Abstract group 2560.eb: $C_2^3\wr C_2\times F_5$

• Label: 2560.eb
• Order: \( 2^{9} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{20} \cdot 3 \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{14} \cdot 3 \cdot 7 \)
• Perm deg.: $17$
• Trans deg.: $80$
• Rank: $5$

Group information

Description:	$C_2^3\wr C_2\times F_5$
Order:	\(2560\)\(\medspace = 2^{9} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$S_3\times C_3^4:C_4^2$, of order \(110100480\)\(\medspace = 2^{20} \cdot 3 \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 9, $C_5$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	4	5	10	20
Elements	1	431	1616	4	284	224	2560
Conjugacy classes	1	73	102	1	36	7	220
Divisions	1	73	58	1	36	7	176
Autjugacy classes	1	7	10	1	3	1	23

Dimension	1	2	4	8
Irr. complex chars.	64	112	16	28	220
Irr. rational chars.	32	72	44	28	176

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$80$
Rank:	$5$
Inequivalent generating 5-tuples:	$297123840$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{4}=b^{10}=c^{4}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $17$ $\langle(13,14,15,17,16), (1,2)(4,6)(10,12)(14,16)(15,17), (1,3)(2,5)(4,7)(6,8) \!\cdots\! \rangle$
Direct product:	$F_5$ $\, \times\, $ $(C_2^3\wr C_2)$
Semidirect product:	$(C_2^4\times F_5)$ $\,\rtimes\,$ $D_4$	$C_2^4$ $\,\rtimes\,$ $(D_4\times F_5)$	$(F_5\times C_2^6)$ $\,\rtimes\,$ $C_2$	$C_2^6$ $\,\rtimes\,$ $(C_2\times F_5)$	all 55
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_5$ . $(C_2^5:D_4)$	$D_{10}$ . $(C_2^4:D_4)$	$(C_2^6\times D_5)$ . $C_2^2$	$D_5$ . $(C_2^3.C_2^5)$	all 45
Aut. group:	$\Aut(C_{10}.C_4^2)$	$\Aut(C_4^2:D_5)$	$\Aut(C_2^3.D_{10})$	$\Aut(D_{10}.D_4)$	all 94

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

Homology

Abelianization:	$C_{2}^{4} \times C_{4} $
Schur multiplier:	$C_{2}^{13}$
Commutator length:	$1$

Subgroups

There are 363196 subgroups in 62904 conjugacy classes, 1886 normal (23 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2^4\times F_5$
Commutator:	$G' \simeq$ $C_2^2\times C_{10}$	$G/G' \simeq$ $C_2^4\times C_4$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^4\times F_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_5\times C_2^6.C_2$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $C_2^3\wr C_2\times F_5$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2^4\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.C_2^5$
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 2560: $C_2^3\wr C_2\times F_5$

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 640: $F_5\times C_2^5$ x 70, $C_2^2\wr C_2\times F_5$ x 56, $(C_2^3\times F_5):C_4$ x 42, $(D_4\times D_{10}):C_4$ 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 320: $C_2^4\times F_5$ x 1345, $D_{10}.C_2^4$ 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, $C_2^4\times D_{10}$ x 70, $C_2^4:D_{10}$ 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, $C_2^4:D_{10}$ 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 160: $C_2^3\times F_5$ x 5690, $C_2^3\times D_{10}$ x 1345, $C_2^3:F_5$ x 210, $D_4\times F_5$ x 112, $C_{10}.C_4^2$ x 84, $D_4\times D_{10}$ x 84, $D_{10}:D_4$ x 84, $C_2^2:D_{20}$ x 84, $D_{10}.D_4$ x 84, $D_{10}.D_4$ x 42, $C_{10}:C_4^2$ x 42, $C_2^2.D_{20}$ x 42, $C_2^4\times C_{10}$ x 35, $C_2^4:C_{10}$ x 28, $C_2^4:D_5$ x 28, $C_2^3:C_{20}$ x 21, $C_2^3.D_{10}$ x 21, $C_2^3:D_{10}$ x 14, $C_{20}:C_2^3$ x 7, $C_2^2\times D_{20}$ x 7, $C_{20}:C_2^3$ x 7

