Abstract group 1280.1114138: $C_2^3:D_4\times F_5$

• Label: 1280.1114138
• Order: \( 2^{8} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{17} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $80$
• Rank: $5$

Group information

Description:	$C_2^3:D_4\times F_5$
Order:	\(1280\)\(\medspace = 2^{8} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	Group of order \(1966080\)\(\medspace = 2^{17} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 8, $C_5$
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	239	784	4	156	96	1280
Conjugacy classes	1	43	68	1	21	6	140
Divisions	1	43	40	1	21	6	112
Autjugacy classes	1	9	10	1	4	1	26

Dimension	1	2	4	8
Irr. complex chars.	64	48	16	12	140
Irr. rational chars.	32	40	28	12	112

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$80$
Rank:	$5$
Inequivalent generating 5-tuples:	$519966720$

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:	${\langle a, b, c, d, e, f \mid a^{4}=b^{10}=c^{4}=d^{2}=e^{2}=f^{2}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(2,3,5,4)(8,9)(10,12)(11,13)(14,15), (6,7)(9,12)(13,15), (9,13)(12,15), (2,4,5,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 5 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 10 \\ 10 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 10 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	$C_2$ $\, \times\, $ $F_5$ $\, \times\, $ $(C_2^2\wr C_2)$
Semidirect product:	$(C_2^3\times F_5)$ $\,\rtimes\,$ $D_4$	$C_2^3$ $\,\rtimes\,$ $(D_4\times F_5)$	$(F_5\times C_2^5)$ $\,\rtimes\,$ $C_2$	$(C_2^5:F_5)$ $\,\rtimes\,$ $C_2$	all 75
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_5$ . $(C_2^4:D_4)$	$(D_4\times D_{10})$ . $C_2^3$	$(D_{10}.D_4)$ . $C_2^3$	$D_{10}$ . $(D_4\times C_2^3)$	all 42
Aut. group:	$\Aut(C_{20}:C_8)$	$\Aut(C_{40}:C_4)$	$\Aut(C_{40}:C_4)$	$\Aut(C_{20}.D_4)$	all 19

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{20}\Z)$.

Homology

Abelianization:	$C_{2}^{4} \times C_{4} $
Schur multiplier:	$C_{2}^{11}$
Commutator length:	$1$

Subgroups

There are 44374 subgroups in 8582 conjugacy classes, 1112 normal (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2^3\times F_5$
Commutator:	$G' \simeq$ $C_2\times C_{10}$	$G/G' \simeq$ $C_2^4\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^4\times F_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^5:C_{10}$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $C_2^3:D_4\times F_5$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2^3\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.C_2^5$
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 1280: $C_2^3:D_4\times F_5$

Order 640: $C_2^2\wr C_2\times F_5$ x 16, $D_{10}.C_2^5$ x 3, $(D_4\times D_{10}):C_4$ x 3, $D_{10}.(C_4\times D_4)$ x 3, $C_2\times D_{10}:C_4^2$ x 3, $F_5\times C_2^5$, $C_2^5:D_{10}$, $C_2^5:F_5$

Order 320: $D_{10}.C_2^4$ x 48, $C_2^4\times F_5$ x 39, $(C_2\times D_{20}):C_4$ x 24, $C_{10}.(C_4\times D_4)$ x 24, $D_{10}:C_4^2$ x 24, $C_2^4:D_{10}$ x 16, $C_2^4:F_5$ x 9, $C_2^4:F_5$ x 8, $D_{10}.C_4^2$ x 6, $C_{20}:C_2^4$ x 3, $D_{10}.C_2^4$ x 3, $D_{10}:C_4^2$ x 3, $C_2^4:D_{10}$ x 3, $C_2^3:D_{20}$ x 3, $C_2^4.D_{10}$ x 3, $C_2^4\times D_{10}$, $C_2^5:C_{10}$, $C_2^4:D_{10}$

