Abstract group 7680.fs: $C_2\wr S_3\times F_5$

• Label: 7680.fs
• Order: \( 2^{9} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $17$
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$C_2\wr S_3\times F_5$
Order:	\(7680\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_5^4.\OD_{16}$, of order \(92160\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5 \)
Composition factors:	$C_2$ x 9, $C_3$, $C_5$
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30
Elements	1	527	32	3568	4	736	348	1280	128	672	384	7680
Conjugacy classes	1	33	1	62	1	7	16	8	1	7	3	140
Divisions	1	33	1	38	1	5	16	3	1	7	2	108
Autjugacy classes	1	19	1	26	1	5	9	4	1	3	2	72

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	16	20	48	4	28	5	12	0	7	140
Irr. rational chars.	8	10	24	9	26	4	19	1	7	108

Minimal presentations

Permutation degree:	$17$
Transitive degree:	not computed
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

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{4}=c^{12}=d^{10}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $17$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17), (2,3)(4,5)(8,9)(10,12) \!\cdots\! \rangle$
Direct product:	$F_5$ $\, \times\, $ $(C_2\wr S_3)$
Semidirect product:	$(F_5\times C_2^6)$ $\,\rtimes\,$ $S_3$	$(C_2^4\times F_5)$ $\,\rtimes\,$ $S_4$	$C_2^6$ $\,\rtimes\,$ $(S_3\times F_5)$	$C_2^4$ $\,\rtimes\,$ $(F_5\times S_4)$	all 37
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^6\times D_5)$ . $D_6$	$D_5$ . $(C_2^6.D_6)$	$(F_5\times C_2^5)$ . $D_6$ (2)	$C_2^5$ . $(D_6\times F_5)$	all 32
Aut. group:	$\Aut(D_5\times C_4^2)$	$\Aut(C_2^4:D_5)$	$\Aut((C_2\times C_4^2).D_6)$

Elements of the group are displayed as permutations of degree 17.

Homology

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

Subgroups

There are 455764 subgroups in 26488 conjugacy classes, 118 normal (64 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^3:S_4\times F_5$
Commutator:	$G' \simeq$ $C_2^4:C_{30}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^3:S_4\times F_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^5\times C_{10}$	$G/\operatorname{Fit} \simeq$ $C_4\times S_3$
Radical:	$R \simeq$ $C_2\wr S_3\times F_5$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_4\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.C_2^5$
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

All subgroups of index up to 40 (order at least 192) are shown, as well as all normal subgroups of any index.

Order 7680: $C_2\wr S_3\times F_5$

Order 3840: $C_2^4.(C_6\times F_5).C_2$, $C_2^4.D_6:F_5$, $C_2^4.(C_2\times C_6):F_5$, $C_2^4.(D_5\times C_3:D_4)$, $C_2^4.(C_6\times F_5).C_2$, $C_2^2\wr C_3\times F_5$, $C_2^3:S_4\times F_5$

Order 2560: $C_5:(C_2^7.C_2^2)$

Order 1920: $C_2^3:A_4\times F_5$ x 4, $F_5\times \GL(2,\mathbb{Z}/4)$ x 3, $(C_2^2\times D_{10}).S_4$ x 2, $F_5\times C_2^2:S_4$ x 2, $C_2^2\times D_5\times C_2^4:C_3$, $C_2^2:S_4\times D_{10}$, $C_2^4.D_{30}.C_2$, $C_2^4.C_3:D_{20}$, $C_2^4.D_{10}:S_3$, $D_5\times C_2^2:(A_4:C_4)$, $C_5\times C_2^4:C_3.D_4$

Order 1536: $C_2^7.D_6$

Order 1280: $C_2^3:D_4\times F_5$ x 7, $C_5:(C_2^6.C_2^2)$ x 7, $C_5:(C_2^6.C_2^2)$ x 7, $C_2^4:C_4\times F_5$ x 7, $C_2^6.C_2\times D_5$, $F_5\times C_2^6$, $C_5:(C_2^7.C_2)$