Order 128: $C_2^5\times C_4$ x 70, $C_2^3.C_2^4$ x 56, $C_2^3.C_4^2$ x 42, $C_2^4.D_4$ x 21, $C_2^4.D_4$ x 21, $C_2^5:C_4$ x 21, $C_2^3:C_4^2$ x 21, $C_2^3\wr C_2$ x 16, $C_2^4:D_4$ x 7, $C_4^2:C_2^3$ x 7, $C_2^7$

Order 80: $C_2^2\times D_{10}$ x 5683, $C_2^2\times F_5$ x 5376, $C_2^3\times C_{10}$ x 344, $D_{10}:C_4$ x 140, $D_4\times D_5$ x 112, $C_{10}:D_4$ x 84, $D_{10}:C_4$ x 84, $D_4\times C_{10}$ x 42, $C_2\times D_{20}$ x 42, $C_4\times D_{10}$ x 42, $C_2^2:C_{20}$ x 42, $C_{10}.D_4$ x 42, $C_{20}:C_4$ x 28, $C_4\times F_5$ x 28, $C_2^2\times C_{20}$ x 7, $C_2^2.D_{10}$ x 7

Order 64: $C_2^4\times C_4$ x 1345, $C_2^4:C_4$ x 112, $C_2^3:D_4$ x 112, $C_2^3.D_4$ x 84, $C_2^3.D_4$ x 84, $C_2^2:C_4^2$ x 84, $C_4^2:C_2^2$ x 84, $C_2^6$ x 71, $C_2^4:C_4$ x 56, $C_2^2.C_4^2$ x 42, $D_4\times C_2^3$ x 7, $C_2^3.D_4$ x 7, $C_2^2\times C_4^2$ x 7

Order 40: $C_2\times D_{10}$ x 5327, $C_2\times F_5$ x 1100, $C_2^2\times C_{10}$ x 726, $C_5:D_4$ x 56, $D_{20}$ x 28, $C_4\times D_5$ x 28, $C_5\times D_4$ x 28, $C_2\times C_{20}$ x 21, $C_{10}:C_4$ x 21

Order 32: $C_2^3\times C_4$ x 5697, $C_2^5$ x 1380, $C_2^3:C_4$ x 378, $C_2^2\wr C_2$ x 224, $C_2^2\times D_4$ x 112, $C_4\times D_4$ x 112, $C_2.C_4^2$ x 84, $C_2^2.D_4$ x 42, $C_2\times C_4^2$ x 42

Order 20: $D_{10}$ x 1051, $C_2\times C_{10}$ x 350, $F_5$ x 44, $C_{20}$ x 7, $C_5:C_4$ x 7

Order 16: $C_2^4$ x 6027, $C_2^2\times C_4$ x 5432, $C_2^2:C_4$ x 308, $C_2\times D_4$ x 280, $C_4:C_4$ x 28, $C_4^2$ x 28

Order 10: $D_5$ x 37, $C_{10}$ x 36

Order 8: $C_2^3$ x 6053, $C_2\times C_4$ x 1170, $D_4$ x 112

Order 5: $C_5$

Order 4: $C_2^2$ x 1401, $C_4$ x 58

Order 2: $C_2$ x 73

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2560: $C_2^3\wr C_2\times F_5$

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

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

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

Order 320: $C_2^4\times F_5$ x 8, $C_2^4\times D_{10}$ x 4, $C_2^4:F_5$ x 3, $C_2^4:D_{10}$ x 2, $C_2^4:F_5$ x 2, $C_2^4:C_{20}$, $C_2^4.D_{10}$, $C_2^3.D_{20}$, $C_2^5\times C_{10}$, $C_{20}:C_2^4$, $D_{10}.C_2^4$, $D_{10}.C_2^4$, $D_{10}:C_4^2$, $C_2^5:C_{10}$, $C_2^4:D_{10}$, $C_2^4:D_{10}$, $C_2^3:D_{20}$, $C_2^4.D_{10}$, $(C_2\times D_{20}):C_4$, $D_{10}.C_4^2$, $C_{10}.(C_4\times D_4)$, $D_{10}:C_4^2$

