Abstract group 256.372: $Q_{32}:C_8$

• Label: 256.372
• Order: \( 2^{8} \)
• Exponent: \( 2^{5} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{11} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $40$
• Trans deg.: $256$
• Rank: $2$

Group information

Description:	$Q_{32}:C_8$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(32\)\(\medspace = 2^{5} \)
Automorphism group:	$C_2^3\times D_{16}:C_8$, of order \(2048\)\(\medspace = 2^{11} \)
Composition factors:	$C_2$ x 8
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	76	80	32	64	256
Conjugacy classes	1	3	12	12	16	32	76
Divisions	1	3	8	4	3	2	21
Autjugacy classes	1	3	7	4	3	1	19

Dimension	1	2	4	8	16	32
Irr. complex chars.	16	60	0	0	0	0	76
Irr. rational chars.	4	4	7	2	3	1	21

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$256$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	20

Constructions

Presentation:	$\langle a, b, c \mid b^{32}=c^{4}=[a,c]=[b,c]=1, a^{2}=b^{16}, b^{a}=b^{15}c \rangle$
Permutation group:	Degree $40$ $\langle(2,7)(3,11)(4,13)(5,17)(8,23)(9,26)(10,22)(12,16)(14,30)(15,20)(18,24)(19,21) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 6 \\ 8 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 3 \\ 0 & 15 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 9 & 10 \\ 16 & 25 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
	$\left\langle \left(\begin{array}{rr} 32 & 0 \\ 0 & 32 \end{array}\right), \left(\begin{array}{rr} 53 & 77 \\ 65 & 81 \end{array}\right), \left(\begin{array}{rr} 85 & 66 \\ 69 & 70 \end{array}\right), \left(\begin{array}{rr} 40 & 6 \\ 57 & 76 \end{array}\right), \left(\begin{array}{rr} 64 & 57 \\ 30 & 34 \end{array}\right), \left(\begin{array}{rr} 61 & 0 \\ 0 & 61 \end{array}\right), \left(\begin{array}{rr} 1 & 62 \\ 62 & 32 \end{array}\right), \left(\begin{array}{rr} 82 & 45 \\ 9 & 73 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/93\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$Q_{32}$ $\,\rtimes\,$ $C_8$ (2)				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $Q_{64}$	$C_4^2$ . $D_8$	$C_4$ . $\SD_{64}$	$C_{16}$ . $\OD_{16}$	all 27

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

Homology

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

Subgroups

There are 187 subgroups in 63 conjugacy classes, 31 normal (29 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $D_{16}$
Commutator:	$G' \simeq$ $C_{16}$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_4\times C_{16}$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $Q_{32}:C_8$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $Q_{32}:C_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $D_8:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{32}:C_8$

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 256: $Q_{32}:C_8$

Order 128: $C_4\times Q_{32}$, $C_4\times C_{32}$, $C_{16}:C_8$

Order 64: $C_2\times C_{32}$, $C_{16}:C_4$, $Q_{16}:C_4$, $C_4\times C_{16}$, $C_2\times Q_{32}$, $C_8:C_8$, $C_4\times Q_{16}$

Order 32: $Q_{32}$ x 3, $C_2\times C_{16}$ x 2, $C_{32}$ x 2, $C_2\times Q_{16}$, $C_4\times C_8$, $C_4\times Q_8$, $C_8:C_4$, $C_4:C_8$, $Q_8:C_4$

Order 16: $Q_{16}$ x 3, $C_2\times C_8$ x 3, $C_{16}$ 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 4, $Q_8$ x 3

Order 4: $C_4$ x 8, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $Q_{32}:C_8$

Order 128: $C_4\times Q_{32}$, $C_4\times C_{32}$, $C_{16}:C_8$

Order 64: $C_2\times C_{32}$, $C_{16}:C_4$, $Q_{16}:C_4$, $C_4\times C_{16}$, $C_2\times Q_{32}$, $C_8:C_8$, $C_4\times Q_{16}$

Order 32: $Q_{32}$ x 2, $C_2\times C_{16}$ x 2, $C_2\times Q_{16}$, $C_4\times C_8$, $C_4\times Q_8$, $C_8:C_4$, $C_4:C_8$, $Q_8:C_4$, $C_{32}$

