Abstract group 3840.eg: $C_2^3:C_4\times S_5$

• Label: 3840.eg
• Order: \( 2^{8} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $13$
• Trans deg.: $40$
• Rank: $3$

Group information

Description:	$C_2^3:C_4\times S_5$
Order:	\(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^4.(D_4\times S_5)$, of order \(15360\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 6, $A_5$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	20
Elements	1	311	20	1480	24	460	264	800	480	3840
Conjugacy classes	1	17	1	26	1	11	5	10	5	77
Divisions	1	17	1	18	1	11	5	6	3	63
Autjugacy classes	1	13	1	12	1	8	4	4	2	46

Dimension	1	2	4	5	6	8	10	12	16	20	24
Irr. complex chars.	16	4	18	16	8	4	4	2	2	2	1	77
Irr. rational chars.	8	8	10	8	4	8	8	4	2	2	1	63

Minimal presentations

Permutation degree:	$13$
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	16	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $13$ $\langle(1,2,4,5,3)(7,9)(8,11), (1,2,3,4,5)(6,7,8,10)(9,11,12,13), (1,3)(6,7,8,10)(9,11,12,13)\rangle$
Transitive group:	40T2805	40T2825	40T2828		more information
Direct product:	$(C_2^3:C_4)$ $\, \times\, $ $S_5$
Semidirect product:	$(C_2^3\times S_5)$ $\,\rtimes\,$ $C_4$	$(C_2^3:S_5)$ $\,\rtimes\,$ $C_4$	$C_2^3$ $\,\rtimes\,$ $(C_4\times S_5)$	$A_5$ $\,\rtimes\,$ $(C_2^4:C_4)$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^2\times S_5)$ . $D_4$ (2)	$C_2^2$ . $(D_4\times S_5)$	$(C_2\times D_4\times S_5)$ . $C_2$	$(C_2^3\times S_5)$ . $C_2^2$	all 16
Aut. group:	$\Aut(C_6^3.C_2^3)$	$\Aut(C_{119}\times \SL(2,5))$

Elements of the group are displayed as permutations of degree 13.

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 48843 subgroups in 1855 conjugacy classes, 53 normal (31 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^2:C_4\times S_5$
Commutator:	$G' \simeq$ $C_2^2\times A_5$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^2\times S_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3:C_4$	$G/\operatorname{Fit} \simeq$ $S_5$
Radical:	$R \simeq$ $C_2^3:C_4$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_2\times A_5$	$G/\operatorname{soc} \simeq$ $C_2^3:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2.D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 3840: $C_2^3:C_4\times S_5$

Order 1920: $C_2^2:C_4\times S_5$ x 2, $(C_2^2\times S_5):C_4$ x 2, $C_2\times D_4\times S_5$, $C_2^3:C_4\times A_5$, $(C_2^3\times A_5):C_4$

Order 960: $D_4\times S_5$ x 4, $C_2\times C_4\times S_5$ x 3, $C_2^2:C_4\times A_5$ x 2, $C_2^3.S_5$ x 2, $(C_2\times C_4):S_5$ x 2, $C_2^3\times S_5$ x 2, $C_2^3:S_5$ x 2, $C_2\times D_4\times A_5$, $C_2\times C_4:S_5$

Order 768: $C_2^3:C_4\times S_4$

Order 640: $C_2^3:C_4\times F_5$

Order 480: $C_2^2\times S_5$ x 13, $C_4\times S_5$ x 6, $C_2^2:S_5$ x 4, $C_2\times C_4\times A_5$ x 3, $C_2^2.S_5$ x 3, $D_4\times A_5$ x 2, $C_4:S_5$ x 2, $C_2^3\times A_5$ x 2

Order 384: $C_2^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_6$ x 2, $C_2^5:C_{12}$, $C_2^2.\GL(2,\mathbb{Z}/4)$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^3:C_4\times D_6$

Order 320: $(C_2^2\times F_5):C_4$ x 2, $D_{10}:C_4^2$ x 2, $C_2^3:C_4\times D_5$, $D_{10}.C_2^4$, $(C_4\times D_{10}):C_4$

Order 256: $C_2^2.D_4^2$

Order 240: $C_2\times S_5$ x 19, $C_2^2\times A_5$ x 6, $C_4\times A_5$ x 3, $A_5:C_4$ x 3

Order 192: $C_2^3:C_4\times S_3$ x 8, $D_4\times S_4$ x 4, $C_2^4.D_6$ x 3, $C_2^4:C_{12}$ x 2, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.D_{12}$ x 2, $C_2^3\times S_4$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.D_6$ x 2, $C_2^4:C_{12}$, $C_2^4.D_6$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^3:D_{12}$

