Abstract group 640.19528: $D_8:C_2\times F_5$

• Label: 640.19528
• Order: \( 2^{7} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $13$
• Trans deg.: $40$
• Rank: $4$

Group information

Description:	$D_8:C_2\times F_5$
Order:	\(640\)\(\medspace = 2^{7} \cdot 5 \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	$C_2\times F_5\times D_4^2$, of order \(2560\)\(\medspace = 2^{9} \cdot 5 \)
Composition factors:	$C_2$ x 7, $C_5$
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	8	10	20	40
Elements	1	95	288	4	128	60	32	32	640
Conjugacy classes	1	11	24	1	8	5	3	2	55
Divisions	1	11	15	1	6	5	3	2	44
Autjugacy classes	1	9	20	1	4	4	3	1	43

Dimension	1	2	4	8	16
Irr. complex chars.	32	8	12	2	1	55
Irr. rational chars.	16	12	12	3	1	44

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$40$
Rank:	$4$
Inequivalent generating quadruples:	$19998720$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	16	16
Arbitrary	8	8	8

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{4}=d^{40}=[a,c]=[b,c]=1, b^{a}=bd^{20}, d^{a}=d^{21}, d^{b}=d^{39}, d^{c}=d^{17} \rangle$
Permutation group:	Degree $13$ $\langle(1,2)(3,6)(4,5)(7,8)(10,11,13,12), (2,5)(4,8)(6,7), (1,3)(2,6)(4,8)(5,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 8 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 19 & 15 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/40\Z)$
Transitive group:	40T523				more information
Direct product:	$F_5$ $\, \times\, $ $(D_8:C_2)$
Semidirect product:	$(D_8\times F_5)$ $\,\rtimes\,$ $C_2$ (2)	$D_8$ $\,\rtimes\,$ $(C_2\times F_5)$ (2)	$(D_4:D_{10})$ $\,\rtimes\,$ $C_4$	$(D_{40}:C_4)$ $\,\rtimes\,$ $C_2$ (2)	all 56
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4\times F_5)$ . $D_4$ (2)	$C_4$ . $(D_4\times F_5)$	$(D_8:D_{10})$ . $C_2$	$D_{10}$ . $(C_4\times D_4)$	all 47
Aut. group:	$\Aut(D_{40})$	$\Aut(C_5:Q_{16})$	$\Aut(C_{40}:C_4)$	$\Aut(C_{40}:C_4)$	all 6

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 2768 subgroups in 548 conjugacy classes, 162 normal (80 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{10}.C_2^4$
Commutator:	$G' \simeq$ $C_{20}$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^3\times F_5$
Fitting:	$\operatorname{Fit} \simeq$ $D_8:C_{10}$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $D_8:C_2\times F_5$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{10}$	$G/\operatorname{soc} \simeq$ $C_4^2:C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.D_4$
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 640: $D_8:C_2\times F_5$

Order 320: $\SD_{16}:F_5$ x 2, $\SD_{16}\times F_5$ x 2, $D_{40}:C_4$ x 2, $D_8\times F_5$ x 2, $D_{10}.C_2^4$, $D_{10}.C_2^4$, $D_8:D_{10}$, $D_{20}:(C_2\times C_4)$, $(C_2\times D_{20}):C_4$, $\OD_{16}:F_5$, $\OD_{16}\times F_5$

Order 160: $D_4\times F_5$ x 10, $D_{20}:C_4$ x 6, $Q_8:F_5$ x 2, $C_{40}:C_4$ x 2, $C_{40}:C_4$ x 2, $C_{40}:C_4$ x 2, $C_8\times F_5$ x 2, $Q_8\times F_5$ x 2, $D_{10}.C_2^3$ x 2, $C_{10}:C_4^2$ x 2, $C_8:D_{10}$ x 2, $D_5\times \SD_{16}$ x 2, $C_8:D_{10}$ x 2, $D_5\times D_8$ x 2, $C_2^3\times F_5$, $D_4:D_{10}$, $D_4\times D_{10}$, $C_2^3:F_5$, $D_{10}.D_4$, $D_5:\OD_{16}$, $D_8:C_{10}$, $D_4:D_{10}$, $D_{20}:C_2^2$, $C_8:D_{10}$, $D_5\times \OD_{16}$

