Abstract group 1280.1114098: $D_4\times C_{10}:C_4^2$

• Label: 1280.1114098
• Order: \( 2^{8} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{18} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{14} \)
• Perm deg.: $15$
• Trans deg.: $160$
• Rank: $5$

Group information

Description:	$D_4\times C_{10}:C_4^2$
Order:	\(1280\)\(\medspace = 2^{8} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	Group of order \(1310720\)\(\medspace = 2^{18} \cdot 5 \)
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	143	880	4	92	160	1280
Conjugacy classes	1	31	128	1	15	24	200
Divisions	1	31	68	1	15	14	130
Autjugacy classes	1	9	14	1	5	4	34

Dimension	1	2	4	8	16
Irr. complex chars.	128	32	32	8	0	200
Irr. rational chars.	32	56	28	12	2	130

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$160$
Rank:	$5$
Inequivalent generating 5-tuples:	$779950080$

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:	${\langle a, b, c, d, e \mid a^{4}=b^{20}=c^{2}=d^{4}=e^{2}=[a,c]=[a,e]=[b,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(2,3,5,4)(6,7)(9,11)(12,13)(14,15), (8,9)(10,11)(12,14,13,15), (8,10)(9,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 3 & 5 \\ 10 & 13 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 19 & 10 \\ 10 & 9 \end{array}\right), \left(\begin{array}{rr} 3 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $D_4$ $\, \times\, $ $F_5$
Semidirect product:	$(D_4\times D_5)$ $\,\rtimes\,$ $C_4^2$	$D_5$ $\,\rtimes\,$ $(D_4\times C_4^2)$	$(D_{10}.C_2^5)$ $\,\rtimes\,$ $C_2$	$(C_2^4.D_{10})$ $\,\rtimes\,$ $C_4$	all 93
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_{10}.D_4)$ . $C_2^3$	$(D_4\times D_{10})$ . $C_2^3$	$(D_{10}.D_4)$ . $C_2^3$	$(D_{10}.D_4)$ . $C_2^3$	all 88
Aut. group:	$\Aut(C_8.D_{20})$	$\Aut(C_8.D_{20})$	$\Aut(C_{40}.D_4)$	$\Aut(C_{16}.D_{10})$	all 40

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

Homology

Abelianization:	$C_{2}^{3} \times C_{4}^{2} $
Schur multiplier:	$C_{2}^{9} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 22838 subgroups in 5694 conjugacy classes, 1832 normal (54 characteristic).

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

Special subgroups

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

Order 640: $D_{20}:C_4^2$ x 16, $C_2\times D_{10}:C_4^2$ x 4, $D_{10}.C_2^5$ x 2, $D_{10}.C_2^5$ x 2, $C_2\times D_{10}:C_4^2$ x 2, $C_2\times C_{20}:C_4^2$ x 2, $C_4\times D_4\times D_{10}$, $C_2\times C_{20}:C_4^2$, $C_{10}:C_4^3$

Order 320: $D_{10}.C_2^4$ x 48, $D_{10}:C_4^2$ x 44, $D_{10}:C_4^2$ x 32, $C_4^2:D_{10}$ x 16, $D_{10}:C_4^2$ x 16, $C_{20}:C_4^2$ x 16, $C_{20}:C_4^2$ x 8, $C_4^2\times F_5$ x 8, $D_{10}.C_4^2$ x 6, $C_2^4\times F_5$ x 4, $C_2^4:F_5$ x 4, $C_{20}:C_2^4$ x 2, $D_{10}.C_2^4$ x 2, $C_2^4.D_{10}$ x 2, $C_2^4.D_{10}$ x 2, $C_2^4.D_{10}$ x 2, $C_{20}:C_2^4$, $C_2\times D_4\times C_{20}$, $C_2^4.D_{10}$, $C_2\times D_{20}:C_4$, $C_4:C_4\times D_{10}$, $C_4^2:D_{10}$, $C_4^2\times D_{10}$