Order 960: $C_2^2:A_4\times D_{10}$ x 4, $C_2^2\times A_4\times F_5$ x 4, $C_2^2:A_4\times F_5$ x 3, $C_2\times F_5\times S_4$ x 3, $D_5\times \GL(2,\mathbb{Z}/4)$ x 3, $D_5.\GL(2,\mathbb{Z}/4)$ x 3, $D_5.\GL(2,\mathbb{Z}/4)$ x 3, $D_5.\GL(2,\mathbb{Z}/4)$ x 3, $A_4:C_4\times F_5$ x 3, $D_5\times C_2^2:S_4$ x 2, $(C_2\times D_{10}).S_4$ x 2, $C_2^5:C_{30}$, $C_2^4:D_{30}$, $C_2^2:S_4\times C_{10}$, $C_2^5.D_{15}$, $C_5\times C_2^3.S_4$

Order 768: $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^6.D_6$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^4.(C_6.Q_8)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_4\times C_2^2:(A_4:C_4)$, $C_2^6:C_{12}$

Order 640: $C_2^2\wr C_2\times F_5$ x 56, $(C_2^3\times F_5):C_4$ x 42, $F_5\times C_2^5$ x 30, $(D_4\times D_{10}):C_4$ x 21, $D_{10}.(C_4\times D_4)$ x 21, $C_2\times D_{10}:C_4^2$ x 21, $C_2^5:F_5$ x 14, $C_5:(C_2^6.C_2)$ x 7, $C_2^5.D_{10}$ x 7, $D_{10}.C_2^5$ x 7, $C_2^5:D_{10}$ x 7, $C_5:(C_2^6.C_2)$ x 7, $C_2^6\times D_5$, $C_5\times C_2^6.C_2$, $C_5:(C_2^6.C_2)$

Order 512: $C_2^4.C_2^5$

Order 480: $C_2\times A_4\times F_5$ x 18, $D_{10}.S_4$ x 6, $F_5\times S_4$ x 6, $C_2\times A_4\times D_{10}$ x 4, $D_{10}:S_4$ x 3, $D_{10}:S_4$ x 3, $D_{10}.S_4$ x 3, $D_5\times C_2^2:A_4$ x 3, $D_{10}\times S_4$ x 3, $C_5:\GL(2,\mathbb{Z}/4)$ x 3, $C_5\times \GL(2,\mathbb{Z}/4)$ x 3, $C_2^4:C_{30}$ x 2, $C_2^4:D_{15}$, $C_5\times C_2^2:S_4$, $C_3:D_4\times F_5$

Order 384: $C_2^5:C_{12}$ x 4, $C_2\wr S_3$ x 4, $C_2^4.S_4$ x 3, $C_4\times \GL(2,\mathbb{Z}/4)$ x 3, $C_2^5.D_6$ x 2, $C_2^5:A_4$, $C_2^2\wr S_3$

Order 320: $C_2^4\times F_5$ x 480, $D_{10}.C_2^4$ x 84, $(C_2\times D_{20}):C_4$ x 84, $C_{10}.(C_4\times D_4)$ x 84, $D_{10}:C_4^2$ x 84, $C_2^4:D_{10}$ x 56, $C_2^4:F_5$ x 56, $C_2^3.D_{20}$ x 42, $D_{10}.C_4^2$ x 42, $C_2^4:F_5$ x 35, $C_2^4\times D_{10}$ x 30, $C_2^4:D_{10}$ x 21, $C_2^3:D_{20}$ x 21, $C_2^4.D_{10}$ x 21, $C_2^4:C_{20}$ x 7, $C_2^4.D_{10}$ x 7, $C_{20}:C_2^4$ x 7, $D_{10}.C_2^4$ x 7, $D_{10}:C_4^2$ x 7, $C_2^5:C_{10}$ x 7, $C_2^4:D_{10}$ x 7, $C_2^5\times C_{10}$

Order 256: $C_2^5.D_4$ x 7, $C_2^5.D_4$ x 7, $C_2^4:C_4^2$ x 7, $C_2^3.C_2^5$ x 7, $C_2^6:C_4$, $C_2^6\times C_4$, $C_2^5:D_4$

Order 240: $A_4\times D_{10}$ x 18, $A_4\times F_5$ x 13, $D_5\times S_4$ x 6, $A_4:F_5$ x 6, $C_2^3:C_{30}$ x 4, $C_{10}:S_4$ x 3, $C_{10}\times S_4$ x 3, $C_{10}.S_4$ x 3, $A_4:C_{20}$ x 3, $D_{10}.D_6$, $D_{30}:C_4$, $C_{15}:C_4^2$, $C_2^4:C_{15}$, $D_{10}:C_{12}$, $D_6\times F_5$, $D_6:D_{10}$, $D_{10}.D_6$