Order 256: $C_2^3.C_2^5$

Order 160: $C_2^3\times F_5$ x 356, $D_4\times F_5$ x 96, $C_2^3:F_5$ x 72, $D_4\times D_{10}$ x 48, $C_2^3\times D_{10}$ x 39, $C_{10}.C_4^2$ x 24, $D_{10}.D_4$ x 24, $C_{10}:C_4^2$ x 24, $D_{10}:D_4$ x 24, $C_2^2:D_{20}$ x 24, $D_{10}.D_4$ x 24, $C_2^4:C_{10}$ x 8, $C_2^4:D_5$ x 8, $C_2^3:D_{10}$ x 6, $C_2^2.D_{20}$ x 6, $C_{20}:C_2^3$ x 3, $C_2^2\times D_{20}$ x 3, $C_{20}:C_2^3$ x 3, $C_2^3:C_{20}$ x 3, $C_2^3.D_{10}$ x 3, $C_2^4\times C_{10}$

Order 128: $C_2^3.C_2^4$ x 16, $C_4^2:C_2^3$ x 3, $C_2^4.D_4$ x 3, $C_2^4.D_4$ x 3, $C_2^3:C_4^2$ x 3, $C_2^5\times C_4$, $C_2^4:D_4$, $C_2^5:C_4$

Order 80: $C_2^2\times F_5$ x 732, $C_2^2\times D_{10}$ x 353, $D_4\times D_5$ x 96, $D_{10}:C_4$ x 72, $C_{10}:D_4$ x 48, $D_4\times C_{10}$ x 24, $C_2\times D_{20}$ x 24, $C_4\times D_{10}$ x 24, $C_{20}:C_4$ x 24, $C_4\times F_5$ x 24, $D_{10}:C_4$ x 24, $C_2^3\times C_{10}$ x 20, $C_2^2:C_{20}$ x 12, $C_{10}.D_4$ x 12, $C_2^2\times C_{20}$ x 3, $C_2^2.D_{10}$ x 3

Order 64: $C_4^2:C_2^2$ x 48, $C_2^4\times C_4$ x 39, $C_2^3.D_4$ x 24, $C_2^3.D_4$ x 24, $C_2^2:C_4^2$ x 24, $C_2^3:D_4$ x 24, $C_2^4:C_4$ x 12, $C_2^4:C_4$ x 8, $C_2^2.C_4^2$ x 6, $D_4\times C_2^3$ x 3, $C_2^3.D_4$ x 3, $C_2^2\times C_4^2$ x 3, $C_2^6$

Order 40: $C_2\times D_{10}$ x 708, $C_2\times F_5$ x 324, $C_2^2\times C_{10}$ x 95, $C_5:D_4$ x 48, $D_{20}$ x 24, $C_4\times D_5$ x 24, $C_5\times D_4$ x 24, $C_2\times C_{20}$ x 12, $C_{10}:C_4$ x 12

Order 32: $C_2^3\times C_4$ x 359, $C_2^3:C_4$ x 108, $C_4\times D_4$ x 96, $C_2^2\wr C_2$ x 64, $C_2^2\times D_4$ x 60, $C_2^5$ x 40, $C_2^2.D_4$ x 24, $C_2\times C_4^2$ x 24, $C_2.C_4^2$ x 24

Order 20: $D_{10}$ x 294, $C_2\times C_{10}$ x 97, $F_5$ x 28, $C_{20}$ x 6, $C_5:C_4$ x 6

Order 16: $C_2^2\times C_4$ x 762, $C_2^4$ x 373, $C_2\times D_4$ x 192, $C_2^2:C_4$ x 120, $C_4:C_4$ x 24, $C_4^2$ x 24

