Abstract group $C_2\times Q_{64}$

• Label: 128.993
• Order: \( 2^{7} \)
• Exponent: \( 2^{5} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{12} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $66$
• Trans deg.: $128$
• Rank: $3$

Group information

Description:	$C_2\times Q_{64}$
Order:	\(128\)\(\medspace = 2^{7} \)
Exponent:	\(32\)\(\medspace = 2^{5} \)
Automorphism group:	$C_8^2.C_8:C_2^3$, of order \(4096\)\(\medspace = 2^{12} \) (generators)
Outer automorphisms:	$C_4^2.C_2^3$, of order \(128\)\(\medspace = 2^{7} \)
Composition factors:	$C_2$ x 7
Nilpotency class:	$5$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	16	32
Elements	1	3	68	8	16	32	128
Conjugacy classes	1	3	6	4	8	16	38
Divisions	1	3	6	2	2	2	16
Autjugacy classes	1	2	3	2	2	1	11

Dimension	1	2	4	8	16
Irr. complex chars.	8	30	0	0	0	38
Irr. rational chars.	8	2	2	2	2	16

Minimal Presentations

Permutation degree:	$66$
Transitive degree:	$128$
Rank:	$3$
Inequivalent generating triples:	$168$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	5	33

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=c^{32}=[a,b]=[a,c]=1, b^{2}=c^{16}, c^{b}=c^{31} \rangle$
Permutation group:	Degree $66$ $\langle(1,2,4,9)(3,13,14,43)(5,21,18,42)(6,26,19,51)(7,28,20,12)(8,30,31,15)(10,36,34,24) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 49 & 18 \\ 48 & 49 \end{array}\right), \left(\begin{array}{rr} 63 & 0 \\ 0 & 63 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 45 \\ 8 & 27 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 32 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/64\Z)$
	$\left\langle \left(\begin{array}{rr} 29 & 36 \\ 30 & 33 \end{array}\right), \left(\begin{array}{rr} 1 & 31 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 53 & 8 \\ 44 & 9 \end{array}\right), \left(\begin{array}{rr} 61 & 32 \\ 52 & 9 \end{array}\right), \left(\begin{array}{rr} 61 & 0 \\ 0 & 61 \end{array}\right), \left(\begin{array}{rr} 17 & 26 \\ 58 & 45 \end{array}\right), \left(\begin{array}{rr} 31 & 20 \\ 48 & 45 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/62\Z)$
Direct product:	$C_2$ $\, \times\, $ $Q_{64}$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_8$ . $D_8$	$C_{16}$ . $D_4$	$C_4$ . $D_{16}$	$Q_{32}$ . $C_2^2$ (4)	all 15

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 140 subgroups in 52 conjugacy classes, 28 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_{16}$
Commutator:	$G' \simeq$ $C_{16}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times Q_{64}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times Q_{64}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $D_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times Q_{64}$

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 128: $C_2\times Q_{64}$

Order 64: $Q_{64}$ x 4, $C_2\times Q_{32}$ x 2, $C_2\times C_{32}$

Order 32: $Q_{32}$ x 6, $C_2\times Q_{16}$ x 2, $C_{32}$ x 2, $C_2\times C_{16}$

Order 16: $Q_{16}$ x 6, $C_2\times Q_8$ x 2, $C_{16}$ x 2, $C_2\times C_8$

Order 8: $Q_8$ x 6, $C_2\times C_4$ x 3, $C_8$ x 2

Order 4: $C_4$ x 6, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 128: $C_2\times Q_{64}$

Order 64: $Q_{64}$, $C_2\times C_{32}$, $C_2\times Q_{32}$

Order 32: $Q_{32}$ x 2, $C_2\times Q_{16}$, $C_2\times C_{16}$, $C_{32}$

Order 16: $Q_{16}$ x 2, $C_{16}$ x 2, $C_2\times C_8$, $C_2\times Q_8$

Order 8: $Q_8$ x 2, $C_2\times C_4$ x 2, $C_8$ x 2

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 128: $C_2\times Q_{64}$ ($C_1$)