Order 256: $C_4^3:C_2^2$

Order 160: $C_{10}:C_4^2$ x 172, $D_4\times F_5$ x 128, $C_2^3\times F_5$ x 88, $C_2^3:F_5$ x 48, $C_{20}:C_2^3$ x 41, $C_{10}.C_4^2$ x 24, $D_4\times D_{10}$ x 24, $D_{10}.D_4$ x 24, $C_{20}:D_4$ x 16, $C_2^3.D_{10}$ x 16, $D_{10}.D_4$ x 16, $C_4\times D_{20}$ x 8, $D_5\times C_4^2$ x 8, $D_4\times C_{20}$ x 8, $C_2^3.D_{10}$ x 8, $D_{20}:C_4$ x 8, $D_{10}.D_4$ x 8, $C_2^2.D_{20}$ x 4, $C_{10}:C_4^2$ x 3, $C_2^3\times D_{10}$ x 2, $C_2^3\times C_{20}$ x 2, $C_2^3:D_{10}$ x 2, $C_{10}.C_2^4$ x 2, $C_2^3:C_{20}$ x 2, $C_2^3.D_{10}$ x 2, $C_2^3.D_{10}$ x 2, $C_{20}:C_2^3$, $C_2^2\times D_{20}$, $C_2\times C_4:C_{20}$, $C_2\times C_4\times C_{20}$, $C_2^2.D_{20}$

Order 128: $D_4\times C_4^2$ x 16, $C_2^3:C_4^2$ x 6, $C_4^2:C_2^3$ x 3, $C_2^3.C_2^4$ x 3, $C_2^3\times C_4^2$ x 2, $C_2\times C_4^3$

Order 80: $C_2^2\times F_5$ x 344, $C_4\times D_{10}$ x 144, $C_4\times F_5$ x 120, $D_4\times D_5$ x 64, $D_{10}:C_4$ x 64, $C_2^2\times D_{10}$ x 43, $C_{20}:C_4$ x 32, $C_{10}:D_4$ x 24, $C_2^2\times C_{20}$ x 21, $C_2^2.D_{10}$ x 21, $D_{10}:C_4$ x 16, $D_4\times C_{10}$ x 12, $C_2\times D_{20}$ x 12, $C_{20}:C_4$ x 12, $C_2^2:C_{20}$ x 8, $C_{10}.D_4$ x 8, $C_{10}.D_4$ x 8, $C_4:C_{20}$ x 4, $C_4\times C_{20}$ x 4, $C_{20}:C_4$ x 4, $C_2^3\times C_{10}$ x 2

Order 64: $C_4^2:C_2^2$ x 72, $C_2^2:C_4^2$ x 48, $C_2^2\times C_4^2$ x 45, $C_4:C_4^2$ x 24, $C_4^3$ x 8, $C_2^2.C_4^2$ x 6, $C_2^4\times C_4$ x 6, $C_2^4:C_4$ x 6, $C_2^3.D_4$ x 3, $D_4\times C_2^3$

Order 40: $C_2\times F_5$ x 280, $C_2\times D_{10}$ x 162, $C_4\times D_5$ x 88, $C_2\times C_{20}$ x 40, $C_{10}:C_4$ x 40, $C_5:D_4$ x 32, $C_2^2\times C_{10}$ x 21, $D_{20}$ x 16, $C_5\times D_4$ x 16

Order 32: $C_4\times D_4$ x 192, $C_2\times C_4^2$ x 184, $C_2^3\times C_4$ x 133, $C_2^3:C_4$ x 72, $C_2^2.D_4$ x 36, $C_2^2\times D_4$ x 28, $C_2.C_4^2$ x 24, $C_2^5$ x 2

Order 20: $D_{10}$ x 124, $F_5$ x 40, $C_2\times C_{10}$ x 39, $C_{20}$ x 14, $C_5:C_4$ x 14

Order 16: $C_2^2\times C_4$ x 530, $C_4^2$ x 136, $C_2\times D_4$ x 112, $C_2^2:C_4$ x 96, $C_4:C_4$ x 48, $C_2^4$ x 45

Order 10: $D_5$ x 16, $C_{10}$ x 15

Order 8: $C_2\times C_4$ x 448, $C_2^3$ x 183, $D_4$ x 64

Order 5: $C_5$

Order 4: $C_2^2$ x 163, $C_4$ x 68

Order 2: $C_2$ x 31

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1280: $D_4\times C_{10}:C_4^2$

Order 640: $D_{10}.C_2^5$, $D_{10}.C_2^5$, $C_4\times D_4\times D_{10}$, $C_2\times D_{10}:C_4^2$, $D_{20}:C_4^2$, $C_2\times C_{20}:C_4^2$, $C_2\times D_{10}:C_4^2$, $C_2\times C_{20}:C_4^2$, $C_{10}:C_4^3$