Order 128: $C_4^2.D_4$

Order 80: $C_2^2\times F_5$ x 13, $D_4\times D_5$ x 8, $D_{10}:C_4$ x 7, $C_{20}:C_4$ x 7, $C_4\times F_5$ x 7, $C_5:D_8$ x 4, $D_{40}$ x 2, $C_{40}:C_2$ x 2, $C_{40}:C_2$ x 2, $D_4:D_5$ x 2, $C_8\times D_5$ x 2, $D_{20}:C_2$ x 2, $C_4\times D_{10}$ x 2, $D_5:C_8$ x 2, $C_5\times \SD_{16}$ x 2, $C_5\times D_8$ x 2, $Q_8:D_5$ x 2, $C_5:\SD_{16}$ x 2, $C_2^2\times D_{10}$, $D_4:C_{10}$, $D_4\times C_{10}$, $C_{10}:D_4$, $D_{20}:C_2$, $Q_8\times D_5$, $C_2\times D_{20}$, $C_5:\OD_{16}$, $C_5:\OD_{16}$, $C_5\times \OD_{16}$, $C_5:\OD_{16}$

Order 64: $D_8:C_4$ x 2, $\SD_{16}:C_4$ x 2, $C_4\times \SD_{16}$ x 2, $C_4\times D_8$ x 2, $C_2^3.D_4$, $C_2^3.D_4$, $C_4\times \OD_{16}$, $C_8:C_2^3$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $\OD_{16}:C_4$

Order 40: $C_2\times F_5$ x 25, $C_2\times D_{10}$ x 11, $C_5:D_4$ x 7, $C_4\times D_5$ x 7, $D_{20}$ x 5, $C_5\times D_4$ x 5, $C_2\times C_{20}$ x 2, $C_{10}:C_4$ x 2, $C_5:Q_8$ x 2, $C_5:C_8$ x 2, $C_{40}$ x 2, $C_5:C_8$ x 2, $C_2^2\times C_{10}$, $C_5\times Q_8$

Order 32: $C_4\times D_4$ x 10, $D_8:C_2$ x 8, $D_4:C_4$ x 6, $C_2\times \SD_{16}$ x 2, $C_8:C_4$ x 2, $C_2\times D_8$ x 2, $C_2\times \OD_{16}$ x 2, $C_4\times C_8$ x 2, $C_4\times Q_8$ x 2, $C_4^2:C_2$ x 2, $C_2\times C_4^2$ x 2, $C_8:C_4$ x 2, $C_8:C_4$ x 2, $Q_8:C_4$ x 2, $D_4:C_2^2$, $C_2^2\times D_4$, $C_2^3\times C_4$, $C_2^2.D_4$, $C_2^3:C_4$

Order 20: $D_{10}$ x 19, $F_5$ x 9, $C_2\times C_{10}$ x 6, $C_{20}$ x 3, $C_5:C_4$ x 3

Order 16: $C_2^2\times C_4$ x 15, $C_2\times D_4$ x 11, $\SD_{16}$ x 8, $D_8$ x 8, $C_4:C_4$ x 7, $C_2^2:C_4$ x 7, $C_4^2$ x 7, $\OD_{16}$ x 6, $D_4:C_2$ x 6, $C_2\times C_8$ x 4, $C_2^4$, $C_2\times Q_8$

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

Order 8: $C_2\times C_4$ x 36, $D_4$ x 17, $C_2^3$ x 12, $C_8$ x 6, $Q_8$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 25, $C_4$ x 15

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 640: $D_8:C_2\times F_5$

Order 320: $D_{10}.C_2^4$, $D_{10}.C_2^4$, $D_8:D_{10}$, $D_{20}:(C_2\times C_4)$, $(C_2\times D_{20}):C_4$, $\SD_{16}:F_5$, $\SD_{16}\times F_5$, $D_{40}:C_4$, $D_8\times F_5$, $\OD_{16}:F_5$, $\OD_{16}\times F_5$