Order 64: $Q_{64}$ ($C_2$) x 4, $C_2\times Q_{32}$ ($C_2$) x 2, $C_2\times C_{32}$ ($C_2$)

Order 32: $Q_{32}$ ($C_2^2$) x 4, $C_{32}$ ($C_2^2$) x 2, $C_2\times C_{16}$ ($C_2^2$)

Order 16: $C_2\times C_8$ ($D_4$), $C_{16}$ ($C_2^3$), $C_{16}$ ($D_4$)

Order 8: $C_2\times C_4$ ($D_8$), $C_8$ ($D_8$), $C_8$ ($C_2\times D_4$)

Order 4: $C_2^2$ ($D_{16}$), $C_4$ ($C_2\times D_8$), $C_4$ ($D_{16}$)

Order 2: $C_2$ ($Q_{64}$) x 2, $C_2$ ($C_2\times D_{16}$)

Order 1: $C_1$ ($C_2\times Q_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 128: $C_2\times Q_{64}$ ($C_1$)

Order 64: $Q_{64}$ ($C_2$), $C_2\times C_{32}$ ($C_2$), $C_2\times Q_{32}$ ($C_2$)

Order 32: $Q_{32}$ ($C_2^2$), $C_2\times C_{16}$ ($C_2^2$), $C_{32}$ ($C_2^2$)

Order 16: $C_2\times C_8$ ($D_4$), $C_{16}$ ($C_2^3$), $C_{16}$ ($D_4$)

Order 8: $C_2\times C_4$ ($D_8$), $C_8$ ($D_8$), $C_8$ ($C_2\times D_4$)

Order 4: $C_2^2$ ($D_{16}$), $C_4$ ($C_2\times D_8$), $C_4$ ($D_{16}$)

Order 2: $C_2$ ($Q_{64}$), $C_2$ ($C_2\times D_{16}$)

Order 1: $C_1$ ($C_2\times Q_{64}$)

Series

Derived series

$C_2\times Q_{64}$

$\rhd$

$C_{16}$

$\rhd$

$C_1$

Chief series

$C_2\times Q_{64}$

$\rhd$

$Q_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2\times Q_{64}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times C_8$

$\lhd$

$C_2\times C_{16}$

$\lhd$

$C_2\times Q_{64}$

Supergroups

This group is a maximal subgroup of 26 larger groups in the database.

This group is a maximal quotient of 16 larger groups in the database.

Character theory

Complex character table

See the $38 \times 38$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	4A	4B	4C	4D	4E	4F	8A	8B	16A	16B	32A	32B
	Size	1	1	1	1	2	2	16	16	16	16	4	4	8	8	16	16
	2 P	1A	1A	1A	1A	2A	2A	2A	2A	2A	2A	4A	4A	8A	8A	16A	16A
	Schur
128.993.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
128.993.1b		$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
128.993.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$
128.993.1d		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$
128.993.1e		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
128.993.1f		$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$
128.993.1g		$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
128.993.1h		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
128.993.2a		$2$	$-2$	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$2$	$-2$	$-2$	$2$	$0$	$0$
128.993.2b		$2$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$2$	$2$	$-2$	$-2$	$0$	$0$
128.993.2c		$4$	$-4$	$4$	$-4$	$4$	$-4$	$0$	$0$	$0$	$0$	$-4$	$4$	$0$	$0$	$0$	$0$
128.993.2d		$4$	$4$	$4$	$4$	$4$	$4$	$0$	$0$	$0$	$0$	$-4$	$-4$	$0$	$0$	$0$	$0$
128.993.2e		$8$	$-8$	$8$	$-8$	$-8$	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.993.2f		$8$	$8$	$8$	$8$	$-8$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.993.2g	2	$16$	$-16$	$-16$	$16$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.993.2h	2	$16$	$16$	$-16$	$-16$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$