Order 16: $C_2\times C_8$ x 3, $C_{16}$ x 3, $Q_{16}$ x 2, $C_4:C_4$ x 2, $C_4^2$ x 2, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 5, $C_8$ x 4, $Q_8$ x 2

Order 4: $C_4$ x 7, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $Q_{32}:C_8$ ($C_1$)

Order 128: $C_4\times Q_{32}$ ($C_2$), $C_4\times C_{32}$ ($C_2$), $C_{16}:C_8$ ($C_2$)

Order 64: $C_{16}:C_4$ ($C_4$), $C_4\times C_{16}$ ($C_2^2$), $C_2\times Q_{32}$ ($C_4$)

Order 32: $Q_{32}$ ($C_8$) x 2, $C_4\times C_8$ ($D_4$), $C_2\times C_{16}$ ($D_4$), $C_2\times C_{16}$ ($C_2\times C_4$)

Order 16: $C_2\times C_8$ ($\SD_{16}$), $C_2\times C_8$ ($C_2^2:C_4$), $C_4^2$ ($D_8$), $C_{16}$ ($\OD_{16}$), $C_{16}$ ($C_2\times C_8$)

Order 8: $C_2\times C_4$ ($D_4:C_4$), $C_2\times C_4$ ($\SD_{32}$), $C_2\times C_4$ ($D_{16}$), $C_8$ ($C_2^2:C_8$), $C_8$ ($C_4\wr C_2$)

Order 4: $C_2^2$ ($D_8:C_4$), $C_4$ ($D_4:C_8$), $C_4$ ($Q_{64}$), $C_4$ ($\SD_{64}$), $C_4$ ($C_4.D_8$)

Order 2: $C_2$ ($D_8:C_8$), $C_2$ ($C_4.D_{16}$), $C_2$ ($Q_{32}:C_4$)

Order 1: $C_1$ ($Q_{32}:C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $Q_{32}:C_8$ ($C_1$)

Order 128: $C_4\times Q_{32}$ ($C_2$), $C_4\times C_{32}$ ($C_2$), $C_{16}:C_8$ ($C_2$)

Order 64: $C_{16}:C_4$ ($C_4$), $C_4\times C_{16}$ ($C_2^2$), $C_2\times Q_{32}$ ($C_4$)

Order 32: $C_4\times C_8$ ($D_4$), $Q_{32}$ ($C_8$), $C_2\times C_{16}$ ($D_4$), $C_2\times C_{16}$ ($C_2\times C_4$)

Order 16: $C_2\times C_8$ ($\SD_{16}$), $C_2\times C_8$ ($C_2^2:C_4$), $C_4^2$ ($D_8$), $C_{16}$ ($\OD_{16}$), $C_{16}$ ($C_2\times C_8$)

Order 8: $C_2\times C_4$ ($D_4:C_4$), $C_2\times C_4$ ($\SD_{32}$), $C_2\times C_4$ ($D_{16}$), $C_8$ ($C_2^2:C_8$), $C_8$ ($C_4\wr C_2$)

Order 4: $C_2^2$ ($D_8:C_4$), $C_4$ ($D_4:C_8$), $C_4$ ($Q_{64}$), $C_4$ ($\SD_{64}$), $C_4$ ($C_4.D_8$)

Order 2: $C_2$ ($D_8:C_8$), $C_2$ ($C_4.D_{16}$), $C_2$ ($Q_{32}:C_4$)

Order 1: $C_1$ ($Q_{32}:C_8$)

Series

Derived series

$Q_{32}:C_8$

$\rhd$

$C_{16}$

$\rhd$

$C_1$

Chief series

$Q_{32}:C_8$

$\rhd$

$C_4\times C_{32}$

$\rhd$

$C_4\times C_{16}$

$\rhd$

$C_4\times C_8$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$Q_{32}:C_8$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_4^2$

$\lhd$

$C_4\times C_8$

$\lhd$

$C_4\times C_{16}$

$\lhd$

$Q_{32}:C_8$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $76 \times 76$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $21 \times 21$ rational character table.