Order 160: $D_4\times F_5$ x 8, $D_{20}:C_4$ x 3, $Q_8\times F_5$ x 2, $D_{10}.C_2^3$ x 2, $C_{10}:C_4^2$ x 2, $Q_8:F_5$, $C_{40}:C_4$, $C_{40}:C_4$, $C_{40}:C_4$, $C_8\times F_5$, $C_2^3\times F_5$, $D_4:D_{10}$, $D_4\times D_{10}$, $C_2^3:F_5$, $D_{10}.D_4$, $D_5:\OD_{16}$, $D_8:C_{10}$, $D_4:D_{10}$, $D_{20}:C_2^2$, $C_8:D_{10}$, $D_5\times \SD_{16}$, $C_8:D_{10}$, $D_5\times D_8$, $C_8:D_{10}$, $D_5\times \OD_{16}$

Order 128: $C_4^2.D_4$

Order 80: $C_2^2\times F_5$ x 8, $C_4\times F_5$ x 7, $D_4\times D_5$ x 6, $C_{20}:C_4$ x 6, $D_{10}:C_4$ x 5, $D_4:D_5$ x 2, $D_{20}:C_2$ x 2, $C_4\times D_{10}$ x 2, $C_5:D_8$ x 2, $D_{40}$, $C_{40}:C_2$, $C_2^2\times D_{10}$, $C_{40}:C_2$, $D_4:C_{10}$, $D_4\times C_{10}$, $C_{10}:D_4$, $D_{20}:C_2$, $Q_8\times D_5$, $C_8\times D_5$, $C_2\times D_{20}$, $C_5:\OD_{16}$, $C_5:\OD_{16}$, $D_5:C_8$, $C_5\times \SD_{16}$, $C_5\times D_8$, $C_5\times \OD_{16}$, $Q_8:D_5$, $C_5:\SD_{16}$, $C_5:\OD_{16}$

Order 64: $C_2^3.D_4$, $C_2^3.D_4$, $C_4\times \OD_{16}$, $C_8:C_2^3$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $D_8:C_4$, $\SD_{16}:C_4$, $C_4\times \SD_{16}$, $C_4\times D_8$, $\OD_{16}:C_4$

Order 40: $C_2\times F_5$ x 14, $C_4\times D_5$ x 7, $C_2\times D_{10}$ x 7, $C_5:D_4$ x 5, $D_{20}$ x 4, $C_5\times D_4$ x 4, $C_2\times C_{20}$ x 2, $C_{10}:C_4$ x 2, $C_5:Q_8$ x 2, $C_5:C_8$, $C_{40}$, $C_2^2\times C_{10}$, $C_5\times Q_8$, $C_5:C_8$

Order 32: $C_4\times D_4$ x 8, $D_8:C_2$ x 6, $D_4:C_4$ x 3, $C_2\times \OD_{16}$ x 2, $C_4\times Q_8$ x 2, $C_4^2:C_2$ x 2, $C_2\times C_4^2$ x 2, $D_4:C_2^2$, $C_2^2\times D_4$, $C_2^3\times C_4$, $C_2\times \SD_{16}$, $C_8:C_4$, $C_2\times D_8$, $C_4\times C_8$, $C_2^2.D_4$, $C_2^3:C_4$, $C_8:C_4$, $C_8:C_4$, $Q_8:C_4$

Order 20: $D_{10}$ x 12, $F_5$ x 7, $C_2\times C_{10}$ x 4, $C_{20}$ x 3, $C_5:C_4$ x 3

Order 16: $C_2^2\times C_4$ x 10, $C_2\times D_4$ x 9, $C_4^2$ x 7, $C_4:C_4$ x 6, $D_4:C_2$ x 6, $\OD_{16}$ x 5, $C_2^2:C_4$ x 5, $\SD_{16}$ x 4, $D_8$ x 4, $C_2\times C_8$ x 2, $C_2^4$, $C_2\times Q_8$

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

Order 8: $C_2\times C_4$ x 25, $D_4$ x 13, $C_2^3$ x 8, $Q_8$ x 3, $C_8$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 16, $C_4$ x 13

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 640: $D_8:C_2\times F_5$ ($C_1$)

