Abstract group $Q_8^2$

• Label: 64.239
• Order: \( 2^{6} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3^{2} \)
• Perm deg.: $16$
• Trans deg.: $64$
• Rank: $4$

Group information

Description:	$Q_8^2$
Order:	\(64\)\(\medspace = 2^{6} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^8.\SOPlus(4,2)$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \) (generators)
Outer automorphisms:	$S_4\wr C_2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
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), metabelian, and rational.

Group statistics

Order	1	2	4
Elements	1	3	60	64
Conjugacy classes	1	3	21	25
Divisions	1	3	21	25
Autjugacy classes	1	2	2	5

Dimension	1	2	4
Irr. complex chars.	16	8	1	25
Irr. rational chars.	16	8	1	25

Minimal Presentations

Permutation degree:	$16$
Transitive degree:	$64$
Rank:	$4$
Inequivalent generating quadruples:	$280$

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

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

Homology

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

Subgroups

There are 133 subgroups in 106 conjugacy classes, 91 normal (4 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^4$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $Q_8^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

Normal subgroups up to automorphism

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See a full page version of the diagram

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Subgroup information

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

Classes of subgroups up to conjugation

Order 64: $Q_8^2$

Order 32: $C_4:Q_8$ x 9, $C_4\times Q_8$ x 6

Order 16: $C_4:C_4$ x 18, $C_4^2$ x 9, $C_2\times Q_8$ x 8

Order 8: $C_2\times C_4$ x 15, $Q_8$ x 14

Order 4: $C_4$ x 21, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 64: $Q_8^2$

Order 32: $C_4:Q_8$, $C_4\times Q_8$

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

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

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 64: $Q_8^2$ ($C_1$)

Order 32: $C_4:Q_8$ ($C_2$) x 9, $C_4\times Q_8$ ($C_2$) x 6

Order 16: $C_4:C_4$ ($C_2^2$) x 18, $C_4^2$ ($C_2^2$) x 9, $C_2\times Q_8$ ($C_2^2$) x 8

Order 8: $C_2\times C_4$ ($C_2^3$) x 15, $Q_8$ ($Q_8$) x 8

Order 4: $C_4$ ($C_2\times Q_8$) x 12, $C_2^2$ ($C_2^4$)

Order 2: $C_2$ ($C_2^2\times Q_8$) x 2, $C_2$ ($D_4:C_2^2$)

Order 1: $C_1$ ($Q_8^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 64: $Q_8^2$ ($C_1$)

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

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

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

Order 4: $C_2^2$ ($C_2^4$), $C_4$ ($C_2\times Q_8$)

Order 2: $C_2$ ($D_4:C_2^2$), $C_2$ ($C_2^2\times Q_8$)

Order 1: $C_1$ ($Q_8^2$)

Series

Derived series

$Q_8^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$Q_8^2$

$\rhd$

$C_4\times Q_8$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$Q_8^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$Q_8^2$

Supergroups

This group is a maximal subgroup of 56 larger groups in the database.

This group is a maximal quotient of 45 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $25 \times 25$ rational character table. Alternatively, you may search for characters of this group with desired properties.