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

Order 256: $C_4^3:C_2^2$

Order 160: $C_{10}:C_4^2$ x 13, $C_{20}:C_2^3$ x 10, $C_2^3\times F_5$ x 7, $D_4\times D_{10}$ x 3, $C_{10}:C_4^2$ x 3, $D_5\times C_4^2$ x 2, $C_{10}.C_4^2$ x 2, $C_2^3:F_5$ x 2, $D_{10}.D_4$ x 2, $C_2^2.D_{20}$ x 2, $D_{10}.D_4$ x 2, $D_{10}.D_4$ x 2, $C_4\times D_{20}$, $C_2^3\times D_{10}$, $C_{20}:C_2^3$, $C_2^3\times C_{20}$, $C_2^3:D_{10}$, $C_{10}.C_2^4$, $C_2^2\times D_{20}$, $D_4\times F_5$, $D_4\times C_{20}$, $C_2\times C_4:C_{20}$, $C_2^3:C_{20}$, $C_2\times C_4\times C_{20}$, $C_2^3.D_{10}$, $C_2^3.D_{10}$, $C_{20}:D_4$, $C_2^2.D_{20}$, $C_2^3.D_{10}$, $D_{20}:C_4$, $C_2^3.D_{10}$

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

Order 80: $C_4\times D_{10}$ x 21, $C_2^2\times F_5$ x 16, $C_2^2\times D_{10}$ x 8, $C_2^2\times C_{20}$ x 6, $C_2^2.D_{10}$ x 6, $C_4\times F_5$ x 6, $C_{20}:C_4$ x 3, $D_4\times C_{10}$ x 2, $C_{10}:D_4$ x 2, $D_4\times D_5$ x 2, $C_2\times D_{20}$ x 2, $D_{10}:C_4$ x 2, $C_2^3\times C_{10}$, $D_{10}:C_4$, $C_{20}:C_4$, $C_4:C_{20}$, $C_2^2:C_{20}$, $C_4\times C_{20}$, $C_{10}.D_4$, $C_{20}:C_4$, $C_{10}.D_4$

Order 64: $C_4^2:C_2^2$ x 10, $C_2^2\times C_4^2$ x 9, $C_4:C_4^2$ x 2, $C_2^2:C_4^2$ x 2, $C_2^2.C_4^2$ x 2, $C_2^4\times C_4$ x 2, $C_2^3.D_4$ x 2, $C_2^4:C_4$ x 2, $C_4^3$, $D_4\times C_2^3$

Order 40: $C_2\times D_{10}$ x 19, $C_2\times F_5$ x 13, $C_4\times D_5$ x 12, $C_2\times C_{20}$ x 9, $C_{10}:C_4$ x 9, $C_2^2\times C_{10}$ x 5, $C_5:D_4$, $D_{20}$, $C_5\times D_4$

Order 32: $C_2^3\times C_4$ x 19, $C_2\times C_4^2$ x 19, $C_2^3:C_4$ x 8, $C_4\times D_4$ x 7, $C_2^2.D_4$ x 7, $C_2^2\times D_4$ x 6, $C_2.C_4^2$ x 2, $C_2^5$

Order 20: $D_{10}$ x 16, $C_2\times C_{10}$ x 8, $C_{20}$ x 4, $C_5:C_4$ x 4, $F_5$ x 3

Order 16: $C_2^2\times C_4$ x 49, $C_4^2$ x 10, $C_2^4$ x 9, $C_2\times D_4$ x 8, $C_2^2:C_4$ x 5, $C_4:C_4$ x 4

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

Order 8: $C_2\times C_4$ x 43, $C_2^3$ x 24, $D_4$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 24, $C_4$ x 11

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1280: $D_4\times C_{10}:C_4^2$ ($C_1$)

Order 640: $D_{20}:C_4^2$ ($C_2$) x 16, $C_2\times D_{10}:C_4^2$ ($C_2$) x 4, $D_{10}.C_2^5$ ($C_2$) x 2, $D_{10}.C_2^5$ ($C_2$) x 2, $C_2\times D_{10}:C_4^2$ ($C_2$) x 2, $C_2\times C_{20}:C_4^2$ ($C_2$) x 2, $C_4\times D_4\times D_{10}$ ($C_2$), $C_2\times C_{20}:C_4^2$ ($C_2$), $C_{10}:C_4^3$ ($C_2$)