Order 320: $\SD_{16}:F_5$ ($C_2$) x 2, $\SD_{16}\times F_5$ ($C_2$) x 2, $D_{40}:C_4$ ($C_2$) x 2, $D_8\times F_5$ ($C_2$) x 2, $D_{10}.C_2^4$ ($C_2$), $D_{10}.C_2^4$ ($C_2$), $D_8:D_{10}$ ($C_2$), $D_{20}:(C_2\times C_4)$ ($C_2$), $(C_2\times D_{20}):C_4$ ($C_2$), $\OD_{16}:F_5$ ($C_2$), $\OD_{16}\times F_5$ ($C_2$)

Order 160: $D_{20}:C_4$ ($C_2^2$) x 6, $D_4\times F_5$ ($C_2^2$) x 6, $Q_8:F_5$ ($C_2^2$) x 2, $C_{40}:C_4$ ($C_2^2$) x 2, $C_{40}:C_4$ ($C_2^2$) x 2, $C_{40}:C_4$ ($C_2^2$) x 2, $C_8\times F_5$ ($C_2^2$) x 2, $Q_8\times F_5$ ($C_2^2$) x 2, $C_8:D_{10}$ ($C_4$) x 2, $D_5\times \SD_{16}$ ($C_2^2$) x 2, $C_8:D_{10}$ ($C_4$) x 2, $D_5\times D_8$ ($C_2^2$) x 2, $D_4:D_{10}$ ($C_2^2$), $D_4\times D_{10}$ ($C_2^2$), $D_{10}.C_2^3$ ($C_2^2$), $D_{10}.D_4$ ($C_2^2$), $C_{10}:C_4^2$ ($C_2^2$), $D_5:\OD_{16}$ ($C_2^2$), $D_8:C_{10}$ ($C_4$), $D_4:D_{10}$ ($C_4$), $D_{20}:C_2^2$ ($C_4$), $C_8:D_{10}$ ($C_4$), $D_5\times \OD_{16}$ ($C_2^2$)

Order 80: $C_{20}:C_4$ ($C_2^3$) x 4, $C_5:D_8$ ($C_2\times C_4$) x 4, $D_4\times D_5$ ($C_2^3$) x 3, $D_{40}$ ($C_2\times C_4$) x 2, $C_{40}:C_2$ ($C_2\times C_4$) x 2, $C_2^2\times F_5$ ($D_4$) x 2, $C_{40}:C_2$ ($C_2\times C_4$) x 2, $C_8\times D_5$ ($C_2^3$) x 2, $D_4\times D_5$ ($C_2\times C_4$) x 2, $C_4\times F_5$ ($C_2^3$) x 2, $C_4\times F_5$ ($D_4$) x 2, $D_5:C_8$ ($C_2^3$) x 2, $C_5\times \SD_{16}$ ($C_2\times C_4$) x 2, $C_5\times D_8$ ($C_2\times C_4$) x 2, $Q_8:D_5$ ($C_2\times C_4$) x 2, $C_5:\SD_{16}$ ($C_2\times C_4$) x 2, $D_4:C_{10}$ ($C_2\times C_4$), $D_4\times C_{10}$ ($C_2\times C_4$), $D_{20}:C_2$ ($C_2\times C_4$), $Q_8\times D_5$ ($C_2^3$), $D_4:D_5$ ($C_2\times C_4$), $D_{20}:C_2$ ($C_2\times C_4$), $C_2\times D_{20}$ ($C_2\times C_4$), $C_4\times D_{10}$ ($C_2^3$), $C_5\times \OD_{16}$ ($C_2\times C_4$), $C_5:\OD_{16}$ ($C_2\times C_4$)

Order 40: $C_2\times F_5$ ($C_2\times D_4$) x 4, $D_{20}$ ($C_2^2\times C_4$) x 3, $C_5\times D_4$ ($C_2^2\times C_4$) x 3, $C_{40}$ ($C_2^2\times C_4$) x 2, $C_5:C_8$ ($C_2^2\times C_4$) x 2, $C_2\times C_{20}$ ($C_2^2\times C_4$), $C_{10}:C_4$ ($D_4:C_2$), $C_4\times D_5$ ($C_2^4$), $C_4\times D_5$ ($D_4:C_2$), $C_4\times D_5$ ($C_2\times D_4$), $C_4\times D_5$ ($C_2^2\times C_4$), $C_5:Q_8$ ($C_2^2\times C_4$), $C_2\times D_{10}$ ($C_2\times D_4$), $C_5\times Q_8$ ($C_2^2\times C_4$)

