Abstract group 1280.1116277: $D_{10}.C_2^6$

• Label: 1280.1116277
• Order: \( 2^{8} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{21} \cdot 3 \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{17} \cdot 3 \cdot 7 \)
• Perm deg.: not computed
• Trans deg.: $160$
• Rank: $6$

Group information

Description:	$D_{10}.C_2^6$
Order:	\(1280\)\(\medspace = 2^{8} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	Group of order \(220200960\)\(\medspace = 2^{21} \cdot 3 \cdot 5 \cdot 7 \)
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	287	736	4	188	64	1280
Conjugacy classes	1	63	96	1	31	8	200
Divisions	1	63	56	1	31	8	160
Autjugacy classes	1	7	8	1	3	1	21

Dimension	1	2	4	8
Irr. complex chars.	128	32	32	8	200
Irr. rational chars.	64	48	40	8	160

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$160$
Rank:	$6$
Inequivalent generating 6-tuples:	$5857125120$

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:	${\langle a, b, c, d, e, f \mid a^{4}=b^{10}=c^{4}=d^{2}=e^{2}=f^{2}=[a,d]= \!\cdots\! \rangle}$
Matrix group:	$\left\langle \left(\begin{array}{rr} 19 & 0 \\ 20 & 19 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 29 & 20 \\ 20 & 19 \end{array}\right), \left(\begin{array}{rr} 29 & 2 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 29 & 20 \\ 20 & 29 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/40\Z)$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $D_4$ $\, \times\, $ $F_5$
Semidirect product:	$(F_5\times C_2^5)$ $\,\rtimes\,$ $C_2$	$(C_2^5:F_5)$ $\,\rtimes\,$ $C_2$	$(C_4\times F_5)$ $\,\rtimes\,$ $C_2^4$	$C_2^5$ $\,\rtimes\,$ $(C_2\times F_5)$	all 65
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}$ . $C_2^6$	$(D_4\times D_5)$ . $C_2^4$	$D_5$ . $(D_4\times C_2^4)$	$D_5$ . $(D_4:C_2^4)$	all 36
Aut. group:	$\Aut(C_{10}.D_8)$	$\Aut(C_{20}.Q_8)$	$\Aut(D_{20}:C_4)$	$\Aut(C_4.D_{20})$	all 40

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{40}\Z)$.

Homology

Abelianization:	$C_{2}^{5} \times C_{4} $
Schur multiplier:	$C_{2}^{15}$
Commutator length:	$1$

Subgroups

There are 81206 subgroups in 19998 conjugacy classes, 6264 normal (23 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^4$	$G/Z \simeq$ $C_2^2\times F_5$
Commutator:	$G' \simeq$ $C_{10}$	$G/G' \simeq$ $C_2^5\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $F_5\times C_2^5$
Fitting:	$\operatorname{Fit} \simeq$ $C_{20}:C_2^4$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $D_{10}.C_2^6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^4$
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 1280: $D_{10}.C_2^6$

Order 640: $D_{10}.C_2^5$ x 56, $F_5\times C_2^5$ x 2, $C_2^5:F_5$ x 2, $C_{20}:C_2^5$, $D_{10}.C_2^5$, $D_{10}.C_2^5$

Order 320: $D_{10}.C_2^4$ x 448, $C_2^4\times F_5$ x 92, $C_{20}:C_2^4$ x 56, $C_2^4:F_5$ x 56, $D_{10}.C_2^4$ x 28, $D_{10}:C_4^2$ x 28, $C_2^4\times D_{10}$ x 2, $C_2^4:D_{10}$ x 2, $C_{20}:C_2^4$, $C_2^3\times D_{20}$, $C_{20}:C_2^4$

Order 256: $C_4^2:C_2^4$

Order 160: $C_2^3\times F_5$ x 768, $D_4\times F_5$ x 512, $D_4\times D_{10}$ x 448, $C_2^3:F_5$ x 224, $D_{10}.D_4$ x 112, $C_{10}:C_4^2$ x 112, $C_2^3\times D_{10}$ x 91, $C_2^3:D_{10}$ x 56, $C_{20}:C_2^3$ x 28, $C_2^2\times D_{20}$ x 28, $C_{20}:C_2^3$ x 28, $C_2^4\times C_{10}$ x 2, $C_2^3\times C_{20}$, $C_{10}.C_2^4$

Order 128: $C_4^2:C_2^3$ x 56, $C_2^5\times C_4$ x 2, $C_2^5:C_4$ x 2, $D_4\times C_2^4$, $C_2^4.D_4$, $C_2^3\times C_4^2$

