($C_2\times D_4$)

Order 28800: $S_5\wr C_2$ ($S_4$), $A_5^2:C_2^3$ ($C_2\times D_6$)

Order 14400: $C_2^2\times A_5^2$ ($S_3\times D_4$), $\POPlus(4,5)$ ($C_2\times S_4$), $S_5^2$ ($C_2\times S_4$), $A_5^2:C_4$ ($C_2\times S_4$)

Order 7200: $\PSOPlus(4,5)$ ($C_2^2\times S_4$)

Order 3600: $A_5^2$ ($D_4\times S_4$)

Order 24: $S_4$ ($A_5^2.D_4$)

Order 12: $A_4$ ($S_5^2:C_2^2$)

Order 4: $C_2^2$ ($S_5^2:D_6$)

Order 1: $C_1$ ($S_4\times S_5\wr C_2$)

Series

Derived series

$S_4\times S_5\wr C_2$

$\rhd$

$A_4\times \PSOPlus(4,5)$

$\rhd$

$C_2^2\times A_5^2$

$\rhd$

$A_5^2$

Chief series

$S_4\times S_5\wr C_2$

$\rhd$

$S_4\times S_5^2$

$\rhd$

$S_4\times \PSOPlus(4,5)$

$\rhd$

$S_4\times A_5^2$

$\rhd$

$S_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$S_4\times S_5\wr C_2$

$\rhd$

$A_4\times \PSOPlus(4,5)$

$\rhd$

$A_4\times A_5^2$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 5 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

See the $175 \times 175$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.