Abstract group 230400.r: $(C_2\times S_5^2).C_2^3$

• Label: 230400.r
• Order: \( 2^{10} \cdot 3^{2} \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{2} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $18$
• Trans deg.: $40$
• Rank: $4$

Group information

Description:	$(C_2\times S_5^2).C_2^3$
Order:	\(230400\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5^{2} \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$D_4^2.A_5^2.D_4$, of order \(1843200\)\(\medspace = 2^{13} \cdot 3^{2} \cdot 5^{2} \)
Composition factors:	$C_2$ x 6, $A_5$ x 2
Derived length:	$2$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	30	60
Elements	1	4607	440	45440	624	31240	28800	23568	55360	960	24960	10560	3840	230400
Conjugacy classes	1	26	2	33	2	36	4	20	24	1	8	7	2	166
Divisions	1	26	2	33	2	36	4	20	24	1	8	7	2	166
Autjugacy classes	1	21	2	18	2	26	1	15	12	1	4	5	1	109

Dimension	1	2	4	8	10	12	16	20	24	25	32	36	40	48	50	60	64	72	80	96	100	120
Irr. complex chars.	16	4	2	16	16	8	20	4	2	16	4	8	16	8	4	8	2	2	4	2	2	2	166
Irr. rational chars.	16	4	2	16	16	8	20	4	2	16	4	8	16	8	4	8	2	2	4	2	2	2	166

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$40$
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $18$ $\langle(1,2,6,9)(3,8,7,4)(11,12)(13,14)(15,18)(16,17), (1,4,6,8,5,3,9,10)(2,7) \!\cdots\! \rangle$
Transitive group:	40T106255	40T106295			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(S_5^2:D_4)$ . $C_2$	$C_2^3$ . $(S_5\wr C_2)$	$(S_5^2:C_4)$ . $C_2^2$	$(C_2\times S_5^2)$ . $C_2^3$	all 40

Elements of the group are displayed as permutations of degree 18.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 17378571 subgroups in 32856 conjugacy classes, 87 normal (31 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^2\times S_5\wr C_2$
Commutator:	$G' \simeq$ $A_5^2:C_2^2$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times S_5\wr C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3$	$G/\operatorname{Fit} \simeq$ $S_5\wr C_2$
Radical:	$R \simeq$ $C_2^3$	$G/R \simeq$ $S_5\wr C_2$
Socle:	$\operatorname{soc} \simeq$ $C_2\times A_5^2$	$G/\operatorname{soc} \simeq$ $C_2^2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.C_2^5.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
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 1024) are shown, as well as all normal subgroups of any index.

Order 230400: $(C_2\times S_5^2).C_2^3$

Order 115200: $(C_2^2\times A_5^2):D_4$ x 4, $C_2^2.A_5^2.D_4$ x 2, $C_2^2.A_5^2.D_4$ x 2, $C_2^2.A_5^2.D_4$ x 2, $S_5^2:D_4$ x 2, $(C_2^3\times A_5).S_5.C_2$, $C_2^3.A_5^2.C_4$, $C_2^3.A_5^2.C_2^2$

Order 57600: $A_5^2:(C_2\times D_4)$ x 6, $(C_2^2\times A_5).S_5.C_2$ x 6, $(C_2^2\times A_5^2):C_4$ x 4, $S_5^2:C_2^2$ x 4, $S_5^2:C_4$ x 4, $(C_2\times A_5^2).D_4$ x 4, $\POPlus(4,5):C_4$ x 4, $A_5^2:C_4\times C_2^2$ x 2, $(C_2^3\times A_5).S_5$, $(C_2^3\times A_5).A_5.C_2$, $(C_2^2\times A_5).S_5.C_2$, $(C_2^2\times A_5).S_5.C_2$, $C_2\times A_5^2:D_4$, $C_2^2\times S_5^2$, $A_5^2:C_2^4$, $C_2\times A_5^2:D_4$, $A_5^2:(C_2^2\times C_4)$

Order 28800: $A_5^2:C_2^3$ x 8, $C_2\times A_5^2:C_4$ x 6, $A_5^2:C_2^3$ x 5, $C_2\times S_5^2$ x 5, $A_5^2:D_4$ x 4, $A_5^2:D_4$ x 4, $S_5\wr C_2$ x 4, $A_5^2:D_4$ x 4, $A_5^2:D_4$ x 4, $A_5^2:(C_2\times C_4)$ x 3, $C_2.S_5^2$ x 3, $(C_2^2\times A_5).A_5.C_2$, $(C_2^3\times A_5).A_5$, $A_5^2:C_2^3$, $A_5^2:C_2^3$, $C_2\times A_5^2:C_4$