Order 320: $D_{10}.C_2^4$ ($C_4$) x 32, $D_{10}:C_4^2$ ($C_2^2$) x 32, $D_{10}:C_4^2$ ($C_2^2$) x 28, $D_{10}.C_2^4$ ($C_2^2$) x 16, $D_{10}:C_4^2$ ($C_2^2$) x 16, $C_{20}:C_4^2$ ($C_2^2$) x 16, $C_4^2:D_{10}$ ($C_2^2$) x 8, $C_4^2:D_{10}$ ($C_4$) x 8, $C_{20}:C_4^2$ ($C_2^2$) x 8, $C_4^2\times F_5$ ($C_2^2$) x 8, $D_{10}.C_4^2$ ($C_2^2$) x 6, $C_2^4\times F_5$ ($C_2^2$) x 4, $C_2^4:F_5$ ($C_2^2$) x 4, $C_{20}:C_2^4$ ($C_2^2$) x 2, $D_{10}.C_2^4$ ($C_2^2$) x 2, $C_2^4.D_{10}$ ($C_4$) x 2, $C_2^4.D_{10}$ ($C_4$) x 2, $C_2^4.D_{10}$ ($C_2^2$) x 2, $C_{20}:C_2^4$ ($C_2^2$), $C_2\times D_4\times C_{20}$ ($C_4$), $C_2^4.D_{10}$ ($C_4$), $C_2\times D_{20}:C_4$ ($C_4$), $C_4:C_4\times D_{10}$ ($C_2^2$), $C_4^2:D_{10}$ ($C_4$), $C_4^2\times D_{10}$ ($C_2^2$)

Order 160: $D_4\times F_5$ ($C_2\times C_4$) x 128, $C_{10}:C_4^2$ ($C_2^3$) x 48, $C_2^3\times F_5$ ($C_2\times C_4$) x 32, $C_2^3:F_5$ ($C_2\times C_4$) x 32, $C_{10}.C_4^2$ ($C_2^3$) x 24, $C_2^3\times F_5$ ($C_2^3$) x 24, $D_4\times D_{10}$ ($C_2\times C_4$) x 20, $C_2^3:F_5$ ($C_2^3$) x 16, $D_{10}.D_4$ ($C_2\times C_4$) x 16, $C_{10}:C_4^2$ ($C_2\times C_4$) x 16, $C_{20}:D_4$ ($C_2\times C_4$) x 16, $C_2^3.D_{10}$ ($C_2\times C_4$) x 16, $C_{20}:C_2^3$ ($C_2^3$) x 13, $C_{20}:C_2^3$ ($C_2\times C_4$) x 12, $C_4\times D_{20}$ ($C_2\times C_4$) x 8, $D_{10}.D_4$ ($C_2^3$) x 8, $C_{10}:C_4^2$ ($D_4$) x 8, $D_4\times C_{20}$ ($C_2\times C_4$) x 8, $C_2^3.D_{10}$ ($C_2\times C_4$) x 8, $D_{20}:C_4$ ($C_2\times C_4$) x 8, $D_{10}.D_4$ ($C_2^3$) x 8, $D_{10}.D_4$ ($C_2\times C_4$) x 8, $D_5\times C_4^2$ ($C_2^3$) x 4, $D_5\times C_4^2$ ($C_2\times C_4$) x 4, $D_4\times D_{10}$ ($C_2^3$) x 4, $C_2^2.D_{20}$ ($C_2\times C_4$) x 4, $D_{10}.D_4$ ($C_2^3$) x 4, $D_{10}.D_4$ ($C_2\times C_4$) x 4, $C_{10}:C_4^2$ ($C_2\times C_4$) x 3, $C_2^3\times D_{10}$ ($C_2^3$) x 2, $C_2^3\times C_{20}$ ($C_2\times C_4$) x 2, $C_2^3:D_{10}$ ($C_2\times C_4$) x 2, $C_{10}.C_2^4$ ($C_2\times C_4$) x 2, $C_2^3:C_{20}$ ($C_2\times C_4$) x 2, $C_2^3.D_{10}$ ($C_2\times C_4$) x 2, $C_2^3.D_{10}$ ($C_2\times C_4$) x 2, $C_{20}:C_2^3$ ($C_2\times C_4$), $C_2^2\times D_{20}$ ($C_2\times C_4$), $C_2\times C_4:C_{20}$ ($C_2\times C_4$), $C_2\times C_4\times C_{20}$ ($C_2\times C_4$), $C_2^2.D_{20}$ ($C_2\times C_4$)