Order 80: $C_2^2\times F_5$ x 1416, $C_2^2\times D_{10}$ x 746, $D_4\times D_5$ x 512, $C_{10}:D_4$ x 224, $D_{10}:C_4$ x 128, $D_4\times C_{10}$ x 112, $C_2\times D_{20}$ x 112, $C_4\times D_{10}$ x 112, $C_{20}:C_4$ x 64, $C_4\times F_5$ x 64, $C_2^3\times C_{10}$ x 45, $C_2^2\times C_{20}$ x 14, $C_2^2.D_{10}$ x 14

Order 64: $C_4^2:C_2^2$ x 448, $C_2^4\times C_4$ x 93, $D_4\times C_2^3$ x 60, $C_2^4:C_4$ x 56, $C_2^3.D_4$ x 28, $C_2^2\times C_4^2$ x 28, $C_2^6$ x 2

Order 40: $C_2\times D_{10}$ x 1332, $C_2\times F_5$ x 568, $C_2^2\times C_{10}$ x 183, $C_5:D_4$ x 128, $D_{20}$ x 64, $C_4\times D_5$ x 64, $C_5\times D_4$ x 64, $C_2\times C_{20}$ x 28, $C_{10}:C_4$ x 28

Order 32: $C_2^3\times C_4$ x 798, $C_2^2\times D_4$ x 560, $C_4\times D_4$ x 512, $C_2^3:C_4$ x 224, $C_2^2.D_4$ x 112, $C_2\times C_4^2$ x 112, $C_2^5$ x 93

Order 20: $D_{10}$ x 504, $C_2\times C_{10}$ x 163, $F_5$ x 40, $C_{20}$ x 8, $C_5:C_4$ x 8

Order 16: $C_2^2\times C_4$ x 1556, $C_2\times D_4$ x 960, $C_2^4$ x 791, $C_2^2:C_4$ x 128, $C_4:C_4$ x 64, $C_4^2$ x 64

Order 10: $D_5$ x 32, $C_{10}$ x 31

Order 8: $C_2^3$ x 1515, $C_2\times C_4$ x 688, $D_4$ x 256

Order 5: $C_5$

Order 4: $C_2^2$ x 667, $C_4$ x 56

Order 2: $C_2$ x 63

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1280: $D_{10}.C_2^6$

Order 640: $F_5\times C_2^5$, $C_{20}:C_2^5$, $C_2^5:F_5$, $D_{10}.C_2^5$, $D_{10}.C_2^5$, $D_{10}.C_2^5$

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

Order 256: $C_4^2:C_2^4$

Order 160: $C_2^3\times F_5$ x 8, $C_2^3\times D_{10}$ x 6, $D_4\times D_{10}$ x 2, $C_{20}:C_2^3$ x 2, $C_2^4\times C_{10}$, $C_{20}:C_2^3$, $C_2^3\times C_{20}$, $C_2^3:D_{10}$, $C_{10}.C_2^4$, $C_2^2\times D_{20}$, $C_2^3:F_5$, $D_4\times F_5$, $D_{10}.D_4$, $C_{10}:C_4^2$

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

Order 80: $C_2^2\times D_{10}$ x 11, $C_2^2\times F_5$ x 8, $C_2^3\times C_{10}$ x 3, $D_4\times D_5$ x 2, $C_4\times D_{10}$ x 2, $D_4\times C_{10}$, $C_2^2\times C_{20}$, $C_{10}:D_4$, $C_2^2.D_{10}$, $C_2\times D_{20}$, $D_{10}:C_4$, $C_{20}:C_4$, $C_4\times F_5$

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

Order 40: $C_2\times D_{10}$ x 12, $C_2\times F_5$ x 7, $C_2^2\times C_{10}$ x 4, $C_4\times D_5$ x 2, $C_2\times C_{20}$, $C_5:D_4$, $C_{10}:C_4$, $D_{20}$, $C_5\times D_4$

Order 32: $C_2^3\times C_4$ x 12, $C_2^5$ x 7, $C_2^2\times D_4$ x 5, $C_4\times D_4$, $C_2^2.D_4$, $C_2^3:C_4$, $C_2\times C_4^2$

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

Order 16: $C_2^4$ x 14, $C_2^2\times C_4$ x 12, $C_2\times D_4$ x 5, $C_4:C_4$, $C_2^2:C_4$, $C_4^2$

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

Order 8: $C_2^3$ x 16, $C_2\times C_4$ x 11, $D_4$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 14, $C_4$ x 5

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1280: $D_{10}.C_2^6$ ($C_1$)