Order 160: $C_2^3:F_5$ x 4, $D_4\times F_5$ x 4, $C_{10}:C_4^2$ x 3, $C_2^3:F_5$ x 2, $C_{10}.C_4^2$ x 2, $D_{10}.D_4$ x 2, $C_2^3\times F_5$ x 2, $C_2^2.D_{20}$ x 2, $D_{10}.D_4$ x 2, $C_2^3:C_{20}$, $C_2^3.D_{10}$, $D_4\times D_{10}$, $D_{10}.D_4$

Order 128: $C_2^4.D_4$ x 4, $C_2^4.C_2^3$ x 2, $C_2^5:C_4$ x 2, $C_2^5:C_4$ x 2, $C_2.D_4^2$ x 2, $C_2^4.C_2^3$, $C_2^3:C_4^2$, $C_2\times D_4^2$

Order 120: $S_5$ x 6, $C_2\times A_5$ x 5

Order 96: $D_4\times D_6$ x 16, $C_2^2\times S_4$ x 13, $D_6.D_4$ x 12, $C_2^2.D_{12}$ x 8, $C_4\times S_4$ x 6, $C_2^3:C_{12}$ x 4, $C_2^3.D_6$ x 4, $\GL(2,\mathbb{Z}/4)$ x 4, $C_{12}:C_2^3$ x 3, $C_2^3:C_{12}$ x 3, $C_2^2.S_4$ x 3, $C_2^3\times D_6$ x 2, $C_2^3\times A_4$ x 2, $C_2^3:D_6$ x 2, $D_4\times A_4$ x 2, $C_4:S_4$ x 2, $C_2^3:C_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^2.D_{12}$ x 2, $C_{12}:C_2^3$, $C_2^2\times D_{12}$

Order 80: $C_2^2\times F_5$ x 16, $D_{10}:C_4$ x 10, $C_4\times F_5$ x 6, $D_4\times D_5$ x 4, $C_4\times D_{10}$ x 3, $C_2^2\times D_{10}$ x 2, $C_{10}:D_4$ x 2, $C_{20}:C_4$ x 2, $C_2^2:C_{20}$ x 2, $C_{10}.D_4$ x 2, $D_{10}:C_4$ x 2, $D_4\times C_{10}$, $C_2\times D_{20}$

Order 64: $C_2^4:C_4$ x 16, $D_4^2$ x 8, $C_2^3.D_4$ x 4, $C_2^4:C_4$ x 4, $D_4\times C_2^3$ x 4, $C_2^2.C_4^2$ x 4, $C_2^3:D_4$ x 4, $C_2^3:D_4$ x 4, $C_4^2:C_2^2$ x 4, $C_2^4:C_4$ x 4, $C_2^3.C_2^3$ x 2, $C_2^3.D_4$ x 2, $C_2^2:C_4^2$ x 2, $C_4^2:C_2^2$

Order 60: $A_5$

Order 48: $S_3\times D_4$ x 32, $C_2^2\times D_6$ x 31, $C_4\times D_6$ x 24, $C_2\times S_4$ x 19, $C_6:D_4$ x 16, $C_6\times D_4$ x 8, $C_2\times D_{12}$ x 8, $D_6:C_4$ x 8, $C_2^2\times A_4$ x 6, $C_2^2:C_{12}$ x 6, $C_6.D_4$ x 6, $C_2^2\times C_{12}$ x 3, $C_6.C_2^3$ x 3, $C_4\times A_4$ x 3, $A_4:C_4$ x 3, $C_2^3\times C_6$ x 2

Order 40: $C_2\times F_5$ x 29, $C_2\times D_{10}$ x 13, $C_4\times D_5$ x 6, $C_5:D_4$ x 4, $C_2\times C_{20}$ x 3, $C_{10}:C_4$ x 3, $D_{20}$ x 2, $C_2^2\times C_{10}$ x 2, $C_5\times D_4$ x 2

Order 32: $C_2^2\times D_4$ x 50, $C_2^3:C_4$ x 34, $C_2^3:C_4$ x 17, $C_4:D_4$ x 16, $C_2^2\wr C_2$ x 16, $C_4\times D_4$ x 16, $C_2^3\times C_4$ x 8, $C_2^5$ x 4, $C_4:D_4$ x 4, $C_2^2.D_4$ x 4, $C_2.C_4^2$ x 4, $C_2\times C_4^2$ x 3