Order 80: $C_2^2\times F_5$ ($C_2^2\times C_4$) x 96, $D_{10}:C_4$ ($C_2^2\times C_4$) x 64, $C_4\times D_{10}$ ($C_2^2\times C_4$) x 34, $D_4\times D_5$ ($C_4^2$) x 32, $D_4\times D_5$ ($C_2^2\times C_4$) x 32, $C_{20}:C_4$ ($C_2^2\times C_4$) x 32, $C_4\times F_5$ ($C_2^2\times C_4$) x 32, $C_2^2\times D_{10}$ ($C_2^2\times C_4$) x 22, $C_2^2\times F_5$ ($C_2^4$) x 16, $C_{10}:D_4$ ($C_4^2$) x 16, $C_4\times F_5$ ($C_2\times D_4$) x 16, $D_{10}:C_4$ ($C_2^2\times C_4$) x 16, $C_2^2\times C_{20}$ ($C_2^2\times C_4$) x 13, $C_2^2.D_{10}$ ($C_2^2\times C_4$) x 13, $C_{20}:C_4$ ($C_2^2\times C_4$) x 12, $C_4\times D_{10}$ ($C_2^4$) x 10, $C_2^2\times F_5$ ($D_4:C_2$) x 8, $C_2^2\times F_5$ ($C_2\times D_4$) x 8, $D_4\times C_{10}$ ($C_4^2$) x 8, $C_{10}:D_4$ ($C_2^2\times C_4$) x 8, $C_2\times D_{20}$ ($C_4^2$) x 8, $C_2^2:C_{20}$ ($C_2^2\times C_4$) x 8, $C_{10}.D_4$ ($C_2^2\times C_4$) x 8, $C_{10}.D_4$ ($C_2^2\times C_4$) x 8, $C_2^2\times D_{10}$ ($C_2^4$) x 5, $D_4\times C_{10}$ ($C_2^2\times C_4$) x 4, $C_2\times D_{20}$ ($C_2^2\times C_4$) x 4, $C_4\times D_{10}$ ($D_4:C_2$) x 4, $C_4\times D_{10}$ ($C_2\times D_4$) x 4, $C_4:C_{20}$ ($C_2^2\times C_4$) x 4, $C_4\times C_{20}$ ($C_2^2\times C_4$) x 4, $C_{20}:C_4$ ($C_2^2\times C_4$) x 4, $C_2^3\times C_{10}$ ($C_2^2\times C_4$) x 2

Order 64: $C_4^2:C_2^2$ ($F_5$)

Order 40: $C_5:D_4$ ($C_2\times C_4^2$) x 32, $C_2\times F_5$ ($C_2^3\times C_4$) x 32, $C_2\times F_5$ ($C_4\times D_4$) x 32, $C_2\times D_{10}$ ($C_2^3\times C_4$) x 25, $C_2\times D_{10}$ ($C_2\times C_4^2$) x 24, $D_{20}$ ($C_2\times C_4^2$) x 16, $C_5\times D_4$ ($C_2\times C_4^2$) x 16, $C_2\times C_{20}$ ($C_2^3\times C_4$) x 10, $C_{10}:C_4$ ($C_2^3\times C_4$) x 10, $C_4\times D_5$ ($C_2^3\times C_4$) x 8, $C_4\times D_5$ ($C_4\times D_4$) x 8, $C_4\times D_5$ ($C_2\times C_4^2$) x 8, $C_2^2\times C_{10}$ ($C_2\times C_4^2$) x 8, $C_2\times F_5$ ($D_4:C_2^2$) x 8, $C_2\times F_5$ ($C_2^2\times D_4$) x 8, $C_2\times D_{10}$ ($D_4:C_2^2$) x 6, $C_2^2\times C_{10}$ ($C_2^3\times C_4$) x 5, $C_2\times C_{20}$ ($C_4\times D_4$) x 4, $C_2\times C_{20}$ ($C_2\times C_4^2$) x 4, $C_{10}:C_4$ ($C_4\times D_4$) x 4, $C_{10}:C_4$ ($C_2\times C_4^2$) x 4, $C_4\times D_5$ ($D_4:C_2^2$) x 4, $C_4\times D_5$ ($C_2^2\times D_4$) x 4, $C_2\times D_{10}$ ($C_2^2\times D_4$) x 2, $C_2\times D_{10}$ ($C_2^5$)