Order 640: $D_{10}.C_2^5$ ($C_2$) x 56, $F_5\times C_2^5$ ($C_2$) x 2, $C_2^5:F_5$ ($C_2$) x 2, $C_{20}:C_2^5$ ($C_2$), $D_{10}.C_2^5$ ($C_2$), $D_{10}.C_2^5$ ($C_2$)

Order 320: $D_{10}.C_2^4$ ($C_2^2$) x 448, $C_2^4\times F_5$ ($C_2^2$) x 60, $C_2^4:F_5$ ($C_2^2$) x 56, $C_{20}:C_2^4$ ($C_2^2$) x 28, $C_{20}:C_2^4$ ($C_4$) x 28, $D_{10}.C_2^4$ ($C_2^2$) x 28, $D_{10}:C_4^2$ ($C_2^2$) x 28, $C_2^4\times D_{10}$ ($C_2^2$) x 2, $C_2^4:D_{10}$ ($C_4$) x 2, $C_{20}:C_2^4$ ($C_4$), $C_2^3\times D_{20}$ ($C_4$), $C_{20}:C_2^4$ ($C_2^2$)

Order 160: $D_4\times F_5$ ($C_2^3$) x 512, $D_4\times D_{10}$ ($C_2\times C_4$) x 336, $C_2^3\times F_5$ ($C_2^3$) x 280, $C_2^3:F_5$ ($C_2^3$) x 224, $D_4\times D_{10}$ ($C_2^3$) x 112, $D_{10}.D_4$ ($C_2^3$) x 112, $C_{10}:C_4^2$ ($C_2^3$) x 112, $C_2^3:D_{10}$ ($C_2\times C_4$) x 56, $C_2^3\times D_{10}$ ($C_2\times C_4$) x 30, $C_2^3\times D_{10}$ ($C_2^3$) x 29, $C_{20}:C_2^3$ ($C_2\times C_4$) x 28, $C_2^2\times D_{20}$ ($C_2\times C_4$) x 28, $C_2^3\times F_5$ ($D_4$) x 16, $C_{20}:C_2^3$ ($C_2^3$) x 14, $C_{20}:C_2^3$ ($C_2\times C_4$) x 14, $C_2^4\times C_{10}$ ($C_2\times C_4$) x 2, $C_2^3\times C_{20}$ ($C_2\times C_4$), $C_{10}.C_2^4$ ($C_2\times C_4$)

Order 80: $D_4\times D_5$ ($C_2^2\times C_4$) x 448, $C_2^2\times F_5$ ($C_2^4$) x 240, $C_{10}:D_4$ ($C_2^2\times C_4$) x 224, $C_2^2\times D_{10}$ ($C_2^2\times C_4$) x 203, $D_{10}:C_4$ ($C_2^4$) x 128, $C_2^2\times F_5$ ($C_2\times D_4$) x 112, $D_4\times C_{10}$ ($C_2^2\times C_4$) x 112, $C_2\times D_{20}$ ($C_2^2\times C_4$) x 112, $C_4\times D_{10}$ ($C_2^2\times C_4$) x 84, $D_4\times D_5$ ($C_2^4$) x 64, $C_{20}:C_4$ ($C_2^4$) x 64, $C_4\times F_5$ ($C_2^4$) x 64, $C_2^2\times D_{10}$ ($C_2^4$) x 63, $C_2^3\times C_{10}$ ($C_2^2\times C_4$) x 29, $C_4\times D_{10}$ ($C_2^4$) x 28, $C_2^2\times C_{20}$ ($C_2^2\times C_4$) x 14, $C_2^2.D_{10}$ ($C_2^2\times C_4$) x 14, $C_2^2\times D_{10}$ ($D_4:C_2$) x 8, $C_2^2\times D_{10}$ ($C_2\times D_4$) x 8

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

Order 40: $C_2\times D_{10}$ ($C_2^3\times C_4$) x 189, $C_5:D_4$ ($C_2^3\times C_4$) x 128, $C_2\times F_5$ ($C_2^2\times D_4$) x 112, $D_{20}$ ($C_2^3\times C_4$) x 64, $C_5\times D_4$ ($C_2^3\times C_4$) x 64, $C_2^2\times C_{10}$ ($C_2^3\times C_4$) x 63, $C_4\times D_5$ ($C_2^3\times C_4$) x 56, $C_2\times D_{10}$ ($C_4\times D_4$) x 56, $C_2\times F_5$ ($C_2^5$) x 32, $C_2\times C_{20}$ ($C_2^3\times C_4$) x 28, $C_{10}:C_4$ ($C_2^3\times C_4$) x 28, $C_2\times D_{10}$ ($D_4:C_2^2$) x 28, $C_2\times D_{10}$ ($C_2^2\times D_4$) x 28, $C_2\times D_{10}$ ($C_2^5$) x 23, $C_4\times D_5$ ($C_2^5$) x 8, $C_2^2\times C_{10}$ ($C_4\times D_4$) x 8