Order 24: $C_2\times D_6$ x 100, $C_4\times S_3$ x 24, $C_3:D_4$ x 16, $C_2^2\times C_6$ x 15, $C_2\times C_{12}$ x 12, $C_6:C_4$ x 12, $D_{12}$ x 8, $C_3\times D_4$ x 8, $S_4$ x 6, $C_2\times A_4$ x 5

Order 20: $D_{10}$ x 19, $F_5$ x 11, $C_2\times C_{10}$ x 6, $C_{20}$ x 3, $C_5:C_4$ x 3

Order 16: $C_2\times D_4$ x 131, $C_2^2:C_4$ x 48, $C_2^2\times C_4$ x 46, $C_2^4$ x 37, $C_4:C_4$ x 8, $C_4^2$ x 6

Order 12: $D_6$ x 78, $C_2\times C_6$ x 25, $C_{12}$ x 6, $C_3:C_4$ x 6, $A_4$

Order 10: $D_5$ x 6, $C_{10}$ x 5

Order 8: $C_2^3$ x 103, $D_4$ x 66, $C_2\times C_4$ x 66

Order 6: $S_3$ x 12, $C_6$ x 11

Order 5: $C_5$

Order 4: $C_2^2$ x 81, $C_4$ x 18

Order 3: $C_3$

Order 2: $C_2$ x 17

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3840: $C_2^3:C_4\times S_5$

Order 1920: $C_2\times D_4\times S_5$, $C_2^2:C_4\times S_5$, $C_2^3:C_4\times A_5$, $(C_2^2\times S_5):C_4$, $(C_2^3\times A_5):C_4$

Order 960: $C_2^3\times S_5$ x 2, $C_2^3:S_5$ x 2, $D_4\times S_5$ x 2, $C_2\times C_4\times S_5$ x 2, $C_2^2:C_4\times A_5$, $C_2^3.S_5$, $(C_2\times C_4):S_5$, $C_2\times D_4\times A_5$, $C_2\times C_4:S_5$

Order 768: $C_2^3:C_4\times S_4$

Order 640: $C_2^3:C_4\times F_5$

Order 480: $C_2^2\times S_5$ x 9, $C_4\times S_5$ x 3, $C_2\times C_4\times A_5$ x 2, $C_2^2.S_5$ x 2, $C_2^2:S_5$ x 2, $C_2^3\times A_5$ x 2, $D_4\times A_5$, $C_4:S_5$

Order 384: $C_2^5:C_{12}$, $C_2^2.\GL(2,\mathbb{Z}/4)$, $C_2^2.\GL(2,\mathbb{Z}/4)$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^5.D_6$, $C_2^3:C_4\times D_6$

Order 320: $C_2^3:C_4\times D_5$, $(C_2^2\times F_5):C_4$, $D_{10}.C_2^4$, $(C_4\times D_{10}):C_4$, $D_{10}:C_4^2$

Order 256: $C_2^2.D_4^2$

Order 240: $C_2\times S_5$ x 11, $C_2^2\times A_5$ x 5, $C_4\times A_5$ x 2, $A_5:C_4$ x 2

Order 192: $C_2^3:C_4\times S_3$ x 6, $C_2^3\times S_4$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $D_4\times S_4$ x 2, $C_2^4.D_6$ x 2, $C_2^4:C_{12}$, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2^4:C_{12}$, $C_2^4.D_6$, $C_2^3.D_{12}$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 160: $C_2^3:F_5$ x 3, $C_2^3\times F_5$ x 2, $D_4\times F_5$ x 2, $C_{10}:C_4^2$ x 2, $C_2^3:F_5$, $C_{10}.C_4^2$, $D_{10}.D_4$, $C_2^3:C_{20}$, $C_2^3.D_{10}$, $D_4\times D_{10}$, $D_{10}.D_4$, $C_2^2.D_{20}$, $D_{10}.D_4$

Order 128: $C_2^4.D_4$ x 2, $C_2^5:C_4$ x 2, $C_2^5:C_4$ x 2, $C_2^4.C_2^3$, $C_2^4.C_2^3$, $C_2^3:C_4^2$, $C_2\times D_4^2$, $C_2.D_4^2$

Order 120: $C_2\times A_5$ x 4, $S_5$ x 4

Order 96: $D_4\times D_6$ x 12, $C_2^2\times S_4$ x 9, $D_6.D_4$ x 6, $C_2^2.D_{12}$ x 4, $C_2^3:C_{12}$ x 3, $C_2^3.D_6$ x 3, $C_4\times S_4$ x 3, $C_2^3\times D_6$ x 2, $C_2^3\times A_4$ x 2, $C_2^3:D_6$ x 2, $C_{12}:C_2^3$ x 2, $C_2^3:C_{12}$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2.S_4$ x 2, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $D_4\times A_4$, $C_4:S_4$, $C_2^3:C_{12}$, $C_2^3.D_6$, $C_2^2.D_{12}$