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

Order 160: $C_2^3\times F_5$ x 15, $C_2^3\times D_{10}$ x 11, $C_2^3:F_5$ x 4, $C_2^4\times C_{10}$ x 2, $C_2^3:D_{10}$ x 2, $D_4\times D_{10}$ x 2, $D_{10}.D_4$ x 2, $C_{10}.C_4^2$, $C_{20}:C_2^3$, $C_2^2\times D_{20}$, $C_{20}:C_2^3$, $D_4\times F_5$, $D_{10}.D_4$, $C_{10}:C_4^2$, $C_2^4:C_{10}$, $C_2^3:C_{20}$, $C_2^4:D_5$, $C_2^3.D_{10}$, $D_{10}:D_4$, $C_2^2.D_{20}$, $C_2^2:D_{20}$

Order 128: $C_2^3\wr C_2$ x 4, $C_2^5\times C_4$ x 3, $C_2^5:C_4$ x 3, $C_2^7$, $C_2^4:D_4$, $C_4^2:C_2^3$, $C_2^3.C_4^2$, $C_2^3.C_2^4$, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^3:C_4^2$

Order 80: $C_2^2\times D_{10}$ x 21, $C_2^2\times F_5$ x 14, $C_2^3\times C_{10}$ x 5, $D_{10}:C_4$ x 3, $C_{10}:D_4$ x 2, $D_4\times D_5$ x 2, $C_4\times D_{10}$ x 2, $D_4\times C_{10}$, $C_2^2\times C_{20}$, $C_2^2.D_{10}$, $C_2\times D_{20}$, $C_{20}:C_4$, $C_4\times F_5$, $C_2^2:C_{20}$, $C_{10}.D_4$, $D_{10}:C_4$

Order 64: $C_2^4\times C_4$ x 8, $C_2^3:D_4$ x 6, $C_2^4:C_4$ x 5, $C_2^6$ x 5, $C_2^4:C_4$ x 4, $C_2^3.D_4$, $C_2^3.D_4$, $C_2^2:C_4^2$, $C_2^2.C_4^2$, $D_4\times C_2^3$, $C_4^2:C_2^2$, $C_2^3.D_4$, $C_2^2\times C_4^2$

Order 40: $C_2\times D_{10}$ x 19, $C_2\times F_5$ x 9, $C_2^2\times C_{10}$ x 7, $C_5:D_4$ x 2, $C_4\times D_5$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$, $D_{20}$, $C_5\times D_4$

Order 32: $C_2^3\times C_4$ x 16, $C_2^5$ x 13, $C_2^3:C_4$ x 9, $C_2^2\times D_4$ x 6, $C_2^2\wr C_2$ x 4, $C_4\times D_4$, $C_2^2.D_4$, $C_2\times C_4^2$, $C_2.C_4^2$

Order 20: $D_{10}$ x 11, $C_2\times C_{10}$ x 5, $F_5$ x 4, $C_{20}$, $C_5:C_4$

Order 16: $C_2^4$ x 26, $C_2^2\times C_4$ x 18, $C_2^2:C_4$ x 6, $C_2\times D_4$ x 6, $C_4:C_4$, $C_4^2$

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 8: $C_2^3$ x 26, $C_2\times C_4$ x 13, $D_4$ x 4

Order 5: $C_5$

Order 4: $C_2^2$ x 16, $C_4$ x 6

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2560: $C_2^3\wr C_2\times F_5$ ($C_1$)

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

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

Order 320: $C_2^4\times F_5$ ($D_4$) x 56, $C_2^3.D_{20}$ ($C_2\times C_4$) x 42, $D_{10}.C_4^2$ ($C_2^3$) x 42, $C_2^4:F_5$ ($C_2^3$) x 35, $C_2^4\times F_5$ ($C_2^3$) x 29, $C_2^4:D_{10}$ ($C_2\times C_4$) x 21, $C_2^3:D_{20}$ ($C_2\times C_4$) x 21, $C_2^4.D_{10}$ ($C_2^3$) x 21, $C_2^4:C_{20}$ ($C_2\times C_4$) x 7, $C_2^4.D_{10}$ ($C_2\times C_4$) x 7, $C_2^4\times D_{10}$ ($C_2^3$) x 7, $C_2^4\times D_{10}$ ($C_2\times C_4$) x 7, $C_{20}:C_2^4$ ($C_2^3$) x 7, $D_{10}.C_2^4$ ($C_2^3$) x 7, $D_{10}:C_4^2$ ($C_2^3$) x 7, $C_2^5:C_{10}$ ($C_2\times C_4$) x 7, $C_2^4:D_{10}$ ($C_2\times C_4$) x 7, $C_2^5\times C_{10}$ ($C_2\times C_4$)