Order 32: $D_8:C_2$ ($F_5$)

Order 20: $F_5$ ($D_8:C_2$) x 2, $C_2\times C_{10}$ ($C_4\times D_4$), $D_{10}$ ($C_2^2\times D_4$), $D_{10}$ ($C_4\times D_4$), $C_{20}$ ($C_2^3\times C_4$), $C_{20}$ ($C_4\times D_4$), $C_5:C_4$ ($D_4:C_2^2$), $C_5:C_4$ ($C_4\times D_4$)

Order 16: $\SD_{16}$ ($C_2\times F_5$) x 2, $D_8$ ($C_2\times F_5$) x 2, $\OD_{16}$ ($C_2\times F_5$), $D_4:C_2$ ($C_2\times F_5$), $C_2\times D_4$ ($C_2\times F_5$)

Order 10: $C_{10}$ ($C_4^2:C_2^2$), $D_5$ ($D_8:C_2^2$), $D_5$ ($C_8:C_2^3$)

Order 8: $D_4$ ($C_2^2\times F_5$) x 3, $C_8$ ($C_2^2\times F_5$) x 2, $Q_8$ ($C_2^2\times F_5$), $C_2\times C_4$ ($C_2^2\times F_5$)

Order 5: $C_5$ ($C_4^2.D_4$)

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

Order 2: $C_2$ ($D_{10}.C_2^4$)

Order 1: $C_1$ ($D_8:C_2\times F_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 640: $D_8:C_2\times F_5$ ($C_1$)

Order 320: $D_{10}.C_2^4$ ($C_2$), $D_{10}.C_2^4$ ($C_2$), $D_8:D_{10}$ ($C_2$), $D_{20}:(C_2\times C_4)$ ($C_2$), $(C_2\times D_{20}):C_4$ ($C_2$), $\SD_{16}:F_5$ ($C_2$), $\SD_{16}\times F_5$ ($C_2$), $D_{40}:C_4$ ($C_2$), $D_8\times F_5$ ($C_2$), $\OD_{16}:F_5$ ($C_2$), $\OD_{16}\times F_5$ ($C_2$)

Order 160: $D_4\times F_5$ ($C_2^2$) x 4, $D_{20}:C_4$ ($C_2^2$) x 3, $Q_8\times F_5$ ($C_2^2$) x 2, $Q_8:F_5$ ($C_2^2$), $C_{40}:C_4$ ($C_2^2$), $C_{40}:C_4$ ($C_2^2$), $C_{40}:C_4$ ($C_2^2$), $C_8\times F_5$ ($C_2^2$), $D_4:D_{10}$ ($C_2^2$), $D_4\times D_{10}$ ($C_2^2$), $D_{10}.C_2^3$ ($C_2^2$), $D_{10}.D_4$ ($C_2^2$), $C_{10}:C_4^2$ ($C_2^2$), $D_5:\OD_{16}$ ($C_2^2$), $D_8:C_{10}$ ($C_4$), $D_4:D_{10}$ ($C_4$), $D_{20}:C_2^2$ ($C_4$), $C_8:D_{10}$ ($C_4$), $D_5\times \SD_{16}$ ($C_2^2$), $C_8:D_{10}$ ($C_4$), $D_5\times D_8$ ($C_2^2$), $C_8:D_{10}$ ($C_4$), $D_5\times \OD_{16}$ ($C_2^2$)