Order 80: $C_2^2\times F_5$ x 11, $D_{10}:C_4$ x 4, $C_4\times F_5$ x 3, $C_2^2\times D_{10}$ x 2, $C_{10}:D_4$ x 2, $D_4\times D_5$ x 2, $C_4\times D_{10}$ x 2, $D_4\times C_{10}$, $C_2\times D_{20}$, $C_{20}:C_4$, $C_2^2:C_{20}$, $C_{10}.D_4$, $D_{10}:C_4$

Order 64: $C_2^4:C_4$ x 11, $D_4\times C_2^3$ x 4, $D_4^2$ x 4, $C_2^3:D_4$ x 4, $C_2^3:D_4$ x 4, $C_4^2:C_2^2$ x 3, $C_2^3.D_4$ x 2, $C_2^4:C_4$ x 2, $C_2^2.C_4^2$ x 2, $C_2^4:C_4$ x 2, $C_2^3.C_2^3$, $C_2^3.D_4$, $C_2^2:C_4^2$, $C_4^2:C_2^2$

Order 60: $A_5$

Order 48: $C_2^2\times D_6$ x 23, $S_3\times D_4$ x 16, $C_4\times D_6$ x 14, $C_6:D_4$ x 12, $C_2\times S_4$ x 11, $C_6\times D_4$ x 6, $C_2\times D_{12}$ x 6, $C_2^2\times A_4$ x 5, $D_6:C_4$ x 4, $C_2^2:C_{12}$ x 3, $C_6.D_4$ x 3, $C_2^3\times C_6$ x 2, $C_2^2\times C_{12}$ x 2, $C_6.C_2^3$ x 2, $C_4\times A_4$ x 2, $A_4:C_4$ x 2

Order 40: $C_2\times F_5$ x 16, $C_2\times D_{10}$ x 11, $C_4\times D_5$ x 4, $C_2\times C_{20}$ x 2, $C_5:D_4$ x 2, $C_{10}:C_4$ x 2, $C_2^2\times C_{10}$ x 2, $D_{20}$, $C_5\times D_4$

Order 32: $C_2^2\times D_4$ x 37, $C_2^3:C_4$ x 21, $C_2^3:C_4$ x 12, $C_4:D_4$ x 8, $C_2^2\wr C_2$ x 8, $C_4\times D_4$ x 8, $C_2^3\times C_4$ x 6, $C_2^5$ x 4, $C_2^2.D_4$ x 3, $C_4:D_4$ x 2, $C_2\times C_4^2$ x 2, $C_2.C_4^2$ x 2

Order 24: $C_2\times D_6$ x 58, $C_4\times S_3$ x 14, $C_2^2\times C_6$ x 11, $C_3:D_4$ x 8, $C_2\times C_{12}$ x 7, $C_6:C_4$ x 7, $D_{12}$ x 4, $C_2\times A_4$ x 4, $S_4$ x 4, $C_3\times D_4$ x 4

Order 20: $D_{10}$ x 14, $F_5$ x 6, $C_2\times C_{10}$ x 5, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 16: $C_2\times D_4$ x 75, $C_2^4$ x 30, $C_2^2\times C_4$ x 30, $C_2^2:C_4$ x 25, $C_4:C_4$ x 4, $C_4^2$ x 3

Order 12: $D_6$ x 45, $C_2\times C_6$ x 16, $C_{12}$ x 4, $C_3:C_4$ x 4, $A_4$

Order 10: $D_5$ x 5, $C_{10}$ x 4

Order 8: $C_2^3$ x 71, $C_2\times C_4$ x 40, $D_4$ x 33

Order 6: $S_3$ x 9, $C_6$ x 8

Order 5: $C_5$

Order 4: $C_2^2$ x 55, $C_4$ x 12

Order 3: $C_3$

Order 2: $C_2$ x 13

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3840: $C_2^3:C_4\times S_5$ ($C_1$)

Order 1920: $C_2^2:C_4\times S_5$ ($C_2$) x 2, $(C_2^2\times S_5):C_4$ ($C_2$) x 2, $C_2\times D_4\times S_5$ ($C_2$), $C_2^3:C_4\times A_5$ ($C_2$), $(C_2^3\times A_5):C_4$ ($C_2$)