Order 10: $D_5$ x 22, $C_{10}$ x 21

Order 8: $C_2^3$ x 803, $C_2\times C_4$ x 372, $D_4$ x 96

Order 5: $C_5$

Order 4: $C_2^2$ x 391, $C_4$ x 40

Order 2: $C_2$ x 43

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1280: $C_2^3:D_4\times F_5$

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

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

Order 256: $C_2^3.C_2^5$

Order 160: $C_2^3\times F_5$ x 14, $C_2^3\times D_{10}$ x 7, $C_2^3:F_5$ x 6, $D_4\times D_{10}$ x 4, $C_2^3:D_{10}$ x 2, $D_{10}.D_4$ x 2, $C_{10}:C_4^2$ x 2, $D_{10}.D_4$ x 2, $C_{10}.C_4^2$, $C_2^4\times C_{10}$, $C_{20}:C_2^3$, $C_2^2\times D_{20}$, $C_{20}:C_2^3$, $D_4\times F_5$, $C_2^4:C_{10}$, $C_2^3:C_{20}$, $C_2^4:D_5$, $C_2^3.D_{10}$, $D_{10}:D_4$, $C_2^2.D_{20}$, $C_2^2:D_{20}$

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

Order 80: $C_2^2\times D_{10}$ x 21, $C_2^2\times F_5$ x 19, $C_2^3\times C_{10}$ x 4, $C_{10}:D_4$ x 4, $C_4\times D_{10}$ x 4, $D_4\times D_5$ x 3, $D_{10}:C_4$ x 3, $D_4\times C_{10}$ x 2, $C_2\times D_{20}$ x 2, $C_2^2\times C_{20}$, $C_2^2.D_{10}$, $C_{20}:C_4$, $C_4\times F_5$, $C_2^2:C_{20}$, $C_{10}.D_4$, $D_{10}:C_4$

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

Order 40: $C_2\times D_{10}$ x 29, $C_2\times F_5$ x 13, $C_2^2\times C_{10}$ x 8, $C_4\times D_5$ x 3, $C_2\times C_{20}$ x 2, $C_5:D_4$ x 2, $C_{10}:C_4$ x 2, $D_{20}$, $C_5\times D_4$

Order 32: $C_2^3\times C_4$ x 15, $C_2^3:C_4$ x 11, $C_2^5$ x 8, $C_2^2\times D_4$ x 8, $C_2^2\wr C_2$ x 4, $C_2^2.D_4$ x 2, $C_2\times C_4^2$ x 2, $C_4\times D_4$, $C_2.C_4^2$

Order 20: $D_{10}$ x 18, $C_2\times C_{10}$ x 8, $F_5$ x 4, $C_{20}$, $C_5:C_4$

Order 16: $C_2^4$ x 25, $C_2^2\times C_4$ x 25, $C_2\times D_4$ x 11, $C_2^2:C_4$ x 6, $C_4:C_4$, $C_4^2$

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

Order 8: $C_2^3$ x 37, $C_2\times C_4$ x 20, $D_4$ x 4

Order 5: $C_5$

Order 4: $C_2^2$ x 26, $C_4$ x 6

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1280: $C_2^3:D_4\times F_5$ ($C_1$)

Order 640: $C_2^2\wr C_2\times F_5$ ($C_2$) x 16, $D_{10}.C_2^5$ ($C_2$) x 3, $(D_4\times D_{10}):C_4$ ($C_2$) x 3, $D_{10}.(C_4\times D_4)$ ($C_2$) x 3, $C_2\times D_{10}:C_4^2$ ($C_2$) x 3, $F_5\times C_2^5$ ($C_2$), $C_2^5:D_{10}$ ($C_2$), $C_2^5:F_5$ ($C_2$)

