Abstract group 64.54: $Q_{64}$

• Label: 64.54
• Order: \( 2^{6} \)
• Exponent: \( 2^{5} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $64$
• Trans deg.: $64$
• Rank: $2$

Group information

Description:	$Q_{64}$
Order:	\(64\)\(\medspace = 2^{6} \)
Exponent:	\(32\)\(\medspace = 2^{5} \)
Automorphism group:	$D_{32}:C_8$, of order \(512\)\(\medspace = 2^{9} \)
Composition factors:	$C_2$ x 6
Nilpotency class:	$5$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	16	32
Elements	1	1	34	4	8	16	64
Conjugacy classes	1	1	3	2	4	8	19
Divisions	1	1	3	1	1	1	8
Autjugacy classes	1	1	2	1	1	1	7

Dimension	1	2	4	8	16
Irr. complex chars.	4	15	0	0	0	19
Irr. rational chars.	4	1	1	1	1	8

Minimal presentations

Permutation degree:	$64$
Transitive degree:	$64$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	32
Arbitrary	2	4	32

Constructions

Groups of Lie type:	$\CSOMinus(2,31)$
Presentation:	$\langle a, b \mid b^{32}=1, a^{2}=b^{16}, b^{a}=b^{31} \rangle$
Permutation group:	Degree $64$ $\langle(1,34,2,33)(3,36,4,35)(5,38,6,37)(7,40,8,39)(9,42,10,41)(11,44,12,43)(13,46,14,45) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 29 & 28 \\ 30 & 29 \end{array}\right), \left(\begin{array}{rr} 8 & 28 \\ 1 & 23 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{31})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_8$ . $D_4$	$C_4$ . $D_8$	$Q_{32}$ . $C_2$ (2)	$C_2$ . $D_{16}$	all 6

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 37 subgroups in 15 conjugacy classes, 9 normal (7 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 64: $Q_{64}$

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

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

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

Order 4: $C_4$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 64: $Q_{64}$

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

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

Order 8: $Q_8$, $C_8$

Order 4: $C_4$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 64: $Q_{64}$ ($C_1$)

Order 32: $Q_{32}$ ($C_2$) x 2, $C_{32}$ ($C_2$)

Order 16: $C_{16}$ ($C_2^2$)

Order 8: $C_8$ ($D_4$)

Order 4: $C_4$ ($D_8$)

Order 2: $C_2$ ($D_{16}$)

Order 1: $C_1$ ($Q_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 64: $Q_{64}$ ($C_1$)

Order 32: $Q_{32}$ ($C_2$), $C_{32}$ ($C_2$)

Order 16: $C_{16}$ ($C_2^2$)

Order 8: $C_8$ ($D_4$)

Order 4: $C_4$ ($D_8$)

Order 2: $C_2$ ($D_{16}$)

Order 1: $C_1$ ($Q_{64}$)

Series

Derived series

$Q_{64}$

$\rhd$

$C_{16}$

$\rhd$

$C_1$

Chief series

$Q_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$Q_{64}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$C_8$

$\lhd$

$C_{16}$

$\lhd$

$Q_{64}$

Supergroups

This group is a maximal subgroup of 41 larger groups in the database.

This group is a maximal quotient of 38 larger groups in the database.

Character theory

Complex character table

See the $19 \times 19$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	4A	4B	4C	8A	16A	32A
	Size	1	1	2	16	16	4	8	16
	2 P	1A	1A	2A	2A	2A	4A	8A	16A
	Schur
64.54.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.54.1b		$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$
64.54.1c		$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$
64.54.1d		$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$
64.54.2a		$2$	$2$	$2$	$0$	$0$	$2$	$-2$	$0$
64.54.2b		$4$	$4$	$4$	$0$	$0$	$-4$	$0$	$0$
64.54.2c		$8$	$8$	$-8$	$0$	$0$	$0$	$0$	$0$
64.54.2d	2	$16$	$-16$	$0$	$0$	$0$	$0$	$0$	$0$