Order 23040: $(C_2\times \GL(2,\mathbb{Z}/4)).S_5$

Order 19200: $(C_2\times C_2^2:F_5).S_5$

Order 14400: $A_5^2:C_2^2$ x 13, $\POPlus(4,5)$ x 6, $S_5^2$ x 6, $C_2^2\times A_5^2$ x 4, $A_5^2:C_2^2$ x 4, $A_5^2:C_2^2$ x 4, $A_5^2:C_4$ x 4, $A_5^2:C_4$ x 2, $A_5^2:C_4$ x 2

Order 11520: $\GL(2,\mathbb{Z}/4):S_5$ x 8, $D_4\times D_6\times S_5$, $C_2^2.(S_4\times S_5)$, $C_2\times A_5:\GL(2,\mathbb{Z}/4)$, $(C_2^2\times S_4):S_5$, $(C_2^3\times A_4).S_5$, $C_2^3\times A_4:S_5$, $C_2^2\times S_4\times S_5$, $C_2\times A_5\times \GL(2,\mathbb{Z}/4)$

Order 9600: $D_{10}:(C_4\times S_5)$ x 8, $(C_2^3\times D_5).S_5$, $D_{10}.C_2^2:S_5$, $D_{10}.C_2^2:S_5$, $(C_2^2\times F_5).S_5$, $(C_2^3\times D_5).S_5$, $D_{10}.(C_2^3\times A_5)$, $C_2^2\times F_5\times S_5$

Order 9216: $C_2^7:\SOPlus(4,2)$

Order 7680: $A_5:(C_2\times D_4^2)$

Order 7200: $\PSOPlus(4,5)$ x 5, $C_2\times A_5^2$ x 4, $A_5\times S_5$ x 2, $\SOPlus(4,4)$ x 2

Order 6400: $(C_2^2\times F_5^2):C_2^2$

Order 5760: $S_3\times D_4\times S_5$ x 16, $C_2^2\times A_4:S_5$ x 10, $C_2\times S_4\times S_5$ x 8, $A_4\times C_2^2:S_5$ x 4, $A_4:(C_4\times S_5)$ x 4, $(C_2\times S_4):S_5$ x 4, $A_5:\GL(2,\mathbb{Z}/4)$ x 4, $A_5\times \GL(2,\mathbb{Z}/4)$ x 4, $S_3\times C_2^3:S_5$ x 2, $C_6:(D_4\times S_5)$ x 2, $C_6:D_4\times S_5$ x 2, $\GL(2,4):C_2^5$ x 2, $C_2^2.S_4\times A_5$, $A_4\times C_2^2.S_5$, $C_2^5:\GL(2,4)$, $C_4:S_5\times D_6$, $C_6\times D_4\times S_5$, $C_2\times D_{12}:S_5$, $C_6:S_5\times D_4$, $C_4\times D_6\times S_5$, $C_2\times D_{12}\times S_5$, $C_2^2\times A_4\times S_5$, $C_2^2\times S_4\times A_5$, $D_4\times D_6\times A_5$

Order 4800: $C_2\times D_{10}.S_5$ x 10, $D_5\times C_2^2:S_5$ x 8, $C_2\times F_5\times S_5$ x 8, $A_5\times D_{10}:C_4$ x 4, $C_{10}.(C_4\times S_5)$ x 4, $(C_{10}\times S_5):C_4$ x 4, $A_5:C_4\times F_5$ x 4, $C_2^2\times F_5\times A_5$ x 2, $C_{10}.C_2^2:S_5$, $C_{10}.C_2^2:S_5$, $C_{10}.C_2^2:S_5$, $(C_2^2\times D_5).S_5$, $(C_2^3\times D_5).A_5$, $C_{10}.C_2^2:S_5$, $C_2\times D_{10}\times S_5$

Order 4608: $C_2^6:\SOPlus(4,2)$ x 4, $C_2^6.\SOPlus(4,2)$ 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, $C_2^3:S_4^2$, $C_2^3:(A_4^2:C_4)$, $C_2^7:S_3^2$

Order 3840: $A_5.D_4^2$ x 16, $(C_2^2\times S_5):D_4$ x 2, $C_2^3:(C_4\times S_5)$ x 2, $C_2^2.(D_4\times S_5)$ x 2, $(C_2^2\times D_4):S_5$ x 2, $C_2^5:S_5$ x 2, $C_2^2\times D_4\times S_5$ x 2, $(C_2^2\times D_4):S_5$, $C_2^3.(D_4\times A_5)$, $C_2\times \GL(2,\mathbb{Z}/4):F_5$, $(C_2^2\times D_4).S_5$

Order 3600: $A_5^2$

Order 3200: $F_5^2:D_4$ x 4, $D_{10}^2.D_4$ x 4, $D_{10}^2.D_4$ x 2, $D_{10}^2.D_4$ x 2, $(C_2\times D_5^2).C_2^4$, $C_2\times C_{10}^2:\OD_{16}$, $D_{10}^2.C_2^3$