Order 160: $C_2^3\times F_5$ ($C_2\times D_4$) x 168, $C_2^2.D_{20}$ ($C_2^2\times C_4$) x 42, $C_2^3\times D_{10}$ ($D_4:C_2$) x 28, $C_2^3\times D_{10}$ ($C_2\times D_4$) x 28, $C_2^3\times D_{10}$ ($C_2^2\times C_4$) x 21, $C_2^3:C_{20}$ ($C_2^2\times C_4$) x 21, $C_2^3.D_{10}$ ($C_2^2\times C_4$) x 21, $C_2^3\times F_5$ ($C_2^4$) x 16, $C_2^3:D_{10}$ ($C_2^2\times C_4$) x 14, $C_2^3\times D_{10}$ ($C_2^4$) x 8, $C_2^4\times C_{10}$ ($C_2^2\times C_4$) x 7, $C_{20}:C_2^3$ ($C_2^2\times C_4$) x 7, $C_2^2\times D_{20}$ ($C_2^2\times C_4$) x 7, $C_{20}:C_2^3$ ($C_2^4$) x 7

Order 128: $C_2^3\wr C_2$ ($F_5$)

Order 80: $C_2^2\times F_5$ ($C_2^2\wr C_2$) x 112, $C_2^2\times D_{10}$ ($C_4\times D_4$) x 84, $C_2^2\times F_5$ ($C_2^2\times D_4$) x 56, $C_2^2\times D_{10}$ ($D_4:C_2^2$) x 42, $C_2^2\times D_{10}$ ($C_2^2\times D_4$) x 42, $C_2^3\times C_{10}$ ($C_4\times D_4$) x 28, $C_2^3\times C_{10}$ ($C_2^3\times C_4$) x 8, $C_2^2\times D_{10}$ ($C_2^3\times C_4$) x 8, $C_2^2\times C_{20}$ ($C_2^3\times C_4$) x 7, $C_2^2.D_{10}$ ($C_2^3\times C_4$) x 7, $C_2^2\times D_{10}$ ($C_2^5$)

Order 64: $C_2^4:C_4$ ($C_2\times F_5$) x 7, $C_2^3:D_4$ ($C_2\times F_5$) x 7, $C_2^6$ ($C_2\times F_5$)

Order 40: $C_2\times D_{10}$ ($C_4^2:C_2^2$) x 84, $C_2\times F_5$ ($C_2^3:D_4$) x 56, $C_2^2\times C_{10}$ ($C_4^2:C_2^2$) x 42, $C_2\times D_{10}$ ($C_4^2:C_2^2$) x 42, $C_2\times D_{10}$ ($C_2^3:D_4$) x 28, $C_2\times D_{10}$ ($D_4:C_2^3$) x 7, $C_2\times D_{10}$ ($D_4\times C_2^3$) x 7, $C_2^2\times C_{10}$ ($C_2^4\times C_4$)

Order 32: $C_2^3:C_4$ ($C_2^2\times F_5$) x 21, $C_2^5$ ($C_2^2\times F_5$) x 7, $C_2^2\times D_4$ ($C_2^2\times F_5$) x 7

Order 20: $C_2\times C_{10}$ ($C_2^3.C_2^4$) x 28, $D_{10}$ ($C_2^3.C_2^4$) x 28, $D_{10}$ ($C_4^2:C_2^3$) x 21, $F_5$ ($C_2^3\wr C_2$) x 8, $C_2\times C_{10}$ ($C_4^2:C_2^3$) x 7, $D_{10}$ ($C_2^4:D_4$) x 7

Order 16: $C_2^4$ ($D_4\times F_5$) x 28, $C_2^4$ ($C_2^3\times F_5$) x 8, $C_2^2\times C_4$ ($C_2^3\times F_5$) x 7

