Abstract group $Q_{16}:C_2$

• Label: 32.44
• Order: \( 2^{5} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $3$

Group information

Description:	$Q_{16}:C_2$
Order:	\(32\)\(\medspace = 2^{5} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$D_4^2$, of order \(64\)\(\medspace = 2^{6} \) (generators)
Outer automorphisms:	$C_2^2$, of order \(4\)\(\medspace = 2^{2} \)
Composition factors:	$C_2$ x 5
Nilpotency class:	$3$
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	8
Elements	1	7	16	8	32
Conjugacy classes	1	3	5	2	11
Divisions	1	3	5	2	11
Autjugacy classes	1	3	4	1	9

Dimension	1	2	4
Irr. complex chars.	8	2	1	11
Irr. rational chars.	8	2	1	11

Minimal Presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$3$
Inequivalent generating triples:	$168$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	8
Arbitrary	4	8	8

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=c^{8}=[a,b]=1, b^{2}=c^{4}, c^{a}=c^{5}, c^{b}=c^{3} \rangle$
Permutation group:	Degree $16$ $\langle(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,6,2,5)(3,8,4,7)(9,16,10,15) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 2 & 1 & 1 & 0 \\ 2 & 2 & 1 & 2 \\ 2 & 2 & 0 & 1 \\ 1 & 0 & 1 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 1 & 0 \\ 1 & 0 & 1 & 2 \\ 0 & 2 & 1 & 1 \\ 1 & 1 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 2 & 1 \\ 1 & 1 & 2 & 0 \\ 1 & 0 & 1 & 1 \\ 2 & 2 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	16T32	16T50	32T18		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$Q_{16}$ $\,\rtimes\,$ $C_2$ (2)	$\SD_{16}$ $\,\rtimes\,$ $C_2$ (2)	$\OD_{16}$ $\,\rtimes\,$ $C_2$	$(C_2\times Q_8)$ $\,\rtimes\,$ $C_2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $D_4$	$Q_8$ . $C_2^2$ (3)	$D_4$ . $C_2^2$	$C_2^2$ . $D_4$	all 9

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 42 subgroups in 30 conjugacy classes, 20 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times D_4$
Commutator:	$G' \simeq$ $C_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $Q_{16}:C_2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $Q_{16}:C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}:C_2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

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

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 32: $Q_{16}:C_2$

Order 16: $Q_{16}$ x 2, $\SD_{16}$ x 2, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3, $D_4$ x 2, $C_8$ x 2

Order 4: $C_4$ x 5, $C_2^2$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 32: $Q_{16}:C_2$

Order 16: $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 8: $Q_8$ x 3, $C_2\times C_4$ x 3, $D_4$ x 2, $C_8$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 32: $Q_{16}:C_2$ ($C_1$)

Order 16: $Q_{16}$ ($C_2$) x 2, $\SD_{16}$ ($C_2$) x 2, $\OD_{16}$ ($C_2$), $D_4:C_2$ ($C_2$), $C_2\times Q_8$ ($C_2$)

Order 8: $Q_8$ ($C_2^2$) x 3, $C_8$ ($C_2^2$) x 2, $D_4$ ($C_2^2$), $C_2\times C_4$ ($C_2^2$)

Order 4: $C_2^2$ ($D_4$), $C_4$ ($C_2^3$), $C_4$ ($D_4$)

Order 2: $C_2$ ($C_2\times D_4$)

Order 1: $C_1$ ($Q_{16}:C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 32: $Q_{16}:C_2$ ($C_1$)

Order 16: $Q_{16}$ ($C_2$), $\SD_{16}$ ($C_2$), $\OD_{16}$ ($C_2$), $D_4:C_2$ ($C_2$), $C_2\times Q_8$ ($C_2$)

Order 8: $Q_8$ ($C_2^2$) x 2, $D_4$ ($C_2^2$), $C_2\times C_4$ ($C_2^2$), $C_8$ ($C_2^2$)

Order 4: $C_2^2$ ($D_4$), $C_4$ ($C_2^3$), $C_4$ ($D_4$)

Order 2: $C_2$ ($C_2\times D_4$)

Order 1: $C_1$ ($Q_{16}:C_2$)

Series

Derived series	$Q_{16}:C_2$	$\rhd$	$C_4$	$\rhd$	$C_1$
Chief series	$Q_{16}:C_2$	$\rhd$	$\SD_{16}$	$\rhd$	$C_8$	$\rhd$	$C_4$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$Q_{16}:C_2$	$\rhd$	$C_4$	$\rhd$	$C_2$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_2$	$\lhd$	$C_2\times C_4$	$\lhd$	$Q_{16}:C_2$

Supergroups

This group is a maximal subgroup of 120 larger groups in the database.

This group is a maximal quotient of 136 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

		1A	2A	2B	2C	4A	4B	4C	4D	4E	8A	8B
	Size	1	1	2	4	2	2	4	4	4	4	4
	2 P	1A	1A	1A	1A	2A	2A	2A	2A	2A	4A	4A
	Schur
32.44.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.44.1b		$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
32.44.1c		$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$
32.44.1d		$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$
32.44.1e		$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$
32.44.1f		$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$
32.44.1g		$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
32.44.1h		$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
32.44.2a		$2$	$2$	$-2$	$0$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$
32.44.2b		$2$	$2$	$2$	$0$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$
32.44.4a	2	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$