Abstract group 320.1591: $D_{10}.C_2^4$

• Label: 320.1591
• Order: \( 2^{6} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $80$
• Rank: $4$

Group information

Description:	$D_{10}.C_2^4$
Order:	\(320\)\(\medspace = 2^{6} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$F_5\times C_2^6:(C_2\times S_4)$, of order \(61440\)\(\medspace = 2^{12} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_5$
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	47	208	4	28	32	320
Conjugacy classes	1	15	24	1	7	8	56
Divisions	1	15	16	1	7	4	44
Autjugacy classes	1	5	4	1	2	1	14

Dimension	1	2	4	8
Irr. complex chars.	32	8	16	0	56
Irr. rational chars.	16	16	8	4	44

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$80$
Rank:	$4$
Inequivalent generating quadruples:	$52080$

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:	$\langle a, b, c, d \mid a^{2}=b^{4}=c^{2}=d^{20}=[a,b]=[a,c]=[a,d]=[b,c]=[c,d]=1, d^{b}=d^{3} \rangle$
Permutation group:	Degree $13$ $\langle(1,2)(3,4)(5,6)(7,8)(10,11,12,13), (1,3)(2,4)(5,7)(6,8), (1,3)(2,4)(5,8)(6,7), (1,4,3,2)(5,8)(6,7), (10,12)(11,13), (1,3)(2,4), (9,10,11,13,12)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 10 \\ 10 & 3 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 9 & 3 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_{20}:C_4)$
Semidirect product:	$(C_4\times D_{10})$ $\,\rtimes\,$ $C_4$	$D_{10}$ $\,\rtimes\,$ $(C_4:C_4)$	$D_5$ $\,\rtimes\,$ $(C_2^2.D_4)$	$(C_2^2\times C_4)$ $\,\rtimes\,$ $F_5$	all 15
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}$ . $C_2^4$	$(C_2\times D_{10})$ . $Q_8$	$D_{10}$ . $(C_2\times Q_8)$	$(C_2\times D_{10})$ . $D_4$	all 24

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:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 1386 subgroups in 418 conjugacy classes, 196 normal (15 characteristic).

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

Special subgroups

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 320: $D_{10}.C_2^4$

Order 160: $D_{10}.D_4$ x 12, $C_2^3\times F_5$ x 2, $C_{20}:C_2^3$

Order 80: $C_2^2\times F_5$ x 20, $C_{20}:C_4$ x 16, $C_4\times D_{10}$ x 12, $C_2^2\times D_{10}$, $C_2^2\times C_{20}$, $C_2^2.D_{10}$

Order 64: $C_2^3.D_4$

Order 40: $C_2\times F_5$ x 32, $C_4\times D_5$ x 16, $C_2\times D_{10}$ x 14, $C_2\times C_{20}$ x 6, $C_{10}:C_4$ x 6, $C_2^2\times C_{10}$

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

Order 20: $D_{10}$ x 28, $F_5$ x 8, $C_2\times C_{10}$ x 7, $C_{20}$ x 4, $C_5:C_4$ x 4

Order 16: $C_2^2\times C_4$ x 34, $C_4:C_4$ x 16, $C_2^4$

Order 10: $D_5$ x 8, $C_{10}$ x 7

Order 8: $C_2\times C_4$ x 60, $C_2^3$ x 15

Order 5: $C_5$

Order 4: $C_2^2$ x 35, $C_4$ x 16

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 320: $D_{10}.C_2^4$

Order 160: $C_2^3\times F_5$, $C_{20}:C_2^3$, $D_{10}.D_4$

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

Order 64: $C_2^3.D_4$

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

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

Order 20: $D_{10}$ x 5, $C_2\times C_{10}$ x 2, $F_5$, $C_{20}$, $C_5:C_4$

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

Order 10: $D_5$ x 3, $C_{10}$ x 2

Order 8: $C_2\times C_4$ x 6, $C_2^3$ x 5

Order 5: $C_5$

Order 4: $C_2^2$ x 7, $C_4$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 320: $D_{10}.C_2^4$ ($C_1$)