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

Order 20: $F_5$ ($C_4^2:C_2^2$) x 16, $C_2\times C_{10}$ ($C_2^2\times C_4^2$) x 10, $D_{10}$ ($C_4^2.C_2^2$) x 10, $D_{10}$ ($C_4^2:C_2^2$) x 10, $D_{10}$ ($C_2^2\times C_4^2$) x 10, $C_{20}$ ($C_4^2:C_2^2$) x 4, $C_{20}$ ($C_2^2\times C_4^2$) x 4, $C_5:C_4$ ($C_4^2:C_2^2$) x 4, $C_5:C_4$ ($C_2^2\times C_4^2$) x 4, $D_{10}$ ($D_4:C_2^3$) x 3, $C_2\times C_{10}$ ($C_4^2.C_2^2$) x 2, $C_2\times C_{10}$ ($C_4^2:C_2^2$) x 2, $D_{10}$ ($C_2^4\times C_4$) x 2, $C_2\times C_{10}$ ($C_2^4\times C_4$), $D_{10}$ ($D_4\times C_2^3$)

Order 16: $C_2^2\times C_4$ ($C_2^2\times F_5$) x 13, $C_2^2:C_4$ ($C_2^2\times F_5$) x 8, $C_2\times D_4$ ($C_4\times F_5$) x 8, $C_4:C_4$ ($C_2^2\times F_5$) x 4, $C_4^2$ ($C_2^2\times F_5$) x 4, $C_2\times D_4$ ($C_2^2\times F_5$) x 4, $C_2^4$ ($C_2^2\times F_5$) x 2

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

Order 8: $D_4$ ($C_{10}:C_4^2$) x 16, $C_2\times C_4$ ($C_2^3\times F_5$) x 10, $C_2^3$ ($C_{10}:C_4^2$) x 8, $C_2^3$ ($C_2^3\times F_5$) x 5, $C_2\times C_4$ ($D_4\times F_5$) x 4, $C_2\times C_4$ ($C_{10}:C_4^2$) x 4

Order 5: $C_5$ ($C_4^3:C_2^2$)

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

Order 2: $C_2$ ($D_{20}:C_4^2$) x 4, $C_2$ ($D_{10}.C_2^5$), $C_2$ ($D_{10}.C_2^5$), $C_2$ ($D_{10}.C_2^5$)

Order 1: $C_1$ ($D_4\times C_{10}:C_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1280: $D_4\times C_{10}:C_4^2$ ($C_1$)

Order 640: $D_{10}.C_2^5$ ($C_2$), $D_{10}.C_2^5$ ($C_2$), $C_4\times D_4\times D_{10}$ ($C_2$), $C_2\times D_{10}:C_4^2$ ($C_2$), $D_{20}:C_4^2$ ($C_2$), $C_2\times C_{20}:C_4^2$ ($C_2$), $C_2\times D_{10}:C_4^2$ ($C_2$), $C_2\times C_{20}:C_4^2$ ($C_2$), $C_{10}:C_4^3$ ($C_2$)

Order 320: $D_{10}:C_4^2$ ($C_2^2$) x 7, $D_{10}.C_4^2$ ($C_2^2$) x 2, $C_2^4\times F_5$ ($C_2^2$), $C_{20}:C_2^4$ ($C_2^2$), $C_{20}:C_2^4$ ($C_2^2$), $C_2^4:F_5$ ($C_2^2$), $D_{10}.C_2^4$ ($C_2^2$), $D_{10}.C_2^4$ ($C_4$), $D_{10}.C_2^4$ ($C_2^2$), $C_2\times D_4\times C_{20}$ ($C_4$), $C_2^4.D_{10}$ ($C_4$), $C_2^4.D_{10}$ ($C_4$), $C_4^2:D_{10}$ ($C_2^2$), $C_4^2:D_{10}$ ($C_4$), $C_2\times D_{20}:C_4$ ($C_4$), $C_4:C_4\times D_{10}$ ($C_2^2$), $C_2^4.D_{10}$ ($C_4$), $C_2^4.D_{10}$ ($C_2^2$), $C_4^2:D_{10}$ ($C_4$), $C_4^2\times D_{10}$ ($C_2^2$), $D_{10}:C_4^2$ ($C_2^2$), $C_{20}:C_4^2$ ($C_2^2$), $D_{10}:C_4^2$ ($C_2^2$), $C_{20}:C_4^2$ ($C_2^2$), $C_4^2\times F_5$ ($C_2^2$)