Order 320: $D_{10}.C_2^4$ ($C_2^2$) x 24, $(C_2\times D_{20}):C_4$ ($C_2^2$) x 24, $C_{10}.(C_4\times D_4)$ ($C_2^2$) x 24, $D_{10}:C_4^2$ ($C_2^2$) x 24, $C_2^4\times F_5$ ($C_2^2$) x 15, $C_2^4:F_5$ ($C_2^2$) x 9, $C_2^4:D_{10}$ ($C_2^2$) x 8, $C_2^4:D_{10}$ ($C_4$) x 8, $C_2^4:F_5$ ($C_2^2$) x 8, $D_{10}.C_4^2$ ($C_2^2$) x 6, $C_{20}:C_2^4$ ($C_2^2$) x 3, $D_{10}.C_2^4$ ($C_2^2$) x 3, $D_{10}:C_4^2$ ($C_2^2$) x 3, $C_2^4:D_{10}$ ($C_4$) x 3, $C_2^3:D_{20}$ ($C_4$) x 3, $C_2^4.D_{10}$ ($C_2^2$) x 3, $C_2^4\times D_{10}$ ($C_2^2$), $C_2^5:C_{10}$ ($C_4$), $C_2^4:D_{10}$ ($C_4$)

Order 160: $C_2^3\times F_5$ ($C_2^3$) x 36, $C_2^3:F_5$ ($C_2^3$) x 36, $C_{10}.C_4^2$ ($C_2^3$) x 24, $C_2^3\times F_5$ ($D_4$) x 24, $D_{10}:D_4$ ($C_2\times C_4$) x 24, $C_2^2:D_{20}$ ($C_2\times C_4$) x 24, $D_4\times D_{10}$ ($C_2^3$) x 12, $D_4\times D_{10}$ ($C_2\times C_4$) x 12, $D_{10}.D_4$ ($C_2^3$) x 12, $C_{10}:C_4^2$ ($C_2^3$) x 12, $D_{10}.D_4$ ($C_2^3$) x 12, $D_{10}.D_4$ ($C_2\times C_4$) x 12, $C_2^3\times D_{10}$ ($C_2^3$) x 8, $C_2^4:C_{10}$ ($C_2\times C_4$) x 8, $C_2^4:D_5$ ($C_2\times C_4$) x 8, $C_2^3\times D_{10}$ ($C_2\times C_4$) x 7, $C_2^3:D_{10}$ ($C_2\times C_4$) x 6, $C_2^2.D_{20}$ ($C_2\times C_4$) x 6, $C_{20}:C_2^3$ ($C_2\times C_4$) x 3, $C_2^2\times D_{20}$ ($C_2\times C_4$) x 3, $C_{20}:C_2^3$ ($C_2^3$) x 3, $C_2^3:C_{20}$ ($C_2\times C_4$) x 3, $C_2^3.D_{10}$ ($C_2\times C_4$) x 3, $C_2^4\times C_{10}$ ($C_2\times C_4$)

Order 80: $C_2^2\times F_5$ ($C_2\times D_4$) x 72, $C_2^2\times D_{10}$ ($C_2^2\times C_4$) x 24, $C_{10}:D_4$ ($C_2^2\times C_4$) x 24, $D_{10}:C_4$ ($C_2^2\times C_4$) x 24, $C_2^2\times F_5$ ($C_2^4$) x 16, $C_2^2\times D_{10}$ ($D_4:C_2$) x 12, $C_2^2\times D_{10}$ ($C_2\times D_4$) x 12, $D_4\times C_{10}$ ($C_2^2\times C_4$) x 12, $C_2\times D_{20}$ ($C_2^2\times C_4$) x 12, $C_2^2:C_{20}$ ($C_2^2\times C_4$) x 12, $C_{10}.D_4$ ($C_2^2\times C_4$) x 12, $C_2^2\times D_{10}$ ($C_2^4$) x 9, $C_2^3\times C_{10}$ ($C_2^2\times C_4$) x 8, $C_4\times D_{10}$ ($C_2^4$) x 6, $C_4\times D_{10}$ ($C_2^2\times C_4$) x 6, $C_2^2\times C_{20}$ ($C_2^2\times C_4$) x 3, $C_2^2.D_{10}$ ($C_2^2\times C_4$) x 3

Order 64: $C_2^3:D_4$ ($F_5$)