Order 2880: $\GL(2,4):C_2^4$ x 39, $C_2\times A_4:S_5$ x 18, $C_3:(D_4\times S_5)$ x 16, $S_3\times C_2^2:S_5$ x 16, $C_3:D_4\times S_5$ x 16, $D_{12}:S_5$ x 8, $S_3\times C_4:S_5$ x 8, $D_{12}\times S_5$ x 8, $S_3\times D_4\times A_5$ x 8, $C_4\times S_3\times S_5$ x 8, $D_4\times C_3:S_5$ x 8, $C_3\times D_4\times S_5$ x 8, $S_4\times S_5$ x 8, $C_2^4:\GL(2,4)$ x 5, $C_2\times S_4\times A_5$ x 4, $C_2\times A_4\times S_5$ x 4, $\GL(2,4):C_2^4$ x 2, $C_6:D_4\times A_5$ x 2, $A_4:C_4\times A_5$ x 2, $\GL(2,4):C_2^4$ x 2, $C_2\times D_6:S_5$ x 2, $C_2\times \GL(2,4):D_4$ x 2, $C_2\times D_6:S_5$ x 2, $C_2\times \GL(2,4):D_4$ x 2, $C_2^2:S_5\times C_6$ x 2, $\GL(2,4):C_2^4$ x 2, $A_4\times A_5:C_4$ x 2, $C_2\times D_{12}\times A_5$, $C_4\times D_6\times A_5$, $A_5:C_4\times D_6$, $C_6\times D_4\times A_5$, $C_2\times C_{12}:S_5$, $C_6:(C_4\times S_5)$, $C_6:C_4\times S_5$, $C_2\times C_{12}:S_5$, $C_2\times C_{12}\times S_5$, $C_2\times C_{12}:S_5$

Order 2400: $D_{10}.S_5$ x 18, $C_2\times D_{10}\times A_5$ x 10, $D_{10}\times S_5$ x 8, $C_2\times F_5\times A_5$ x 8, $F_5\times S_5$ x 8, $D_{10}.S_5$ x 4, $D_{10}:S_5$ x 4, $D_{10}:S_5$ x 4, $C_5\times C_2^2:S_5$ x 4, $(C_2\times C_{10}):S_5$ x 4, $C_5.(C_2^3\times A_5)$, $C_{10}.A_5:C_4$, $C_{10}.A_5:C_4$, $C_2\times C_{10}:S_5$, $C_2\times C_{10}\times S_5$

Order 2304: $\GL(2,\mathbb{Z}/4):S_4$ x 6, $C_2\wr S_3^2$ x 6, $C_2^5.\SOPlus(4,2)$ x 4, $\POPlus(4,3):C_4$ x 4, $S_4^2:C_4$ x 4, $(C_2^2\times A_4^2):C_4$ x 4, $S_4^2:C_2^2$ x 4, $A_4^2:C_4\times C_2^2$ x 2, $D_6^2.C_2^4$, $C_2^2.S_4^2$, $A_4^2:C_2^4$, $C_2^4:D_6^2$, $C_2^2.S_4^2$, $C_2\times A_4\times \GL(2,\mathbb{Z}/4)$, $C_2\times A_4^2:D_4$, $C_2^6.S_3^2$, $C_2^4:D_6^2$, $C_2^6.S_3^2$, $C_2^2\times S_4^2$

Order 1920: $C_2^4:S_5$ x 34, $C_2\times D_4\times S_5$ x 33, $(C_2\times S_5):D_4$ x 16, $(C_2\times D_4):S_5$ x 16, $C_2.(D_4\times S_5)$ x 16, $C_2^2:(C_4\times S_5)$ x 16, $(C_2\times D_4):S_5$ x 8, $A_5\times C_2^2\wr C_2$ x 8, $(C_2\times D_4).S_5$ x 8, $\GL(2,\mathbb{Z}/4):F_5$ x 8, $(C_2^2\times C_4):S_5$ x 4, $C_2^4\times S_5$ x 3, $C_2^2\times D_4\times A_5$ x 3, $C_2^2\times C_4:S_5$ x 3, $C_2^3:C_4\times A_5$ x 3, $C_4\times C_2^2\times S_5$ x 2, $C_2^4.S_5$ x 2, $C_2^2.(C_4\times S_5)$ x 2, $C_2^5\times A_5$, $C_2^4.S_5$, $C_2\times A_5:C_4^2$, $C_2^3\times A_4:F_5$, $C_2^2\times F_5\times S_4$, $D_{10}\times \GL(2,\mathbb{Z}/4)$, $A_4\times C_2^3:F_5$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $(C_{10}\times A_4):C_4^2$, $D_4\times D_6\times F_5$

