Abstract group 64.4: $C_2^3:C_8$

• Label: 64.4
• Order: \( 2^{6} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $2$

Group information

Description:	$C_2^3:C_8$
Order:	\(64\)\(\medspace = 2^{6} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^5:D_4$, of order \(256\)\(\medspace = 2^{8} \)
Composition factors:	$C_2$ x 6
Nilpotency class:	$3$
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
Elements	1	15	16	32	64
Conjugacy classes	1	7	6	8	22
Divisions	1	7	3	2	13
Autjugacy classes	1	6	2	1	10

Dimension	1	2	4
Irr. complex chars.	16	4	2	22
Irr. rational chars.	4	4	5	13

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	5	6	8

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{8}=c^{2}=d^{2}=[a,c]=[a,d]=[b,c]=[c,d]=1, b^{a}=bcd, d^{b}=cd \rangle$
Permutation group:	Degree $16$ $\langle(1,14,6,10,2,13,5,9)(3,16,8,12,4,15,7,11), (1,2)(3,4)(5,8)(6,7)(13,15)(14,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 3 & 4 \\ 4 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 2 \\ 2 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/8\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 15 \\ 12 & 7 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 15 & 2 \\ 2 & 21 \end{array}\right), \left(\begin{array}{rr} 9 & 10 \\ 10 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$
Transitive group:	16T84	16T85	32T81	32T82	all 5
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_2^3$ $\,\rtimes\,$ $C_8$ (2)	$(C_2^2:C_8)$ $\,\rtimes\,$ $C_2$ (2)			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $C_4$	$C_2^2$ . $\OD_{16}$	$(C_2\times C_4)$ . $D_4$ (2)	$(C_2^3:C_4)$ . $C_2$	all 12

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{8}\Z)$.

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 109 subgroups in 49 conjugacy classes, 19 normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^2:C_4$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_2^2\times C_4$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3:C_8$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^3:C_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^2:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3:C_8$

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: $C_2^3:C_8$

Order 32: $C_2^2:C_8$ x 2, $C_2^3:C_4$

Order 16: $C_2\times C_8$ x 2, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_2^4$

Order 8: $C_2^3$ x 7, $C_2\times C_4$ x 5, $C_8$ x 2

Order 4: $C_2^2$ x 13, $C_4$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 64: $C_2^3:C_8$

Order 32: $C_2^2:C_8$, $C_2^3:C_4$

Order 16: $C_2^2\times C_4$ x 2, $C_2\times C_8$, $C_2^2:C_4$, $C_2^4$

Order 8: $C_2^3$ x 6, $C_2\times C_4$ x 4, $C_8$

Order 4: $C_2^2$ x 10, $C_4$ x 2

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 64: $C_2^3:C_8$ ($C_1$)

Order 32: $C_2^2:C_8$ ($C_2$) x 2, $C_2^3:C_4$ ($C_2$)

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

Order 8: $C_2^3$ ($C_8$) x 2, $C_2\times C_4$ ($D_4$) x 2, $C_2^3$ ($C_2\times C_4$)

Order 4: $C_2^2$ ($\OD_{16}$), $C_2^2$ ($C_2\times C_8$), $C_2^2$ ($C_2^2:C_4$)

Order 2: $C_2$ ($\OD_{16}:C_2$), $C_2$ ($C_2^3:C_4$), $C_2$ ($C_2^2:C_8$)

Order 1: $C_1$ ($C_2^3:C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 64: $C_2^3:C_8$ ($C_1$)

Order 32: $C_2^2:C_8$ ($C_2$), $C_2^3:C_4$ ($C_2$)

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

Order 8: $C_2^3$ ($C_2\times C_4$), $C_2^3$ ($C_8$), $C_2\times C_4$ ($D_4$)

Order 4: $C_2^2$ ($\OD_{16}$), $C_2^2$ ($C_2\times C_8$), $C_2^2$ ($C_2^2:C_4$)

Order 2: $C_2$ ($\OD_{16}:C_2$), $C_2$ ($C_2^3:C_4$), $C_2$ ($C_2^2:C_8$)

Order 1: $C_1$ ($C_2^3:C_8$)

Series

Derived series

$C_2^3:C_8$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^3:C_8$

$\rhd$

$C_2^2:C_8$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3:C_8$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_2^3:C_8$

Supergroups

This group is a maximal subgroup of 82 larger groups in the database.

This group is a maximal quotient of 85 larger groups in the database.

Character theory

Complex character table

See the $22 \times 22$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	8A	8B
	Size	1	1	1	1	2	2	4	4	4	4	8	16	16
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	2C	2C	2B	4A	4B
64.4.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.4.1b		$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$
64.4.1c		$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
64.4.1d		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
64.4.1e		$2$	$2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$-2$	$-2$	$2$	$0$	$0$
64.4.1f		$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$-2$	$0$	$0$
64.4.1g		$4$	$-4$	$-4$	$4$	$4$	$-4$	$-4$	$4$	$0$	$0$	$0$	$0$	$0$
64.4.1h		$4$	$-4$	$-4$	$4$	$4$	$-4$	$4$	$-4$	$0$	$0$	$0$	$0$	$0$
64.4.2a		$2$	$2$	$2$	$2$	$-2$	$-2$	$0$	$0$	$-2$	$2$	$0$	$0$	$0$
64.4.2b		$2$	$2$	$2$	$2$	$-2$	$-2$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$
64.4.2c		$4$	$-4$	$-4$	$4$	$-4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.4.4a		$4$	$4$	$-4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.4.4b		$4$	$-4$	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$