Order 80: $C_{20}:C_4$ ($C_2^3$) x 3, $C_2^2\times F_5$ ($D_4$) x 2, $D_4\times D_5$ ($C_2^3$) x 2, $C_4\times F_5$ ($C_2^3$) x 2, $C_4\times F_5$ ($D_4$) x 2, $C_5:D_8$ ($C_2\times C_4$) x 2, $D_{40}$ ($C_2\times C_4$), $C_{40}:C_2$ ($C_2\times C_4$), $C_{40}:C_2$ ($C_2\times C_4$), $D_4:C_{10}$ ($C_2\times C_4$), $D_4\times C_{10}$ ($C_2\times C_4$), $D_{20}:C_2$ ($C_2\times C_4$), $Q_8\times D_5$ ($C_2^3$), $D_4:D_5$ ($C_2\times C_4$), $C_8\times D_5$ ($C_2^3$), $D_4\times D_5$ ($C_2\times C_4$), $D_{20}:C_2$ ($C_2\times C_4$), $C_2\times D_{20}$ ($C_2\times C_4$), $C_4\times D_{10}$ ($C_2^3$), $D_5:C_8$ ($C_2^3$), $C_5\times \SD_{16}$ ($C_2\times C_4$), $C_5\times D_8$ ($C_2\times C_4$), $C_5\times \OD_{16}$ ($C_2\times C_4$), $Q_8:D_5$ ($C_2\times C_4$), $C_5:\SD_{16}$ ($C_2\times C_4$), $C_5:\OD_{16}$ ($C_2\times C_4$)

Order 40: $C_2\times F_5$ ($C_2\times D_4$) x 4, $D_{20}$ ($C_2^2\times C_4$) x 2, $C_5\times D_4$ ($C_2^2\times C_4$) x 2, $C_2\times C_{20}$ ($C_2^2\times C_4$), $C_{10}:C_4$ ($D_4:C_2$), $C_4\times D_5$ ($C_2^4$), $C_4\times D_5$ ($D_4:C_2$), $C_4\times D_5$ ($C_2\times D_4$), $C_4\times D_5$ ($C_2^2\times C_4$), $C_5:Q_8$ ($C_2^2\times C_4$), $C_{40}$ ($C_2^2\times C_4$), $C_2\times D_{10}$ ($C_2\times D_4$), $C_5\times Q_8$ ($C_2^2\times C_4$), $C_5:C_8$ ($C_2^2\times C_4$)

Order 32: $D_8:C_2$ ($F_5$)

Order 20: $C_2\times C_{10}$ ($C_4\times D_4$), $D_{10}$ ($C_2^2\times D_4$), $D_{10}$ ($C_4\times D_4$), $F_5$ ($D_8:C_2$), $C_{20}$ ($C_2^3\times C_4$), $C_{20}$ ($C_4\times D_4$), $C_5:C_4$ ($D_4:C_2^2$), $C_5:C_4$ ($C_4\times D_4$)

Order 16: $\SD_{16}$ ($C_2\times F_5$), $D_8$ ($C_2\times F_5$), $\OD_{16}$ ($C_2\times F_5$), $D_4:C_2$ ($C_2\times F_5$), $C_2\times D_4$ ($C_2\times F_5$)

Order 10: $C_{10}$ ($C_4^2:C_2^2$), $D_5$ ($D_8:C_2^2$), $D_5$ ($C_8:C_2^3$)

Order 8: $D_4$ ($C_2^2\times F_5$) x 2, $Q_8$ ($C_2^2\times F_5$), $C_2\times C_4$ ($C_2^2\times F_5$), $C_8$ ($C_2^2\times F_5$)

Order 5: $C_5$ ($C_4^2.D_4$)

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

Order 2: $C_2$ ($D_{10}.C_2^4$)

Order 1: $C_1$ ($D_8:C_2\times F_5$)

Series

Derived series

$D_8:C_2\times F_5$

$\rhd$

$C_{20}$

$\rhd$

$C_1$

Chief series

$D_8:C_2\times F_5$

$\rhd$

$D_{10}.C_2^4$

$\rhd$

$D_4:D_{10}$

$\rhd$

$C_4\times D_{10}$

$\rhd$

$C_4\times D_5$

$\rhd$

$D_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$D_8:C_2\times F_5$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

$\lhd$

$D_8:C_2$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 6 larger groups in the database.

Character theory

Complex character table

See the $55 \times 55$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $44 \times 44$ rational character table.