Order 160: $C_{10}:C_4^2$ ($C_2^3$) x 6, $C_{20}:C_2^3$ ($C_2^3$) x 5, $C_2^3\times F_5$ ($C_2^3$) x 4, $C_{20}:C_2^3$ ($C_2\times C_4$) x 3, $C_{10}:C_4^2$ ($C_2\times C_4$) x 3, $C_{10}.C_4^2$ ($C_2^3$) x 2, $D_4\times D_{10}$ ($C_2\times C_4$) x 2, $C_2^2.D_{20}$ ($C_2\times C_4$) x 2, $C_4\times D_{20}$ ($C_2\times C_4$), $D_5\times C_4^2$ ($C_2^3$), $D_5\times C_4^2$ ($C_2\times C_4$), $C_2^3\times D_{10}$ ($C_2^3$), $C_2^3\times F_5$ ($C_2\times C_4$), $C_{20}:C_2^3$ ($C_2\times C_4$), $C_2^3\times C_{20}$ ($C_2\times C_4$), $C_2^3:D_{10}$ ($C_2\times C_4$), $C_{10}.C_2^4$ ($C_2\times C_4$), $D_4\times D_{10}$ ($C_2^3$), $C_2^2\times D_{20}$ ($C_2\times C_4$), $C_2^3:F_5$ ($C_2^3$), $C_2^3:F_5$ ($C_2\times C_4$), $D_4\times F_5$ ($C_2\times C_4$), $D_{10}.D_4$ ($C_2^3$), $D_{10}.D_4$ ($C_2\times C_4$), $C_{10}:C_4^2$ ($D_4$), $C_{10}:C_4^2$ ($C_2\times C_4$), $D_4\times C_{20}$ ($C_2\times C_4$), $C_2\times C_4:C_{20}$ ($C_2\times C_4$), $C_2^3:C_{20}$ ($C_2\times C_4$), $C_2\times C_4\times C_{20}$ ($C_2\times C_4$), $C_2^3.D_{10}$ ($C_2\times C_4$), $C_2^3.D_{10}$ ($C_2\times C_4$), $C_{20}:D_4$ ($C_2\times C_4$), $C_2^2.D_{20}$ ($C_2\times C_4$), $C_2^3.D_{10}$ ($C_2\times C_4$), $D_{20}:C_4$ ($C_2\times C_4$), $D_{10}.D_4$ ($C_2^3$), $D_{10}.D_4$ ($C_2\times C_4$), $C_2^3.D_{10}$ ($C_2\times C_4$), $D_{10}.D_4$ ($C_2^3$), $D_{10}.D_4$ ($C_2\times C_4$)

Order 80: $C_4\times D_{10}$ ($C_2^2\times C_4$) x 7, $C_2^2\times C_{20}$ ($C_2^2\times C_4$) x 5, $C_2^2.D_{10}$ ($C_2^2\times C_4$) x 5, $C_2^2\times F_5$ ($C_2^2\times C_4$) x 4, $C_4\times D_{10}$ ($C_2^4$) x 4, $C_2^2\times D_{10}$ ($C_2^2\times C_4$) x 3, $C_2^2\times F_5$ ($C_2^4$) x 3, $C_{20}:C_4$ ($C_2^2\times C_4$) x 3, $C_2^2\times D_{10}$ ($C_2^4$) x 2, $D_{10}:C_4$ ($C_2^2\times C_4$) x 2, $C_2^3\times C_{10}$ ($C_2^2\times C_4$), $C_2^2\times F_5$ ($D_4:C_2$), $C_2^2\times F_5$ ($C_2\times D_4$), $D_4\times C_{10}$ ($C_4^2$), $D_4\times C_{10}$ ($C_2^2\times C_4$), $C_{10}:D_4$ ($C_4^2$), $C_{10}:D_4$ ($C_2^2\times C_4$), $D_4\times D_5$ ($C_4^2$), $D_4\times D_5$ ($C_2^2\times C_4$), $C_2\times D_{20}$ ($C_4^2$), $C_2\times D_{20}$ ($C_2^2\times C_4$), $C_4\times D_{10}$ ($D_4:C_2$), $C_4\times D_{10}$ ($C_2\times D_4$), $D_{10}:C_4$ ($C_2^2\times C_4$), $C_{20}:C_4$ ($C_2^2\times C_4$), $C_4\times F_5$ ($C_2\times D_4$), $C_4\times F_5$ ($C_2^2\times C_4$), $C_4:C_{20}$ ($C_2^2\times C_4$), $C_2^2:C_{20}$ ($C_2^2\times C_4$), $C_4\times C_{20}$ ($C_2^2\times C_4$), $C_{10}.D_4$ ($C_2^2\times C_4$), $C_{20}:C_4$ ($C_2^2\times C_4$), $C_{10}.D_4$ ($C_2^2\times C_4$)