Order 32: $C_2^2\times D_4$ ($C_2\times F_5$) x 28, $C_2^5$ ($C_2\times F_5$) x 2, $C_2^3\times C_4$ ($C_2\times F_5$)

Order 20: $D_{10}$ ($C_4^2:C_2^2$) x 84, $C_2\times C_{10}$ ($C_4^2:C_2^2$) x 28, $C_2\times C_{10}$ ($C_2^4\times C_4$) x 23, $D_{10}$ ($C_2^4\times C_4$) x 23, $F_5$ ($D_4\times C_2^3$) x 16, $D_{10}$ ($D_4:C_2^3$) x 14, $D_{10}$ ($D_4\times C_2^3$) x 14, $C_{20}$ ($C_2^4\times C_4$) x 8, $C_5:C_4$ ($C_2^4\times C_4$) x 8, $D_{10}$ ($C_2^6$)

Order 16: $C_2\times D_4$ ($C_2^2\times F_5$) x 112, $C_2^4$ ($C_2^2\times F_5$) x 29, $C_2^2\times C_4$ ($C_2^2\times F_5$) x 14

Order 10: $C_{10}$ ($C_4^2:C_2^3$) x 14, $D_5$ ($C_4^2:C_2^3$) x 14, $C_{10}$ ($C_2^5\times C_4$), $D_5$ ($D_4:C_2^4$), $D_5$ ($D_4\times C_2^4$)

Order 8: $D_4$ ($C_2^3\times F_5$) x 64, $C_2^3$ ($C_2^3\times F_5$) x 63, $C_2\times C_4$ ($C_2^3\times F_5$) x 28, $C_2^3$ ($D_4\times F_5$) x 8

Order 5: $C_5$ ($C_4^2:C_2^4$)

Order 4: $C_2^2$ ($D_{10}.C_2^4$) x 28, $C_2^2$ ($C_2^4\times F_5$) x 23, $C_4$ ($C_2^4\times F_5$) x 8

Order 2: $C_2$ ($D_{10}.C_2^5$) x 14, $C_2$ ($F_5\times C_2^5$)

Order 1: $C_1$ ($D_{10}.C_2^6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1280: $D_{10}.C_2^6$ ($C_1$)

Order 640: $F_5\times C_2^5$ ($C_2$), $C_{20}:C_2^5$ ($C_2$), $C_2^5:F_5$ ($C_2$), $D_{10}.C_2^5$ ($C_2$), $D_{10}.C_2^5$ ($C_2$), $D_{10}.C_2^5$ ($C_2$)

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

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

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

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

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

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

Order 20: $C_2\times C_{10}$ ($C_2^4\times C_4$) x 2, $D_{10}$ ($C_2^4\times C_4$) x 2, $C_2\times C_{10}$ ($C_4^2:C_2^2$), $D_{10}$ ($C_2^6$), $D_{10}$ ($D_4:C_2^3$), $D_{10}$ ($D_4\times C_2^3$), $D_{10}$ ($C_4^2:C_2^2$), $F_5$ ($D_4\times C_2^3$), $C_{20}$ ($C_2^4\times C_4$), $C_5:C_4$ ($C_2^4\times C_4$)

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

Order 10: $C_{10}$ ($C_2^5\times C_4$), $C_{10}$ ($C_4^2:C_2^3$), $D_5$ ($D_4:C_2^4$), $D_5$ ($D_4\times C_2^4$), $D_5$ ($C_4^2:C_2^3$)

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

Order 5: $C_5$ ($C_4^2:C_2^4$)

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

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

Order 1: $C_1$ ($D_{10}.C_2^6$)

Series

Derived series

$D_{10}.C_2^6$

$\rhd$

$C_{10}$

$\rhd$

$C_1$

Chief series

$D_{10}.C_2^6$

$\rhd$

$D_{10}.C_2^5$

$\rhd$

$C_{20}:C_2^4$

$\rhd$

$C_2^3\times C_{20}$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$D_{10}.C_2^6$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2^4$

$\lhd$

$D_4\times C_2^3$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 3 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 $160 \times 160$ rational character table (warning: may be slow to load).