Order 1600: $D_{10}^2.C_2^2$ x 12, $F_5^2:C_4$ x 8, $F_5^2:C_2^2$ x 8, $C_{10}^2:\OD_{16}$ x 6, $C_{10}^2:C_4^2$ x 6, $C_{10}^2:\OD_{16}$ x 2, $C_2^2\times F_5^2$ x 2, $D_{10}^2.C_2^2$, $C_2\times C_{10}^2:C_8$, $D_{10}^2.C_2^2$, $D_{10}^2.C_2^2$, $D_{10}^2.C_2^2$, $D_{10}^2.C_2^2$, $D_{10}^2.C_2^2$, $D_{10}^2.C_4$, $C_{10}^2.C_4^2$, $D_{10}^2:C_4$, $D_{10}^2.C_2^2$, $C_2\times C_{10}^2:Q_8$, $D_{10}^2.C_2^2$, $D_{10}^2:C_2^2$, $D_{10}^2:C_4$

Order 1536: $C_2^7:D_6$, $C_2^7:D_6$

Order 1440: $D_6\times S_5$ x 124, $\GL(2,4):C_2^3$ x 19, $\GL(2,4):C_2^3$ x 19, $\GL(2,4):C_2^3$ x 19, $\GL(2,4):D_4$ x 8, $\GL(2,4):D_4$ x 8, $\GL(2,4):D_4$ x 8, $D_6:S_5$ x 8, $\GL(2,4):D_4$ x 8, $D_6:S_5$ x 8, $A_4:S_5$ x 6, $C_2^3:\GL(2,4)$ x 5, $D_4\times \GL(2,4)$ x 4, $D_{12}\times A_5$ x 4, $C_4\times S_3\times A_5$ x 4, $C_{12}:S_5$ x 4, $C_{12}:S_5$ x 4, $C_{12}:S_5$ x 4, $C_{12}\times S_5$ x 4, $C_3:C_4\times S_5$ x 4, $D_6.S_5$ x 4, $C_3:(C_4\times S_5)$ x 4, $C_2^3\times \GL(2,4)$ x 2, $S_4\times A_5$ x 2, $A_4\times S_5$ x 2, $C_2\times C_{12}\times A_5$, $C_6:C_4\times A_5$, $C_2\times \GL(2,4):C_4$, $A_5:C_4\times C_6$

Order 1280: $D_{10}.D_4^2$

Order 1200: $D_{10}\times A_5$ x 18, $D_5\times S_5$ x 8, $A_5:F_5$ x 6, $C_2\times C_{10}\times A_5$ x 5, $C_{10}\times S_5$ x 4, $C_{10}:S_5$ x 4, $F_5\times A_5$ x 4, $C_{10}.S_5$ x 2, $A_5:C_{20}$ x 2

Order 1152: $\GL(2,\mathbb{Z}/4):D_6$ x 16, $D_6^2:D_4$ x 8, $A_4^2:C_2^3$ x 8, $C_2\times A_4^2:C_4$ x 6, $A_4^2:C_2^3$ x 5, $C_2\times S_4^2$ x 5, $C_6^2.(C_2^2\times D_4)$ x 4, $D_6^2:D_4$ x 4, $S_4\wr C_2$ x 4, $A_4^2:D_4$ x 4, $A_4\times \GL(2,\mathbb{Z}/4)$ x 4, $A_4^2:D_4$ x 4, $A_4:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^5.S_3^2$ x 3, $C_2.S_4^2$ x 3, $D_6^2.C_2^3$ x 2, $D_6^2:D_4$ x 2, $D_6^2.C_2^3$ x 2, $C_6^2.(C_2^2\times D_4)$ x 2, $C_2^3:D_6^2$ x 2, $\GL(2,\mathbb{Z}/4):D_6$ x 2, $C_2^5:S_3^2$ x 2, $D_6\times \GL(2,\mathbb{Z}/4)$ x 2, $C_3^2.(C_2^3.D_4).C_2$, $C_6^2.(C_2^3\times C_4)$, $C_2\times C_3^2.(C_2^5.C_2)$, $C_2^3\times A_4^2$, $A_4^2:C_2^3$, $A_4^2:C_2^3$, $D_4\times A_4\times D_6$, $C_6:S_4\times D_4$, $C_6\times D_4\times S_4$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2\times A_4^2:C_4$, $C_2\times A_4^2:C_4$, $D_4\times D_6^2$, $C_2\times D_6:D_6.C_2^2$, $C_6^2.C_2^5$, $D_6^2:D_4$

Order 1024: $C_2^4.C_2^5.C_2$

Order 25: $C_5^2$

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 4: $C_2^2$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 225 (order at least 1024) are shown, as well as all normal subgroups of any index.