Order 64: $C_4^2:C_2^2$ ($F_5$)

Order 40: $C_2\times C_{20}$ ($C_2^3\times C_4$) x 4, $C_{10}:C_4$ ($C_2^3\times C_4$) x 4, $C_2\times D_{10}$ ($C_2^3\times C_4$) x 4, $C_2\times F_5$ ($C_2^3\times C_4$) x 3, $C_2^2\times C_{10}$ ($C_2^3\times C_4$) x 2, $C_2\times D_{10}$ ($D_4:C_2^2$) x 2, $C_2\times D_{10}$ ($C_2\times C_4^2$) x 2, $C_2\times C_{20}$ ($C_4\times D_4$), $C_2\times C_{20}$ ($C_2\times C_4^2$), $C_5:D_4$ ($C_2\times C_4^2$), $C_{10}:C_4$ ($C_4\times D_4$), $C_{10}:C_4$ ($C_2\times C_4^2$), $D_{20}$ ($C_2\times C_4^2$), $C_4\times D_5$ ($D_4:C_2^2$), $C_4\times D_5$ ($C_2^2\times D_4$), $C_4\times D_5$ ($C_2^3\times C_4$), $C_4\times D_5$ ($C_4\times D_4$), $C_4\times D_5$ ($C_2\times C_4^2$), $C_2^2\times C_{10}$ ($C_2\times C_4^2$), $C_2\times D_{10}$ ($C_2^5$), $C_2\times D_{10}$ ($C_2^2\times D_4$), $C_2\times F_5$ ($D_4:C_2^2$), $C_2\times F_5$ ($C_2^2\times D_4$), $C_2\times F_5$ ($C_4\times D_4$), $C_5\times D_4$ ($C_2\times C_4^2$)

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

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

Order 16: $C_2^2\times C_4$ ($C_2^2\times F_5$) x 5, $C_4:C_4$ ($C_2^2\times F_5$), $C_2^2:C_4$ ($C_2^2\times F_5$), $C_4^2$ ($C_2^2\times F_5$), $C_2^4$ ($C_2^2\times F_5$), $C_2\times D_4$ ($C_2^2\times F_5$), $C_2\times D_4$ ($C_4\times F_5$)

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

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

Order 5: $C_5$ ($C_4^3:C_2^2$)

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

Order 2: $C_2$ ($D_{10}.C_2^5$), $C_2$ ($D_{10}.C_2^5$), $C_2$ ($D_{10}.C_2^5$), $C_2$ ($D_{20}:C_4^2$)

Order 1: $C_1$ ($D_4\times C_{10}:C_4^2$)

Series

Derived series

$D_4\times C_{10}:C_4^2$

$\rhd$

$C_{10}$

$\rhd$

$C_1$

Chief series

$D_4\times C_{10}:C_4^2$

$\rhd$

$D_{10}.C_2^5$

$\rhd$

$C_{20}:C_2^4$

$\rhd$

$D_4\times D_{10}$

$\rhd$

$D_4\times C_{10}$

$\rhd$

$C_2\times D_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$D_4\times C_{10}:C_4^2$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_4^2:C_2^2$

Supergroups

This group is a maximal subgroup of 5 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 $200 \times 200$ 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 $130 \times 130$ rational character table (warning: may be slow to load).