Order 192: $C_2^2\wr C_3$ x 5, $C_2^3:S_4$ x 4, $C_2^4:C_{12}$ x 4, $C_2^3.S_4$ x 4, $C_2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2.\GL(2,\mathbb{Z}/4)$ x 3, $A_4:C_4^2$ x 3, $C_2^4:C_{12}$ x 3, $C_2\times \GL(2,\mathbb{Z}/4)$ x 3, $C_2^4.D_6$ x 3

Order 160: $D_4\times F_5$ x 64, $C_2^3\times F_5$ x 6, $C_2^3\times D_{10}$ x 5, $C_2^4\times C_{10}$

Order 120: $S_3\times F_5$ x 2

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

Order 80: $C_2^2\times F_5$ x 7, $C_2^2\times D_{10}$ x 6, $C_2^3\times C_{10}$ x 4

Order 64: $C_2^6$

Order 48: $C_2^2:A_4$

Order 40: $C_2\times F_5$ x 182, $C_2\times D_{10}$ x 7, $C_2^2\times C_{10}$ x 3

Order 32: $C_4\times D_4$ x 64, $C_2^5$

Order 24: $S_4$ x 6, $C_3:D_4$ x 4, $C_4\times S_3$ x 2

Order 20: $F_5$ x 24, $C_2\times C_{10}$ x 4, $D_{10}$ x 2

Order 16: $C_2^4$ x 4

Order 10: $D_5$ x 2, $C_{10}$

Order 8: $C_2\times C_4$ x 180, $D_4$ x 64, $C_2^3$ x 3

Order 6: $S_3$ x 2

Order 5: $C_5$

Order 4: $C_4$ x 24, $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 40 (order at least 192) are shown, as well as all normal subgroups of any index.

Order 7680: $C_2\wr S_3\times F_5$

Order 3840: $C_2^4.(C_6\times F_5).C_2$, $C_2^4.D_6:F_5$, $C_2^4.(C_2\times C_6):F_5$, $C_2^4.(D_5\times C_3:D_4)$, $C_2^4.(C_6\times F_5).C_2$, $C_2^2\wr C_3\times F_5$, $C_2^3:S_4\times F_5$

Order 2560: $C_5:(C_2^7.C_2^2)$

Order 1920: $C_2^3:A_4\times F_5$ x 3, $(C_2^2\times D_{10}).S_4$ x 2, $C_2^2\times D_5\times C_2^4:C_3$, $C_2^2:S_4\times D_{10}$, $F_5\times C_2^2:S_4$, $C_2^4.D_{30}.C_2$, $C_2^4.C_3:D_{20}$, $C_2^4.D_{10}:S_3$, $D_5\times C_2^2:(A_4:C_4)$, $F_5\times \GL(2,\mathbb{Z}/4)$, $C_5\times C_2^4:C_3.D_4$

Order 1536: $C_2^7.D_6$

Order 1280: $C_2^3:D_4\times F_5$ x 3, $C_5:(C_2^6.C_2^2)$ x 3, $C_5:(C_2^6.C_2^2)$ x 3, $C_2^4:C_4\times F_5$ x 3, $C_2^6.C_2\times D_5$, $F_5\times C_2^6$, $C_5:(C_2^7.C_2)$

Order 960: $C_2^2:A_4\times D_{10}$ x 4, $D_5\times C_2^2:S_4$ x 2, $C_2^2:A_4\times F_5$ x 2, $C_2^2\times A_4\times F_5$ x 2, $(C_2\times D_{10}).S_4$ x 2, $C_2^5:C_{30}$, $C_2^4:D_{30}$, $C_2^2:S_4\times C_{10}$, $C_2\times F_5\times S_4$, $C_2^5.D_{15}$, $C_5\times C_2^3.S_4$, $D_5\times \GL(2,\mathbb{Z}/4)$, $D_5.\GL(2,\mathbb{Z}/4)$, $D_5.\GL(2,\mathbb{Z}/4)$, $D_5.\GL(2,\mathbb{Z}/4)$, $A_4:C_4\times F_5$