Order 40: $C_2\times D_{10}$ ($C_4\times D_4$) x 36, $C_2\times F_5$ ($C_2^2\times D_4$) x 24, $C_2\times D_{10}$ ($D_4:C_2^2$) x 18, $C_2\times D_{10}$ ($C_2^2\times D_4$) x 18, $C_2\times F_5$ ($C_2^2\wr C_2$) x 16, $C_2^2\times C_{10}$ ($C_4\times D_4$) x 12, $C_2^2\times C_{10}$ ($C_2^3\times C_4$) x 9, $C_2\times D_{10}$ ($C_2^3\times C_4$) x 9, $C_2\times C_{20}$ ($C_2^3\times C_4$) x 6, $C_{10}:C_4$ ($C_2^3\times C_4$) x 6, $C_2\times D_{10}$ ($C_2^5$)

Order 32: $C_2^2\wr C_2$ ($C_2\times F_5$) x 8, $C_2^2\times D_4$ ($C_2\times F_5$) x 3, $C_2^3:C_4$ ($C_2\times F_5$) x 3, $C_2^5$ ($C_2\times F_5$)

Order 20: $C_2\times C_{10}$ ($C_4^2:C_2^2$) x 18, $D_{10}$ ($C_4^2:C_2^2$) x 18, $D_{10}$ ($C_4^2:C_2^2$) x 12, $F_5$ ($C_2^3:D_4$) x 8, $D_{10}$ ($C_2^3:D_4$) x 4, $D_{10}$ ($D_4:C_2^3$) x 3, $D_{10}$ ($D_4\times C_2^3$) x 3, $C_2\times C_{10}$ ($C_2^4\times C_4$)

Order 16: $C_2^2:C_4$ ($C_2^2\times F_5$) x 12, $C_2\times D_4$ ($C_2^2\times F_5$) x 12, $C_2^4$ ($C_2^2\times F_5$) x 8, $C_2^2\times C_4$ ($C_2^2\times F_5$) x 3

Order 10: $C_{10}$ ($C_2^3.C_2^4$) x 4, $D_5$ ($C_2^3.C_2^4$) x 4, $C_{10}$ ($C_4^2:C_2^3$) x 3, $D_5$ ($C_4^2:C_2^3$) x 3, $D_5$ ($C_2^4:D_4$)

Order 8: $C_2^3$ ($D_4\times F_5$) x 12, $C_2^3$ ($C_2^3\times F_5$) x 9, $C_2\times C_4$ ($C_2^3\times F_5$) x 6

Order 5: $C_5$ ($C_2^3.C_2^5$)

Order 4: $C_2^2$ ($D_{10}.C_2^4$) x 18, $C_2^2$ ($C_2^4\times F_5$)

Order 2: $C_2$ ($C_2^2\wr C_2\times F_5$) x 4, $C_2$ ($D_{10}.C_2^5$) x 3

Order 1: $C_1$ ($C_2^3:D_4\times F_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1280: $C_2^3:D_4\times F_5$ ($C_1$)

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

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

Order 160: $C_2^3\times F_5$ ($C_2^3$) x 7, $C_2^3\times D_{10}$ ($C_2^3$) x 3, $C_2^3:F_5$ ($C_2^3$) x 3, $C_2^3\times D_{10}$ ($C_2\times C_4$) x 2, $C_2^3:D_{10}$ ($C_2\times C_4$) x 2, $C_{10}.C_4^2$ ($C_2^3$), $C_2^4\times C_{10}$ ($C_2\times C_4$), $C_2^3\times F_5$ ($D_4$), $C_{20}:C_2^3$ ($C_2\times C_4$), $D_4\times D_{10}$ ($C_2^3$), $D_4\times D_{10}$ ($C_2\times C_4$), $C_2^2\times D_{20}$ ($C_2\times C_4$), $C_{20}:C_2^3$ ($C_2^3$), $D_{10}.D_4$ ($C_2^3$), $C_{10}:C_4^2$ ($C_2^3$), $C_2^4:C_{10}$ ($C_2\times C_4$), $C_2^3:C_{20}$ ($C_2\times C_4$), $C_2^4:D_5$ ($C_2\times C_4$), $C_2^3.D_{10}$ ($C_2\times C_4$), $D_{10}:D_4$ ($C_2\times C_4$), $C_2^2.D_{20}$ ($C_2\times C_4$), $C_2^2:D_{20}$ ($C_2\times C_4$), $D_{10}.D_4$ ($C_2^3$), $D_{10}.D_4$ ($C_2\times C_4$)