Order 230400: $(C_2\times S_5^2).C_2^3$

Order 115200: $(C_2^3\times A_5).S_5.C_2$, $C_2^2.A_5^2.D_4$, $C_2^2.A_5^2.D_4$, $C_2^2.A_5^2.D_4$, $C_2^3.A_5^2.C_4$, $S_5^2:D_4$, $C_2^3.A_5^2.C_2^2$, $(C_2^2\times A_5^2):D_4$

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

Order 28800: $A_5^2:C_2^3$ x 7, $A_5^2:C_2^3$ x 3, $C_2\times S_5^2$ x 3, $A_5^2:D_4$ x 2, $A_5^2:D_4$ x 2, $A_5^2:(C_2\times C_4)$ x 2, $C_2\times A_5^2:C_4$ x 2, $C_2.S_5^2$ x 2, $A_5^2:D_4$ x 2, $A_5^2:D_4$ x 2, $(C_2^2\times A_5).A_5.C_2$, $(C_2^3\times A_5).A_5$, $S_5\wr C_2$, $A_5^2:C_2^3$, $A_5^2:C_2^3$, $C_2\times A_5^2:C_4$

Order 23040: $(C_2\times \GL(2,\mathbb{Z}/4)).S_5$

Order 19200: $(C_2\times C_2^2:F_5).S_5$

Order 14400: $A_5^2:C_2^2$ x 10, $C_2^2\times A_5^2$ x 4, $A_5^2:C_2^2$ x 2, $\POPlus(4,5)$ x 2, $S_5^2$ x 2, $A_5^2:C_2^2$ x 2, $A_5^2:C_4$, $A_5^2:C_4$, $A_5^2:C_4$

Order 11520: $\GL(2,\mathbb{Z}/4):S_5$ x 4, $D_4\times D_6\times S_5$, $C_2^2.(S_4\times S_5)$, $C_2\times A_5:\GL(2,\mathbb{Z}/4)$, $(C_2^2\times S_4):S_5$, $(C_2^3\times A_4).S_5$, $C_2^3\times A_4:S_5$, $C_2^2\times S_4\times S_5$, $C_2\times A_5\times \GL(2,\mathbb{Z}/4)$

Order 9600: $D_{10}:(C_4\times S_5)$ x 4, $(C_2^3\times D_5).S_5$, $D_{10}.C_2^2:S_5$, $D_{10}.C_2^2:S_5$, $(C_2^2\times F_5).S_5$, $(C_2^3\times D_5).S_5$, $D_{10}.(C_2^3\times A_5)$, $C_2^2\times F_5\times S_5$

Order 9216: $C_2^7:\SOPlus(4,2)$

Order 7680: $A_5:(C_2\times D_4^2)$

Order 7200: $\PSOPlus(4,5)$ x 4, $C_2\times A_5^2$ x 4, $A_5\times S_5$, $\SOPlus(4,4)$

Order 6400: $(C_2^2\times F_5^2):C_2^2$

Order 5760: $S_3\times D_4\times S_5$ x 8, $C_2^2\times A_4:S_5$ x 7, $C_2\times S_4\times S_5$ x 3, $S_3\times C_2^3:S_5$ x 2, $C_6:(D_4\times S_5)$ x 2, $C_6:D_4\times S_5$ x 2, $\GL(2,4):C_2^5$ x 2, $A_4\times C_2^2:S_5$ x 2, $A_4:(C_4\times S_5)$ x 2, $(C_2\times S_4):S_5$ x 2, $A_5:\GL(2,\mathbb{Z}/4)$ x 2, $A_5\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^2.S_4\times A_5$, $A_4\times C_2^2.S_5$, $C_2^5:\GL(2,4)$, $C_4:S_5\times D_6$, $C_6\times D_4\times S_5$, $C_2\times D_{12}:S_5$, $C_6:S_5\times D_4$, $C_4\times D_6\times S_5$, $C_2\times D_{12}\times S_5$, $C_2^2\times A_4\times S_5$, $C_2^2\times S_4\times A_5$, $D_4\times D_6\times A_5$

Order 4800: $C_2\times D_{10}.S_5$ x 7, $D_5\times C_2^2:S_5$ x 4, $C_2\times F_5\times S_5$ x 3, $A_5\times D_{10}:C_4$ x 2, $C_{10}.(C_4\times S_5)$ x 2, $(C_{10}\times S_5):C_4$ x 2, $A_5:C_4\times F_5$ x 2, $C_2^2\times F_5\times A_5$ x 2, $C_{10}.C_2^2:S_5$, $C_{10}.C_2^2:S_5$, $C_{10}.C_2^2:S_5$, $(C_2^2\times D_5).S_5$, $(C_2^3\times D_5).A_5$, $C_{10}.C_2^2:S_5$, $C_2\times D_{10}\times S_5$

Order 4608: $C_2^3:S_4^2$, $C_2^3:(A_4^2:C_4)$, $C_2^7:S_3^2$, $C_2^6.\SOPlus(4,2)$, $C_2^6:\SOPlus(4,2)$, $C_2^6:\SOPlus(4,2)$, $C_2^6.\SOPlus(4,2)$, $S_4^2:D_4$

