Abstract group 64.127: $C_8:Q_8$

• Label: 64.127
• Order: \( 2^{6} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $16$
• Trans deg.: $64$
• Rank: $3$

Group information

Description:	$C_8:Q_8$
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:	$2$
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	3	28	32	64
Conjugacy classes	1	3	12	12	28
Divisions	1	3	9	5	18
Autjugacy classes	1	3	6	3	13

Dimension	1	2	4	8
Irr. complex chars.	16	12	0	0	28
Irr. rational chars.	8	6	3	1	18

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$64$
Rank:	$3$
Inequivalent generating triples:	$336$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	8	8

Constructions

Presentation:	$\langle a, b, c \mid b^{4}=c^{8}=[a,c]=1, a^{2}=b^{2}, b^{a}=b^{3}, c^{b}=c^{5} \rangle$
Permutation group:	Degree $16$ $\langle(1,2,5,4,3,6,8,7)(9,10,12,14)(11,15,16,13), (1,3)(2,6)(4,7)(5,8)(9,11,12,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 4 & 3 & 2 & 3 \\ 2 & 3 & 3 & 4 \\ 2 & 3 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 1 & 3 & 3 & 3 \\ 1 & 4 & 2 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 3 & 1 & 4 & 0 \\ 1 & 4 & 2 & 3 \\ 2 & 3 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 4 & 4 \\ 3 & 0 & 1 & 1 \\ 1 & 1 & 0 & 4 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 2 & 3 & 0 & 0 \\ 1 & 0 & 3 & 1 \\ 0 & 0 & 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_8$ $\,\rtimes\,$ $Q_8$ (2)				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $\OD_{16}$ (2)	$C_4^2$ . $C_2^2$ (3)	$(C_4\times Q_8)$ . $C_2$	$(C_2\times Q_8)$ . $C_4$	all 17

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

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 57 subgroups in 47 conjugacy classes, 37 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_2^3$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_8:Q_8$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_8:Q_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8:Q_8$

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: $C_8:Q_8$

Order 32: $C_4:C_8$ x 3, $C_8:C_4$ x 2, $C_4\times C_8$, $C_4\times Q_8$

Order 16: $C_2\times C_8$ x 4, $C_4:C_4$ x 3, $C_4^2$ x 3, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 7, $C_8$ x 5, $Q_8$ x 2

Order 4: $C_4$ x 9, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 64: $C_8:Q_8$

Order 32: $C_4:C_8$ x 2, $C_8:C_4$, $C_4\times C_8$, $C_4\times Q_8$

Order 16: $C_2\times C_8$ x 3, $C_4:C_4$ x 2, $C_4^2$ x 2, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 5, $C_8$ x 3, $Q_8$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 64: $C_8:Q_8$ ($C_1$)

Order 32: $C_4:C_8$ ($C_2$) x 3, $C_8:C_4$ ($C_2$) x 2, $C_4\times C_8$ ($C_2$), $C_4\times Q_8$ ($C_2$)

Order 16: $C_2\times C_8$ ($C_2^2$) x 4, $C_4:C_4$ ($C_4$) x 3, $C_4^2$ ($C_2^2$) x 3, $C_2\times Q_8$ ($C_4$)

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

Order 4: $C_4$ ($\OD_{16}$) x 2, $C_2^2$ ($C_2^2\times C_4$), $C_4$ ($D_4:C_2$), $C_4$ ($C_2\times Q_8$)

Order 2: $C_2$ ($\OD_{16}:C_2$), $C_2$ ($C_2\times \OD_{16}$), $C_2$ ($C_4\times Q_8$)

Order 1: $C_1$ ($C_8:Q_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 64: $C_8:Q_8$ ($C_1$)

Order 32: $C_4:C_8$ ($C_2$) x 2, $C_8:C_4$ ($C_2$), $C_4\times C_8$ ($C_2$), $C_4\times Q_8$ ($C_2$)

Order 16: $C_2\times C_8$ ($C_2^2$) x 3, $C_4:C_4$ ($C_4$) x 2, $C_4^2$ ($C_2^2$) x 2, $C_2\times Q_8$ ($C_4$)

Order 8: $C_2\times C_4$ ($C_2\times C_4$) x 4, $C_2\times C_4$ ($C_2^3$), $C_8$ ($Q_8$)

Order 4: $C_2^2$ ($C_2^2\times C_4$), $C_4$ ($\OD_{16}$), $C_4$ ($D_4:C_2$), $C_4$ ($C_2\times Q_8$)

Order 2: $C_2$ ($\OD_{16}:C_2$), $C_2$ ($C_2\times \OD_{16}$), $C_2$ ($C_4\times Q_8$)

Order 1: $C_1$ ($C_8:Q_8$)

Series

Derived series

$C_8:Q_8$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_8:Q_8$

$\rhd$

$C_8:C_4$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_8:Q_8$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_8:Q_8$

Supergroups

This group is a maximal subgroup of 146 larger groups in the database.

This group is a maximal quotient of 88 larger groups in the database.

Character theory

Complex character table

See the $28 \times 28$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $18 \times 18$ rational character table.