Order 80: $C_2^2\times D_{10}$ ($C_2^2\times C_4$) x 5, $C_2^2\times F_5$ ($C_2^4$) x 4, $C_2^3\times C_{10}$ ($C_2^2\times C_4$) x 3, $C_2^2\times D_{10}$ ($C_2^4$) x 3, $C_2^2\times F_5$ ($C_2\times D_4$) x 3, $C_{10}:D_4$ ($C_2^2\times C_4$) x 2, $C_2^2\times D_{10}$ ($D_4:C_2$), $C_2^2\times D_{10}$ ($C_2\times D_4$), $D_4\times C_{10}$ ($C_2^2\times C_4$), $C_2^2\times C_{20}$ ($C_2^2\times C_4$), $C_2^2.D_{10}$ ($C_2^2\times C_4$), $C_2\times D_{20}$ ($C_2^2\times C_4$), $C_4\times D_{10}$ ($C_2^4$), $C_4\times D_{10}$ ($C_2^2\times C_4$), $C_2^2:C_{20}$ ($C_2^2\times C_4$), $C_{10}.D_4$ ($C_2^2\times C_4$), $D_{10}:C_4$ ($C_2^2\times C_4$)

Order 64: $C_2^3:D_4$ ($F_5$)

Order 40: $C_2^2\times C_{10}$ ($C_2^3\times C_4$) x 3, $C_2\times D_{10}$ ($C_2^3\times C_4$) x 3, $C_2\times D_{10}$ ($D_4:C_2^2$) x 2, $C_2\times D_{10}$ ($C_2^2\times D_4$) x 2, $C_2\times D_{10}$ ($C_4\times D_4$) x 2, $C_2\times F_5$ ($C_2^2\times D_4$) x 2, $C_2\times C_{20}$ ($C_2^3\times C_4$), $C_{10}:C_4$ ($C_2^3\times C_4$), $C_2^2\times C_{10}$ ($C_4\times D_4$), $C_2\times D_{10}$ ($C_2^5$), $C_2\times F_5$ ($C_2^2\wr C_2$)

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

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

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

Order 10: $C_{10}$ ($C_4^2:C_2^3$), $C_{10}$ ($C_2^3.C_2^4$), $D_5$ ($C_4^2:C_2^3$), $D_5$ ($C_2^4:D_4$), $D_5$ ($C_2^3.C_2^4$)

Order 8: $C_2^3$ ($C_2^3\times F_5$) x 3, $C_2^3$ ($D_4\times F_5$), $C_2\times C_4$ ($C_2^3\times F_5$)

Order 5: $C_5$ ($C_2^3.C_2^5$)

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

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

Order 1: $C_1$ ($C_2^3:D_4\times F_5$)

Series

Derived series

$C_2^3:D_4\times F_5$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_1$

Chief series

$C_2^3:D_4\times F_5$

$\rhd$

$D_{10}.(C_4\times D_4)$

$\rhd$

$C_2^4.D_{10}$

$\rhd$

$C_2^3:C_{20}$

$\rhd$

$C_2^3:C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3:D_4\times F_5$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^3:D_4$

Supergroups

This group is a maximal subgroup of 14 larger groups in the database.

This group is a maximal quotient of 7 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 $112 \times 112$ rational character table.