Order 768: $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^6.D_6$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^4.(C_6.Q_8)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_4\times C_2^2:(A_4:C_4)$, $C_2^6:C_{12}$

Order 640: $F_5\times C_2^5$ x 13, $C_2^2\wr C_2\times F_5$ x 12, $(C_2^3\times F_5):C_4$ x 10, $C_2^5:F_5$ x 6, $(D_4\times D_{10}):C_4$ x 6, $D_{10}.(C_4\times D_4)$ x 6, $C_2\times D_{10}:C_4^2$ x 6, $C_5:(C_2^6.C_2)$ x 3, $C_2^5.D_{10}$ x 3, $D_{10}.C_2^5$ x 3, $C_2^5:D_{10}$ x 3, $C_5:(C_2^6.C_2)$ x 3, $C_2^6\times D_5$, $C_5\times C_2^6.C_2$, $C_5:(C_2^6.C_2)$

Order 512: $C_2^4.C_2^5$

Order 480: $C_2\times A_4\times F_5$ x 6, $D_5\times C_2^2:A_4$ x 3, $C_2^4:C_{30}$ x 2, $C_2\times A_4\times D_{10}$ x 2, $D_{10}.S_4$ x 2, $D_{10}:S_4$, $D_{10}:S_4$, $D_{10}.S_4$, $C_2^4:D_{15}$, $C_5\times C_2^2:S_4$, $D_{10}\times S_4$, $F_5\times S_4$, $C_5:\GL(2,\mathbb{Z}/4)$, $C_5\times \GL(2,\mathbb{Z}/4)$, $C_3:D_4\times F_5$

Order 384: $C_2\wr S_3$ x 4, $C_2^5:C_{12}$ x 3, $C_2^4.S_4$ x 3, $C_2^5:A_4$, $C_2^2\wr S_3$, $C_2^5.D_6$, $C_4\times \GL(2,\mathbb{Z}/4)$

Order 320: $C_2^4\times F_5$ x 86, $D_{10}.C_2^4$ x 17, $(C_2\times D_{20}):C_4$ x 17, $C_{10}.(C_4\times D_4)$ x 17, $D_{10}:C_4^2$ x 17, $C_2^4\times D_{10}$ x 16, $C_2^4:D_{10}$ x 16, $C_2^4:F_5$ x 16, $C_2^4:F_5$ x 12, $C_2^3.D_{20}$ x 10, $D_{10}.C_4^2$ x 10, $C_2^4:D_{10}$ x 6, $C_2^3:D_{20}$ x 6, $C_2^4.D_{10}$ x 6, $C_2^4:C_{20}$ x 3, $C_2^4.D_{10}$ x 3, $C_{20}:C_2^4$ x 3, $D_{10}.C_2^4$ x 3, $D_{10}:C_4^2$ x 3, $C_2^5:C_{10}$ x 3, $C_2^4:D_{10}$ x 3, $C_2^5\times C_{10}$

Order 256: $C_2^5.D_4$ x 3, $C_2^5.D_4$ x 3, $C_2^4:C_4^2$ x 3, $C_2^3.C_2^5$ x 3, $C_2^6:C_4$, $C_2^6\times C_4$, $C_2^5:D_4$

Order 240: $A_4\times D_{10}$ x 8, $A_4\times F_5$ x 4, $C_2^3:C_{30}$ x 2, $D_5\times S_4$ x 2, $A_4:F_5$ x 2, $D_{10}.D_6$, $D_{30}:C_4$, $C_{15}:C_4^2$, $C_2^4:C_{15}$, $D_{10}:C_{12}$, $C_{10}:S_4$, $C_{10}\times S_4$, $D_6\times F_5$, $D_6:D_{10}$, $D_{10}.D_6$, $C_{10}.S_4$, $A_4:C_{20}$

Order 192: $C_2^2\wr C_3$ x 5, $C_2^3:S_4$ x 4, $C_2^3.S_4$ x 4, $C_2^4:C_{12}$ x 2, $C_2^4:C_{12}$ x 2, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2.\GL(2,\mathbb{Z}/4)$, $A_4:C_4^2$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^4.D_6$