Order 160: $D_{10}.D_4$ ($C_2$) x 12, $C_2^3\times F_5$ ($C_2$) x 2, $C_{20}:C_2^3$ ($C_2$)

Order 80: $C_{20}:C_4$ ($C_2^2$) x 16, $C_2^2\times F_5$ ($C_2^2$) x 12, $C_4\times D_{10}$ ($C_2^2$) x 6, $C_4\times D_{10}$ ($C_4$) x 6, $C_2^2\times D_{10}$ ($C_2^2$), $C_2^2\times C_{20}$ ($C_4$), $C_2^2.D_{10}$ ($C_4$)

Order 40: $C_4\times D_5$ ($C_2\times C_4$) x 12, $C_2\times F_5$ ($C_2^3$) x 8, $C_2\times C_{20}$ ($C_2\times C_4$) x 6, $C_{10}:C_4$ ($C_2\times C_4$) x 6, $C_4\times D_5$ ($C_2^3$) x 4, $C_2\times D_{10}$ ($Q_8$) x 4, $C_2\times D_{10}$ ($D_4$) x 4, $C_2\times D_{10}$ ($C_2^3$) x 3, $C_2\times D_{10}$ ($C_2\times C_4$) x 3, $C_2^2\times C_{10}$ ($C_2\times C_4$)

Order 20: $D_{10}$ ($C_4:C_4$) x 12, $D_{10}$ ($C_2\times Q_8$) x 6, $D_{10}$ ($C_2\times D_4$) x 6, $C_2\times C_{10}$ ($C_4:C_4$) x 4, $C_{20}$ ($C_2^2\times C_4$) x 4, $C_5:C_4$ ($C_2^2\times C_4$) x 4, $C_2\times C_{10}$ ($C_2^2\times C_4$) x 3, $D_{10}$ ($C_2^2\times C_4$) x 3, $D_{10}$ ($C_2^4$)

Order 16: $C_2^2\times C_4$ ($F_5$)

Order 10: $C_{10}$ ($C_2^2.D_4$) x 6, $D_5$ ($C_2^2.D_4$) x 6, $C_{10}$ ($C_2^3\times C_4$), $D_5$ ($C_2^2\times Q_8$), $D_5$ ($C_2^2\times D_4$)

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

Order 5: $C_5$ ($C_2^3.D_4$)

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

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

Order 1: $C_1$ ($D_{10}.C_2^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 320: $D_{10}.C_2^4$ ($C_1$)

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

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

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

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

Order 16: $C_2^2\times C_4$ ($F_5$)

Order 10: $C_{10}$ ($C_2^3\times C_4$), $C_{10}$ ($C_2^2.D_4$), $D_5$ ($C_2^2\times Q_8$), $D_5$ ($C_2^2\times D_4$), $D_5$ ($C_2^2.D_4$)

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

Order 5: $C_5$ ($C_2^3.D_4$)

Order 4: $C_2^2$ ($C_2^2\times F_5$), $C_2^2$ ($C_{20}:C_4$), $C_4$ ($C_2^2\times F_5$)

Order 2: $C_2$ ($C_2^3\times F_5$), $C_2$ ($D_{10}.D_4$)

Order 1: $C_1$ ($D_{10}.C_2^4$)

Series

Derived series

$D_{10}.C_2^4$

$\rhd$

$C_{10}$

$\rhd$

$C_1$

Chief series

$D_{10}.C_2^4$

$\rhd$

$C_2^3\times F_5$

$\rhd$

$C_2^2\times F_5$

$\rhd$

$C_2\times D_{10}$

$\rhd$

$D_{10}$

$\rhd$

$D_5$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$D_{10}.C_2^4$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2\times C_4$

Supergroups

This group is a maximal subgroup of 34 larger groups in the database.

This group is a maximal quotient of 22 larger groups in the database.

Character theory

Complex character table

See the $56 \times 56$ 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.