Abstract group 320.178: $C_5\times Q_{64}$

• Label: 320.178
• Order: \( 2^{6} \cdot 5 \)
• Exponent: \( 2^{5} \cdot 5 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{11} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $69$
• Trans deg.: $320$
• Rank: $2$

Group information

Description:	$C_5\times Q_{64}$
Order:	\(320\)\(\medspace = 2^{6} \cdot 5 \)
Exponent:	\(160\)\(\medspace = 2^{5} \cdot 5 \)
Automorphism group:	$C_4\times D_{32}:C_8$, of order \(2048\)\(\medspace = 2^{11} \)
Composition factors:	$C_2$ x 6, $C_5$
Nilpotency class:	$5$
Derived length:	$2$

This group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).

Group statistics

Order	1	2	4	5	8	10	16	20	32	40	80	160
Elements	1	1	34	4	4	4	8	136	16	16	32	64	320
Conjugacy classes	1	1	3	4	2	4	4	12	8	8	16	32	95
Divisions	1	1	3	1	1	1	1	3	1	1	1	1	16
Autjugacy classes	1	1	2	1	1	1	1	2	1	1	1	1	14

Dimension	1	2	4	8	16	32	64
Irr. complex chars.	20	75	0	0	0	0	0	95
Irr. rational chars.	4	1	5	2	2	1	1	16

Minimal presentations

Permutation degree:	$69$
Transitive degree:	$320$
Rank:	$2$
Inequivalent generating pairs:	$18$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	64
Arbitrary	2	4	36

Constructions

Presentation:	$\langle a, b \mid b^{160}=1, a^{2}=b^{80}, b^{a}=b^{31} \rangle$
Permutation group:	Degree $69$ $\langle(1,34,2,33)(3,36,4,35)(5,38,6,37)(7,40,8,39)(9,42,10,41)(11,44,12,43)(13,46,14,45) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 21 & 3 \\ 1 & 21 \end{array}\right), \left(\begin{array}{rr} 7 & 3 \\ 30 & 24 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{31})$
Direct product:	$C_5$ $\, \times\, $ $Q_{64}$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{40}$ . $D_4$	$C_{20}$ . $D_8$	$Q_{32}$ . $C_{10}$ (2)	$C_{10}$ . $D_{16}$	all 12

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

Homology

Abelianization:	$C_{2} \times C_{10} \simeq C_{2}^{2} \times C_{5}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 74 subgroups in 30 conjugacy classes, 18 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $D_{16}$
Commutator:	$G' \simeq$ $C_{16}$	$G/G' \simeq$ $C_2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_2\times C_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_5\times Q_{64}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_5\times Q_{64}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{10}$	$G/\operatorname{soc} \simeq$ $D_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{64}$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 320: $C_5\times Q_{64}$

Order 160: $C_5\times Q_{32}$ x 2, $C_{160}$

Order 80: $C_5\times Q_{16}$ x 2, $C_{80}$

Order 64: $Q_{64}$

Order 40: $C_5\times Q_8$ x 2, $C_{40}$

Order 32: $Q_{32}$ x 2, $C_{32}$

Order 20: $C_{20}$ x 3

Order 16: $Q_{16}$ x 2, $C_{16}$

Order 10: $C_{10}$

Order 8: $Q_8$ x 2, $C_8$

Order 5: $C_5$

Order 4: $C_4$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 320: $C_5\times Q_{64}$

Order 160: $C_5\times Q_{32}$, $C_{160}$

Order 80: $C_5\times Q_{16}$, $C_{80}$

Order 64: $Q_{64}$

Order 40: $C_{40}$, $C_5\times Q_8$

Order 32: $Q_{32}$, $C_{32}$

Order 20: $C_{20}$ x 2

Order 16: $Q_{16}$, $C_{16}$

Order 10: $C_{10}$

Order 8: $Q_8$, $C_8$

Order 5: $C_5$

Order 4: $C_4$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 320: $C_5\times Q_{64}$ ($C_1$)

Order 160: $C_5\times Q_{32}$ ($C_2$) x 2, $C_{160}$ ($C_2$)

Order 80: $C_{80}$ ($C_2^2$)

Order 64: $Q_{64}$ ($C_5$)

Order 40: $C_{40}$ ($D_4$)

Order 32: $Q_{32}$ ($C_{10}$) x 2, $C_{32}$ ($C_{10}$)

Order 20: $C_{20}$ ($D_8$)

Order 16: $C_{16}$ ($C_2\times C_{10}$)