Order 160: $D_4\times F_5$ x 9, $C_2^3\times D_{10}$ x 3, $C_2^3\times F_5$ x 2, $C_2^4\times C_{10}$

Order 120: $S_3\times F_5$

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

Order 80: $C_2^3\times C_{10}$ x 2, $C_2^2\times D_{10}$ x 2, $C_2^2\times F_5$ x 2

Order 64: $C_2^6$

Order 48: $C_2^2:A_4$

Order 40: $C_2\times F_5$ x 20, $C_2\times D_{10}$ x 3, $C_2^2\times C_{10}$

Order 32: $C_4\times D_4$ x 9, $C_2^5$

Order 24: $C_3:D_4$ x 4, $S_4$ x 2, $C_4\times S_3$

Order 20: $F_5$ x 10, $C_2\times C_{10}$ x 2, $D_{10}$ x 2

Order 16: $C_2^4$ x 2

Order 10: $D_5$ x 2, $C_{10}$

Order 8: $C_2\times C_4$ x 18, $D_4$ x 16, $C_2^3$

Order 6: $S_3$ x 2

Order 5: $C_5$

Order 4: $C_4$ x 10, $C_2^2$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 7680: $C_2\wr S_3\times F_5$ ($C_1$)

Order 3840: $C_2^4.(C_6\times F_5).C_2$ ($C_2$), $C_2^4.D_6:F_5$ ($C_2$), $C_2^4.(C_2\times C_6):F_5$ ($C_2$), $C_2^4.(D_5\times C_3:D_4)$ ($C_2$), $C_2^4.(C_6\times F_5).C_2$ ($C_2$), $C_2^2\wr C_3\times F_5$ ($C_2$), $C_2^3:S_4\times F_5$ ($C_2$)

Order 1920: $C_2^3:A_4\times F_5$ ($C_2^2$) x 2, $(C_2^2\times D_{10}).S_4$ ($C_2^2$) x 2, $C_2^2\times D_5\times C_2^4:C_3$ ($C_2^2$), $C_2^2:S_4\times D_{10}$ ($C_2^2$), $C_2^4.D_{30}.C_2$ ($C_4$), $C_2^4.C_3:D_{20}$ ($C_4$), $C_2^4.D_{10}:S_3$ ($C_4$), $D_5\times C_2^2:(A_4:C_4)$ ($C_2^2$), $C_5\times C_2^4:C_3.D_4$ ($C_4$)

Order 1280: $F_5\times C_2^6$ ($S_3$)

Order 960: $C_2^2:A_4\times F_5$ ($D_4$) x 2, $C_2^5:C_{30}$ ($C_2\times C_4$), $C_2^2:A_4\times D_{10}$ ($C_2^3$), $C_2^2:A_4\times D_{10}$ ($C_2\times C_4$), $C_2^4:D_{30}$ ($C_2\times C_4$), $C_2^2:S_4\times C_{10}$ ($C_2\times C_4$), $C_2^5.D_{15}$ ($C_2\times C_4$), $C_5\times C_2^3.S_4$ ($C_2\times C_4$)

Order 640: $F_5\times C_2^5$ ($D_6$) x 2, $C_2^6\times D_5$ ($D_6$)

Order 480: $C_2^4:C_{30}$ ($C_2^2\times C_4$), $D_5\times C_2^2:A_4$ ($D_4:C_2$), $D_5\times C_2^2:A_4$ ($C_2\times D_4$)

Order 384: $C_2\wr S_3$ ($F_5$)

Order 320: $C_2^4\times F_5$ ($S_4$) x 3, $C_2^4\times F_5$ ($C_3:D_4$) x 2, $C_2^5\times C_{10}$ ($C_4\times S_3$), $C_2^4\times D_{10}$ ($C_4\times S_3$), $C_2^4\times D_{10}$ ($C_2\times D_6$)

Order 240: $C_2^4:C_{15}$ ($C_4\times D_4$)

Order 192: $C_2^2\wr C_3$ ($C_2\times F_5$), $C_2^3:S_4$ ($C_2\times F_5$), $C_2^3.S_4$ ($C_2\times F_5$)

Order 160: $C_2^3\times F_5$ ($C_2\times S_4$) x 6, $C_2^3\times D_{10}$ ($C_2\times S_4$) x 3, $C_2^4\times C_{10}$ ($C_4\times D_6$), $C_2^3\times D_{10}$ ($C_6:D_4$), $C_2^3\times D_{10}$ ($D_{12}:C_2$)