Order 3840: $A_5.D_4^2$ x 8, $(C_2^2\times S_5):D_4$ x 2, $C_2^3:(C_4\times S_5)$ x 2, $C_2^2.(D_4\times S_5)$ x 2, $(C_2^2\times D_4):S_5$ x 2, $C_2^5:S_5$ x 2, $C_2^2\times D_4\times S_5$ x 2, $(C_2^2\times D_4):S_5$, $C_2^3.(D_4\times A_5)$, $C_2\times \GL(2,\mathbb{Z}/4):F_5$, $(C_2^2\times D_4).S_5$

Order 3600: $A_5^2$

Order 3200: $F_5^2:D_4$ x 2, $(C_2\times D_5^2).C_2^4$, $C_2\times C_{10}^2:\OD_{16}$, $D_{10}^2.D_4$, $D_{10}^2.D_4$, $D_{10}^2.D_4$, $D_{10}^2.C_2^3$

Order 2880: $\GL(2,4):C_2^4$ x 23, $C_2\times A_4:S_5$ x 10, $C_3:(D_4\times S_5)$ x 8, $S_3\times C_2^2:S_5$ x 8, $C_3:D_4\times S_5$ x 8, $D_{12}:S_5$ x 4, $S_3\times C_4:S_5$ x 4, $D_{12}\times S_5$ x 4, $C_2^4:\GL(2,4)$ x 4, $S_3\times D_4\times A_5$ x 4, $C_4\times S_3\times S_5$ x 4, $D_4\times C_3:S_5$ x 4, $C_3\times D_4\times S_5$ x 4, $\GL(2,4):C_2^4$ x 2, $C_6:D_4\times A_5$ x 2, $\GL(2,4):C_2^4$ x 2, $C_2\times D_6:S_5$ x 2, $C_2\times \GL(2,4):D_4$ x 2, $C_2\times D_6:S_5$ x 2, $C_2\times \GL(2,4):D_4$ x 2, $C_2^2:S_5\times C_6$ x 2, $\GL(2,4):C_2^4$ x 2, $C_2\times S_4\times A_5$ x 2, $C_2\times A_4\times S_5$ x 2, $S_4\times S_5$ x 2, $C_2\times D_{12}\times A_5$, $C_4\times D_6\times A_5$, $A_5:C_4\times D_6$, $A_4:C_4\times A_5$, $C_6\times D_4\times A_5$, $C_2\times C_{12}:S_5$, $C_6:(C_4\times S_5)$, $C_6:C_4\times S_5$, $C_2\times C_{12}:S_5$, $C_2\times C_{12}\times S_5$, $C_2\times C_{12}:S_5$, $A_4\times A_5:C_4$

Order 2400: $D_{10}.S_5$ x 10, $C_2\times D_{10}\times A_5$ x 8, $D_{10}\times S_5$ x 4, $C_2\times F_5\times A_5$ x 4, $D_{10}.S_5$ x 2, $D_{10}:S_5$ x 2, $D_{10}:S_5$ x 2, $C_5\times C_2^2:S_5$ x 2, $(C_2\times C_{10}):S_5$ x 2, $F_5\times S_5$ x 2, $C_5.(C_2^3\times A_5)$, $C_{10}.A_5:C_4$, $C_{10}.A_5:C_4$, $C_2\times C_{10}:S_5$, $C_2\times C_{10}\times S_5$

Order 2304: $\GL(2,\mathbb{Z}/4):S_4$ x 4, $C_2\wr S_3^2$ x 4, $(C_2^2\times A_4^2):C_4$ x 2, $D_6^2.C_2^4$, $C_2^2.S_4^2$, $A_4^2:C_2^4$, $C_2^4:D_6^2$, $C_2^2.S_4^2$, $C_2\times A_4\times \GL(2,\mathbb{Z}/4)$, $C_2\times A_4^2:D_4$, $A_4^2:C_4\times C_2^2$, $C_2^6.S_3^2$, $C_2^4:D_6^2$, $C_2^6.S_3^2$, $C_2^5.\SOPlus(4,2)$, $\POPlus(4,3):C_4$, $C_2^2\times S_4^2$, $S_4^2:C_4$, $S_4^2:C_2^2$