Order 10: $C_{10}$ ($D_{16}$)

Order 8: $C_8$ ($C_5\times D_4$)

Order 5: $C_5$ ($Q_{64}$)

Order 4: $C_4$ ($C_5\times D_8$)

Order 2: $C_2$ ($C_5\times D_{16}$)

Order 1: $C_1$ ($C_5\times Q_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 320: $C_5\times Q_{64}$ ($C_1$)

Order 160: $C_5\times Q_{32}$ ($C_2$), $C_{160}$ ($C_2$)

Order 80: $C_{80}$ ($C_2^2$)

Order 64: $Q_{64}$ ($C_5$)

Order 40: $C_{40}$ ($D_4$)

Order 32: $Q_{32}$ ($C_{10}$), $C_{32}$ ($C_{10}$)

Order 20: $C_{20}$ ($D_8$)

Order 16: $C_{16}$ ($C_2\times C_{10}$)

Order 10: $C_{10}$ ($D_{16}$)

Order 8: $C_8$ ($C_5\times D_4$)

Order 5: $C_5$ ($Q_{64}$)

Order 4: $C_4$ ($C_5\times D_8$)

Order 2: $C_2$ ($C_5\times D_{16}$)

Order 1: $C_1$ ($C_5\times Q_{64}$)

Series

Derived series

$C_5\times Q_{64}$

$\rhd$

$C_{16}$

$\rhd$

$C_1$

Chief series

$C_5\times Q_{64}$

$\rhd$

$C_5\times Q_{32}$

$\rhd$

$C_{80}$

$\rhd$

$C_{40}$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_5\times Q_{64}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_{20}$

$\lhd$

$C_{40}$

$\lhd$

$C_{80}$

$\lhd$

$C_5\times Q_{64}$

Supergroups

This group is a maximal subgroup of 11 larger groups in the database.

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

Character theory

Complex character table

See the $95 \times 95$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	4A	4B	4C	5A	8A	10A	16A	20A	20B	20C	32A	40A	80A	160A
	Size	1	1	2	16	16	4	4	4	8	8	64	64	16	16	32	64
	2 P	1A	1A	2A	2A	2A	5A	4A	5A	8A	10A	10A	10A	16A	20A	40A	80A
	5 P	1A	2A	4A	4B	4C	1A	8A	2A	16A	4A	4B	4C	32A	8A	16A	32A
	Schur
320.178.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
320.178.1b		$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
320.178.1c		$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$
320.178.1d		$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
320.178.1e		$4$	$4$	$4$	$4$	$4$	$-1$	$4$	$-1$	$4$	$-1$	$-1$	$-1$	$4$	$-1$	$-1$	$-1$
320.178.1f		$4$	$4$	$4$	$-4$	$-4$	$-1$	$4$	$-1$	$4$	$-1$	$1$	$1$	$4$	$-1$	$-1$	$-1$
320.178.1g		$4$	$4$	$4$	$-4$	$4$	$-1$	$4$	$-1$	$4$	$-1$	$1$	$-1$	$-4$	$-1$	$-1$	$1$
320.178.1h		$4$	$4$	$4$	$4$	$-4$	$-1$	$4$	$-1$	$4$	$-1$	$-1$	$1$	$-4$	$-1$	$-1$	$1$
320.178.2a		$2$	$2$	$2$	$0$	$0$	$2$	$2$	$2$	$-2$	$2$	$0$	$0$	$0$	$2$	$-2$	$0$
320.178.2b		$4$	$4$	$4$	$0$	$0$	$4$	$-4$	$4$	$0$	$4$	$0$	$0$	$0$	$-4$	$0$	$0$
320.178.2c		$8$	$8$	$-8$	$0$	$0$	$8$	$0$	$8$	$0$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$
320.178.2d		$8$	$8$	$8$	$0$	$0$	$-2$	$8$	$-2$	$-8$	$-2$	$0$	$0$	$0$	$-2$	$2$	$0$
320.178.2e		$16$	$16$	$16$	$0$	$0$	$-4$	$-16$	$-4$	$0$	$-4$	$0$	$0$	$0$	$4$	$0$	$0$
320.178.2f	2	$16$	$-16$	$0$	$0$	$0$	$16$	$0$	$-16$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
320.178.2g		$32$	$32$	$-32$	$0$	$0$	$-8$	$0$	$-8$	$0$	$8$	$0$	$0$	$0$	$0$	$0$	$0$
320.178.2h		$64$	$-64$	$0$	$0$	$0$	$-16$	$0$	$16$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$