Order 96: $C_2^3:A_4$ ($C_2^2\times F_5$)

Order 80: $C_2^2\times F_5$ ($\GL(2,\mathbb{Z}/4)$) x 6, $C_2^3\times C_{10}$ ($C_4\times S_4$) x 3, $C_2^2\times D_{10}$ ($C_2^2\times S_4$) x 3, $C_2^2\times D_{10}$ ($C_4\times S_4$) x 3, $C_2^3\times C_{10}$ ($C_{12}:D_4$), $C_2^2\times F_5$ ($C_2^2:S_4$)

Order 64: $C_2^6$ ($S_3\times F_5$)

Order 48: $C_2^2:A_4$ ($D_4\times F_5$)

Order 40: $C_2^2\times C_{10}$ ($C_2^4.D_6$) x 3, $C_2\times D_{10}$ ($C_2\times \GL(2,\mathbb{Z}/4)$) x 3, $C_2\times D_{10}$ ($\GL(2,\mathbb{Z}/4):C_2$) x 3, $C_2\times F_5$ ($C_2^3:S_4$) x 2, $C_2\times D_{10}$ ($C_2^3:S_4$)

Order 32: $C_2^5$ ($D_6\times F_5$)

Order 20: $C_2\times C_{10}$ ($C_4\times \GL(2,\mathbb{Z}/4)$) x 3, $F_5$ ($C_2\wr S_3$) x 2, $C_2\times C_{10}$ ($C_2^5.D_6$), $D_{10}$ ($C_2^2\wr S_3$), $D_{10}$ ($C_2^5.D_6$)

Order 16: $C_2^4$ ($F_5\times S_4$) x 3, $C_2^4$ ($C_3:D_4\times F_5$)

Order 10: $C_{10}$ ($C_2^6.D_6$), $D_5$ ($C_2^6.D_6$), $D_5$ ($C_2^3:\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_2\times F_5\times S_4$) x 3

Order 5: $C_5$ ($C_2^7.D_6$)

Order 4: $C_2^2$ ($F_5\times \GL(2,\mathbb{Z}/4)$) x 3, $C_2^2$ ($F_5\times C_2^2:S_4$)

Order 2: $C_2$ ($C_2^3:S_4\times F_5$)

Order 1: $C_1$ ($C_2\wr S_3\times F_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 7680: $C_2\wr S_3\times F_5$ ($C_1$)

Order 3840: $C_2^4.(C_6\times F_5).C_2$ ($C_2$), $C_2^4.D_6:F_5$ ($C_2$), $C_2^4.(C_2\times C_6):F_5$ ($C_2$), $C_2^4.(D_5\times C_3:D_4)$ ($C_2$), $C_2^4.(C_6\times F_5).C_2$ ($C_2$), $C_2^2\wr C_3\times F_5$ ($C_2$), $C_2^3:S_4\times F_5$ ($C_2$)

Order 1920: $C_2^3:A_4\times F_5$ ($C_2^2$) x 2, $(C_2^2\times D_{10}).S_4$ ($C_2^2$) x 2, $C_2^2\times D_5\times C_2^4:C_3$ ($C_2^2$), $C_2^2:S_4\times D_{10}$ ($C_2^2$), $C_2^4.D_{30}.C_2$ ($C_4$), $C_2^4.C_3:D_{20}$ ($C_4$), $C_2^4.D_{10}:S_3$ ($C_4$), $D_5\times C_2^2:(A_4:C_4)$ ($C_2^2$), $C_5\times C_2^4:C_3.D_4$ ($C_4$)

Order 1280: $F_5\times C_2^6$ ($S_3$)

Order 960: $C_2^5:C_{30}$ ($C_2\times C_4$), $C_2^2:A_4\times D_{10}$ ($C_2^3$), $C_2^2:A_4\times D_{10}$ ($C_2\times C_4$), $C_2^4:D_{30}$ ($C_2\times C_4$), $C_2^2:S_4\times C_{10}$ ($C_2\times C_4$), $C_2^2:A_4\times F_5$ ($D_4$), $C_2^5.D_{15}$ ($C_2\times C_4$), $C_5\times C_2^3.S_4$ ($C_2\times C_4$)