Order 10: $C_{10}$ ($C_2^3.C_2^5$) x 7, $D_5$ ($C_2^3.C_2^5$) x 7, $D_5$ ($C_2^5:D_4$)

Order 8: $C_2^3$ ($D_{10}.C_2^4$) x 42, $C_2^3$ ($C_2^4\times F_5$)

Order 5: $C_5$ ($C_2^4.C_2^5$)

Order 4: $C_2^2$ ($C_2^2\wr C_2\times F_5$) x 28, $C_2^2$ ($D_{10}.C_2^5$) x 7

Order 2: $C_2$ ($C_2^3:D_4\times F_5$) x 7

Order 1: $C_1$ ($C_2^3\wr C_2\times F_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2560: $C_2^3\wr C_2\times F_5$ ($C_1$)

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

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

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

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

Order 128: $C_2^3\wr C_2$ ($F_5$)

Order 80: $C_2^3\times C_{10}$ ($C_2^3\times C_4$) x 2, $C_2^2\times D_{10}$ ($C_2^3\times C_4$) x 2, $C_2^2\times F_5$ ($C_2^2\times D_4$) x 2, $C_2^3\times C_{10}$ ($C_4\times D_4$), $C_2^2\times D_{10}$ ($C_2^5$), $C_2^2\times D_{10}$ ($D_4:C_2^2$), $C_2^2\times D_{10}$ ($C_2^2\times D_4$), $C_2^2\times D_{10}$ ($C_4\times D_4$), $C_2^2\times F_5$ ($C_2^2\wr C_2$), $C_2^2\times C_{20}$ ($C_2^3\times C_4$), $C_2^2.D_{10}$ ($C_2^3\times C_4$)

Order 64: $C_2^4:C_4$ ($C_2\times F_5$), $C_2^6$ ($C_2\times F_5$), $C_2^3:D_4$ ($C_2\times F_5$)

Order 40: $C_2\times F_5$ ($C_2^3:D_4$) x 2, $C_2^2\times C_{10}$ ($C_2^4\times C_4$), $C_2^2\times C_{10}$ ($C_4^2:C_2^2$), $C_2\times D_{10}$ ($D_4:C_2^3$), $C_2\times D_{10}$ ($D_4\times C_2^3$), $C_2\times D_{10}$ ($C_4^2:C_2^2$), $C_2\times D_{10}$ ($C_2^3:D_4$), $C_2\times D_{10}$ ($C_4^2:C_2^2$)

Order 32: $C_2^5$ ($C_2^2\times F_5$), $C_2^2\times D_4$ ($C_2^2\times F_5$), $C_2^3:C_4$ ($C_2^2\times F_5$)

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

Order 16: $C_2^4$ ($C_2^3\times F_5$) x 2, $C_2^4$ ($D_4\times F_5$), $C_2^2\times C_4$ ($C_2^3\times F_5$)

Order 10: $C_{10}$ ($C_2^3.C_2^5$), $D_5$ ($C_2^3.C_2^5$), $D_5$ ($C_2^5:D_4$)

Order 8: $C_2^3$ ($C_2^4\times F_5$), $C_2^3$ ($D_{10}.C_2^4$)

Order 5: $C_5$ ($C_2^4.C_2^5$)

Order 4: $C_2^2$ ($D_{10}.C_2^5$), $C_2^2$ ($C_2^2\wr C_2\times F_5$)

Order 2: $C_2$ ($C_2^3:D_4\times F_5$)

Order 1: $C_1$ ($C_2^3\wr C_2\times F_5$)

Series

Derived series

$C_2^3\wr C_2\times F_5$

$\rhd$

$C_2^2\times C_{10}$

$\rhd$

$C_1$

Chief series

$C_2^3\wr C_2\times F_5$

$\rhd$

$F_5\times C_2^6$

$\rhd$

$F_5\times C_2^5$

$\rhd$

$C_2^4\times F_5$

$\rhd$

$C_2^3\times F_5$

$\rhd$

$C_2^2\times D_{10}$

$\rhd$

$C_2^2\times C_{10}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_2^3\wr C_2\times F_5$

$\rhd$

$C_2^2\times C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^3\wr C_2$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $220 \times 220$ 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 $176 \times 176$ rational character table (warning: may be slow to load).