Order 1920: $C_2^4:S_5$ x 24, $C_2\times D_4\times S_5$ x 17, $(C_2\times S_5):D_4$ x 8, $(C_2\times D_4):S_5$ x 8, $C_2.(D_4\times S_5)$ x 8, $C_2^2:(C_4\times S_5)$ x 8, $(C_2\times D_4):S_5$ x 4, $A_5\times C_2^2\wr C_2$ x 4, $(C_2\times D_4).S_5$ x 4, $(C_2^2\times C_4):S_5$ x 4, $\GL(2,\mathbb{Z}/4):F_5$ x 4, $C_2^4\times S_5$ x 3, $C_2^2\times D_4\times A_5$ x 3, $C_2^2\times C_4:S_5$ x 3, $C_2^3:C_4\times A_5$ x 3, $C_4\times C_2^2\times S_5$ x 2, $C_2^4.S_5$ x 2, $C_2^2.(C_4\times S_5)$ x 2, $C_2^5\times A_5$, $C_2^4.S_5$, $C_2\times A_5:C_4^2$, $C_2^3\times A_4:F_5$, $C_2^2\times F_5\times S_4$, $D_{10}\times \GL(2,\mathbb{Z}/4)$, $A_4\times C_2^3:F_5$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $(C_{10}\times A_4):C_4^2$, $D_4\times D_6\times F_5$

Order 1600: $D_{10}^2.C_2^2$ x 8, $C_{10}^2:\OD_{16}$ x 4, $C_{10}^2:C_4^2$ x 4, $F_5^2:C_4$ x 2, $C_2^2\times F_5^2$ x 2, $F_5^2:C_2^2$ x 2, $D_{10}^2.C_2^2$, $C_2\times C_{10}^2:C_8$, $D_{10}^2.C_2^2$, $D_{10}^2.C_2^2$, $D_{10}^2.C_2^2$, $D_{10}^2.C_2^2$, $D_{10}^2.C_2^2$, $D_{10}^2.C_4$, $C_{10}^2.C_4^2$, $D_{10}^2:C_4$, $D_{10}^2.C_2^2$, $C_{10}^2:\OD_{16}$, $C_2\times C_{10}^2:Q_8$, $D_{10}^2.C_2^2$, $D_{10}^2:C_2^2$, $D_{10}^2:C_4$

Order 1536: $C_2^7:D_6$, $C_2^7:D_6$

Order 1440: $D_6\times S_5$ x 50, $\GL(2,4):C_2^3$ x 13, $\GL(2,4):C_2^3$ x 11, $\GL(2,4):C_2^3$ x 11, $C_2^3:\GL(2,4)$ x 4, $A_4:S_5$ x 4, $\GL(2,4):D_4$ x 4, $\GL(2,4):D_4$ x 4, $\GL(2,4):D_4$ x 4, $D_6:S_5$ x 4, $\GL(2,4):D_4$ x 4, $D_6:S_5$ x 4, $C_2^3\times \GL(2,4)$ x 2, $D_4\times \GL(2,4)$ x 2, $D_{12}\times A_5$ x 2, $C_4\times S_3\times A_5$ x 2, $C_{12}:S_5$ x 2, $C_{12}:S_5$ x 2, $C_{12}:S_5$ x 2, $C_{12}\times S_5$ x 2, $C_3:C_4\times S_5$ x 2, $D_6.S_5$ x 2, $C_3:(C_4\times S_5)$ x 2, $S_4\times A_5$, $A_4\times S_5$, $C_2\times C_{12}\times A_5$, $C_6:C_4\times A_5$, $C_2\times \GL(2,4):C_4$, $A_5:C_4\times C_6$

Order 1280: $D_{10}.D_4^2$

Order 1200: $D_{10}\times A_5$ x 13, $C_2\times C_{10}\times A_5$ x 4, $A_5:F_5$ x 4, $D_5\times S_5$ x 3, $C_{10}\times S_5$ x 2, $C_{10}:S_5$ x 2, $F_5\times A_5$ x 2, $C_{10}.S_5$, $A_5:C_{20}$

Order 1152: $\GL(2,\mathbb{Z}/4):D_6$ x 8, $A_4^2:C_2^3$ x 7, $A_4^2:C_2^3$ x 3, $C_2\times S_4^2$ x 3, $D_6^2:D_4$ x 2, $C_6^2.(C_2^2\times D_4)$ x 2, $D_6^2:D_4$ x 2, $C_2^3:D_6^2$ x 2, $C_2\times A_4^2:C_4$ x 2, $\GL(2,\mathbb{Z}/4):D_6$ x 2, $C_2^5:S_3^2$ x 2, $D_6\times \GL(2,\mathbb{Z}/4)$ x 2, $A_4^2:D_4$ x 2, $A_4\times \GL(2,\mathbb{Z}/4)$ x 2, $A_4^2:D_4$ x 2, $C_2^5.S_3^2$ x 2, $A_4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2.S_4^2$ x 2, $D_6^2.C_2^3$, $D_6^2:D_4$, $D_6^2.C_2^3$, $C_6^2.(C_2^2\times D_4)$, $C_3^2.(C_2^3.D_4).C_2$, $C_6^2.(C_2^3\times C_4)$, $C_2\times C_3^2.(C_2^5.C_2)$, $C_2^3\times A_4^2$, $A_4^2:C_2^3$, $A_4^2:C_2^3$, $S_4\wr C_2$, $D_4\times A_4\times D_6$, $C_6:S_4\times D_4$, $C_6\times D_4\times S_4$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2\times A_4^2:C_4$, $C_2\times A_4^2:C_4$, $D_4\times D_6^2$, $C_2\times D_6:D_6.C_2^2$, $C_6^2.C_2^5$, $D_6^2:D_4$