Order 640: $F_5\times C_2^5$ ($D_6$) x 2, $C_2^6\times D_5$ ($D_6$)

Order 480: $C_2^4:C_{30}$ ($C_2^2\times C_4$), $D_5\times C_2^2:A_4$ ($D_4:C_2$), $D_5\times C_2^2:A_4$ ($C_2\times D_4$)

Order 384: $C_2\wr S_3$ ($F_5$)

Order 320: $C_2^5\times C_{10}$ ($C_4\times S_3$), $C_2^4\times D_{10}$ ($C_4\times S_3$), $C_2^4\times D_{10}$ ($C_2\times D_6$), $C_2^4\times F_5$ ($C_3:D_4$), $C_2^4\times F_5$ ($S_4$)

Order 240: $C_2^4:C_{15}$ ($C_4\times D_4$)

Order 192: $C_2^2\wr C_3$ ($C_2\times F_5$), $C_2^3:S_4$ ($C_2\times F_5$), $C_2^3.S_4$ ($C_2\times F_5$)

Order 160: $C_2^3\times F_5$ ($C_2\times S_4$) x 2, $C_2^4\times C_{10}$ ($C_4\times D_6$), $C_2^3\times D_{10}$ ($C_2\times S_4$), $C_2^3\times D_{10}$ ($C_6:D_4$), $C_2^3\times D_{10}$ ($D_{12}:C_2$)

Order 96: $C_2^3:A_4$ ($C_2^2\times F_5$)

Order 80: $C_2^3\times C_{10}$ ($C_4\times S_4$), $C_2^3\times C_{10}$ ($C_{12}:D_4$), $C_2^2\times D_{10}$ ($C_2^2\times S_4$), $C_2^2\times D_{10}$ ($C_4\times S_4$), $C_2^2\times F_5$ ($C_2^2:S_4$), $C_2^2\times F_5$ ($\GL(2,\mathbb{Z}/4)$)

Order 64: $C_2^6$ ($S_3\times F_5$)

Order 48: $C_2^2:A_4$ ($D_4\times F_5$)

Order 40: $C_2\times F_5$ ($C_2^3:S_4$) x 2, $C_2^2\times C_{10}$ ($C_2^4.D_6$), $C_2\times D_{10}$ ($C_2^3:S_4$), $C_2\times D_{10}$ ($C_2\times \GL(2,\mathbb{Z}/4)$), $C_2\times D_{10}$ ($\GL(2,\mathbb{Z}/4):C_2$)

Order 32: $C_2^5$ ($D_6\times F_5$)

Order 20: $C_2\times C_{10}$ ($C_2^5.D_6$), $C_2\times C_{10}$ ($C_4\times \GL(2,\mathbb{Z}/4)$), $D_{10}$ ($C_2^2\wr S_3$), $D_{10}$ ($C_2^5.D_6$), $F_5$ ($C_2\wr S_3$)

Order 16: $C_2^4$ ($F_5\times S_4$), $C_2^4$ ($C_3:D_4\times F_5$)

Order 10: $C_{10}$ ($C_2^6.D_6$), $D_5$ ($C_2^6.D_6$), $D_5$ ($C_2^3:\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_2\times F_5\times S_4$)

Order 5: $C_5$ ($C_2^7.D_6$)

Order 4: $C_2^2$ ($F_5\times C_2^2:S_4$), $C_2^2$ ($F_5\times \GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_2^3:S_4\times F_5$)

Order 1: $C_1$ ($C_2\wr S_3\times F_5$)

Series

Derived series

$C_2\wr S_3\times F_5$

$\rhd$

$C_2^4:C_{30}$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_2\wr S_3\times F_5$

$\rhd$

$C_2^2\wr C_3\times F_5$

$\rhd$

$C_2^3:A_4\times F_5$

$\rhd$

$C_2^2:A_4\times D_{10}$

$\rhd$

$C_2^4:C_{30}$

$\rhd$

$C_2^4:C_{15}$

$\rhd$

$C_2^3\times C_{10}$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2\wr S_3\times F_5$

$\rhd$

$C_2^4:C_{30}$

$\rhd$

$C_2^4:C_{15}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $140 \times 140$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $108 \times 108$ rational character table.