Order 960: $C_2^2:C_4\times A_5$ ($C_2^2$) x 2, $C_2^3.S_5$ ($C_2^2$) x 2, $C_2^3\times S_5$ ($C_2^2$), $C_2^3\times S_5$ ($C_4$), $C_2\times D_4\times A_5$ ($C_2^2$), $C_2^3:S_5$ ($C_2^2$), $C_2^3:S_5$ ($C_4$), $C_2\times C_4:S_5$ ($C_4$), $C_2\times C_4\times S_5$ ($C_4$)

Order 480: $C_2^2\times S_5$ ($D_4$) x 4, $C_2^2\times S_5$ ($C_2\times C_4$) x 3, $C_2\times C_4\times A_5$ ($C_2\times C_4$), $C_2^2.S_5$ ($C_2\times C_4$), $C_2^3\times A_5$ ($C_2^3$), $C_2^3\times A_5$ ($C_2\times C_4$)

Order 240: $C_2\times S_5$ ($C_2^2:C_4$) x 4, $C_2^2\times A_5$ ($C_2\times D_4$) x 2, $C_2^2\times A_5$ ($C_2^2\times C_4$)

Order 120: $S_5$ ($C_2^3:C_4$) x 2, $C_2\times A_5$ ($C_2^3:C_4$)

Order 60: $A_5$ ($C_2^4:C_4$)

Order 32: $C_2^3:C_4$ ($S_5$)

Order 16: $C_2^2:C_4$ ($C_2\times S_5$) x 2, $C_2\times D_4$ ($C_2\times S_5$)

Order 8: $C_2^3$ ($C_4\times S_5$), $C_2^3$ ($C_2^2\times S_5$), $C_2\times C_4$ ($C_4\times S_5$)

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

Order 2: $C_2$ ($C_2^2:C_4\times S_5$)

Order 1: $C_1$ ($C_2^3:C_4\times S_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3840: $C_2^3:C_4\times S_5$ ($C_1$)

Order 1920: $C_2\times D_4\times S_5$ ($C_2$), $C_2^2:C_4\times S_5$ ($C_2$), $C_2^3:C_4\times A_5$ ($C_2$), $(C_2^2\times S_5):C_4$ ($C_2$), $(C_2^3\times A_5):C_4$ ($C_2$)

Order 960: $C_2^2:C_4\times A_5$ ($C_2^2$), $C_2^3.S_5$ ($C_2^2$), $C_2^3\times S_5$ ($C_2^2$), $C_2^3\times S_5$ ($C_4$), $C_2\times D_4\times A_5$ ($C_2^2$), $C_2^3:S_5$ ($C_2^2$), $C_2^3:S_5$ ($C_4$), $C_2\times C_4:S_5$ ($C_4$), $C_2\times C_4\times S_5$ ($C_4$)

Order 480: $C_2^2\times S_5$ ($C_2\times C_4$) x 3, $C_2^2\times S_5$ ($D_4$) x 2, $C_2\times C_4\times A_5$ ($C_2\times C_4$), $C_2^2.S_5$ ($C_2\times C_4$), $C_2^3\times A_5$ ($C_2^3$), $C_2^3\times A_5$ ($C_2\times C_4$)

Order 240: $C_2\times S_5$ ($C_2^2:C_4$) x 3, $C_2^2\times A_5$ ($C_2\times D_4$), $C_2^2\times A_5$ ($C_2^2\times C_4$)

Order 120: $C_2\times A_5$ ($C_2^3:C_4$), $S_5$ ($C_2^3:C_4$)

Order 60: $A_5$ ($C_2^4:C_4$)

Order 32: $C_2^3:C_4$ ($S_5$)

Order 16: $C_2^2:C_4$ ($C_2\times S_5$), $C_2\times D_4$ ($C_2\times S_5$)

Order 8: $C_2^3$ ($C_4\times S_5$), $C_2^3$ ($C_2^2\times S_5$), $C_2\times C_4$ ($C_4\times S_5$)

Order 4: $C_2^2$ ($D_4\times S_5$), $C_2^2$ ($C_2\times C_4\times S_5$)

Order 2: $C_2$ ($C_2^2:C_4\times S_5$)

Order 1: $C_1$ ($C_2^3:C_4\times S_5$)

Series

Derived series

$C_2^3:C_4\times S_5$

$\rhd$

$C_2^2\times A_5$

$\rhd$

$A_5$

Chief series

$C_2^3:C_4\times S_5$

$\rhd$

$C_2^3:C_4\times A_5$

$\rhd$

$C_2^3:C_4$

$\rhd$

$C_2\times D_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3:C_4\times S_5$

$\rhd$

$C_2^2\times A_5$

$\rhd$

$C_2\times A_5$

$\rhd$

$A_5$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^3:C_4$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $77 \times 77$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $63 \times 63$ rational character table.