Order 1024: $C_2^4.C_2^5.C_2$

Order 25: $C_5^2$

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 4: $C_2^2$ x 3

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 230400: $(C_2\times S_5^2).C_2^3$ ($C_1$)

Order 115200: $(C_2^2\times A_5^2):D_4$ ($C_2$) x 4, $C_2^2.A_5^2.D_4$ ($C_2$) x 2, $C_2^2.A_5^2.D_4$ ($C_2$) x 2, $C_2^2.A_5^2.D_4$ ($C_2$) x 2, $S_5^2:D_4$ ($C_2$) x 2, $(C_2^3\times A_5).S_5.C_2$ ($C_2$), $C_2^3.A_5^2.C_4$ ($C_2$), $C_2^3.A_5^2.C_2^2$ ($C_2$)

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

Order 28800: $C_2\times A_5^2:C_4$ ($C_2^3$) x 4, $A_5^2:C_2^3$ ($C_2^3$) x 3, $A_5^2:C_2^3$ ($D_4$) x 3, $A_5^2:C_2^3$ ($C_2^3$) x 2, $A_5^2:(C_2\times C_4)$ ($C_2^3$) x 2, $C_2.S_5^2$ ($C_2^3$) x 2, $C_2\times S_5^2$ ($C_2^3$) x 2, $(C_2^3\times A_5).A_5$ ($D_4$)

Order 14400: $C_2^2\times A_5^2$ ($C_2\times D_4$) x 3, $A_5^2:C_2^2$ ($C_2\times D_4$) x 3, $A_5^2:C_2^2$ ($C_2^4$)

Order 7200: $\PSOPlus(4,5)$ ($D_4:C_2^2$) x 2, $C_2\times A_5^2$ ($C_2^2\times D_4$)

Order 3600: $A_5^2$ ($C_2^3:D_4$)

Order 8: $C_2^3$ ($S_5\wr C_2$)

Order 4: $C_2^2$ ($C_2.A_5^2.D_4$) x 2, $C_2^2$ ($S_5^2:C_2^2$)

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

Order 1: $C_1$ ($(C_2\times S_5^2).C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 230400: $(C_2\times S_5^2).C_2^3$ ($C_1$)

Order 115200: $(C_2^3\times A_5).S_5.C_2$ ($C_2$), $C_2^2.A_5^2.D_4$ ($C_2$), $C_2^2.A_5^2.D_4$ ($C_2$), $C_2^2.A_5^2.D_4$ ($C_2$), $C_2^3.A_5^2.C_4$ ($C_2$), $S_5^2:D_4$ ($C_2$), $C_2^3.A_5^2.C_2^2$ ($C_2$), $(C_2^2\times A_5^2):D_4$ ($C_2$)

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

Order 28800: $A_5^2:C_2^3$ ($C_2^3$) x 3, $A_5^2:C_2^3$ ($D_4$) x 3, $(C_2^3\times A_5).A_5$ ($D_4$), $A_5^2:C_2^3$ ($C_2^3$), $A_5^2:(C_2\times C_4)$ ($C_2^3$), $C_2\times A_5^2:C_4$ ($C_2^3$), $C_2.S_5^2$ ($C_2^3$), $C_2\times S_5^2$ ($C_2^3$)

Order 14400: $C_2^2\times A_5^2$ ($C_2\times D_4$) x 3, $A_5^2:C_2^2$ ($C_2\times D_4$) x 3, $A_5^2:C_2^2$ ($C_2^4$)

Order 7200: $\PSOPlus(4,5)$ ($D_4:C_2^2$), $C_2\times A_5^2$ ($C_2^2\times D_4$)

Order 3600: $A_5^2$ ($C_2^3:D_4$)

Order 8: $C_2^3$ ($S_5\wr C_2$)

Order 4: $C_2^2$ ($C_2.A_5^2.D_4$) x 2, $C_2^2$ ($S_5^2:C_2^2$)

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

Order 1: $C_1$ ($(C_2\times S_5^2).C_2^3$)

Series

Derived series

$(C_2\times S_5^2).C_2^3$

$\rhd$

$A_5^2:C_2^2$

$\rhd$

$A_5^2$

Chief series

$(C_2\times S_5^2).C_2^3$

$\rhd$

$(C_2^3\times A_5).S_5.C_2$

$\rhd$

$(C_2^3\times A_5).S_5$

$\rhd$

$(C_2^3\times A_5).A_5$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_2\times S_5^2).C_2^3$

$\rhd$

$A_5^2:C_2^2$

$\rhd$

$A_5^2$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 1 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 $166 \times 166$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.