Abstract group 64.255: $Q_{16}:C_2^2$

• Label: 64.255
• Order: \( 2^{6} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $18$
• Trans deg.: $32$
• Rank: $4$

Group information

Description:	$Q_{16}:C_2^2$
Order:	\(64\)\(\medspace = 2^{6} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^7:C_2^3$, of order \(1024\)\(\medspace = 2^{10} \)
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), metabelian, and rational.

Group statistics

Order	1	2	4	8
Elements	1	15	32	16	64
Conjugacy classes	1	7	10	4	22
Divisions	1	7	10	4	22
Autjugacy classes	1	4	5	1	11

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

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$32$
Rank:	$4$
Inequivalent generating quadruples:	$5040$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=d^{8}=[a,b]=[a,c]=[a,d]=[b,c]=1, c^{2}=d^{4}, d^{b}=d^{5}, d^{c}=d^{3} \rangle$
Permutation group:	Degree $18$ $\langle(1,2)(3,9)(4,7)(5,13)(6,11)(8,12)(10,16)(14,15)(17,18), (1,3,4,10)(2,6,7,15) \!\cdots\! \rangle$
Transitive group:	32T278	32T369			more information
Direct product:	$C_2$ $\, \times\, $ $(Q_{16}:C_2)$
Semidirect product:	$Q_{16}$ $\,\rtimes\,$ $C_2^2$	$\SD_{16}$ $\,\rtimes\,$ $C_2^2$	$\OD_{16}$ $\,\rtimes\,$ $C_2^2$	$(C_2\times Q_{16})$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_8$ . $C_2^3$ (2)	$C_2^3$ . $D_4$	$D_4$ . $C_2^3$	$C_8$ . $C_2^3$	all 16

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 185 subgroups in 129 conjugacy classes, 81 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2\times D_4$
Commutator:	$G' \simeq$ $C_4$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $Q_{16}:C_2^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $Q_{16}:C_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}:C_2^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

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.

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

Subgroup information

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

Classes of subgroups up to conjugation

Order 64: $Q_{16}:C_2^2$

Order 32: $Q_{16}:C_2$ x 8, $C_2\times Q_{16}$ x 2, $C_2\times \SD_{16}$ x 2, $D_4:C_2^2$, $C_2^2\times Q_8$, $C_2\times \OD_{16}$

Order 16: $C_2\times Q_8$ x 10, $Q_{16}$ x 8, $\SD_{16}$ x 8, $D_4:C_2$ x 6, $\OD_{16}$ x 4, $C_2^2\times C_4$ x 3, $C_2\times C_8$ x 2, $C_2\times D_4$ x 2

Order 8: $C_2\times C_4$ x 17, $Q_8$ x 13, $D_4$ x 7, $C_8$ x 4, $C_2^3$ x 2

Order 4: $C_4$ x 10, $C_2^2$ x 9

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 64: $Q_{16}:C_2^2$

Order 32: $D_4:C_2^2$, $C_2^2\times Q_8$, $Q_{16}:C_2$, $C_2\times Q_{16}$, $C_2\times \SD_{16}$, $C_2\times \OD_{16}$

Order 16: $C_2\times Q_8$ x 5, $C_2^2\times C_4$ x 3, $D_4:C_2$ x 2, $C_2\times D_4$ x 2, $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $C_2\times C_8$

Order 8: $C_2\times C_4$ x 8, $Q_8$ x 5, $D_4$ x 3, $C_2^3$ x 2, $C_8$

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

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 64: $Q_{16}:C_2^2$ ($C_1$)

Order 32: $Q_{16}:C_2$ ($C_2$) x 8, $C_2\times Q_{16}$ ($C_2$) x 2, $C_2\times \SD_{16}$ ($C_2$) x 2, $D_4:C_2^2$ ($C_2$), $C_2^2\times Q_8$ ($C_2$), $C_2\times \OD_{16}$ ($C_2$)

Order 16: $Q_{16}$ ($C_2^2$) x 8, $\SD_{16}$ ($C_2^2$) x 8, $C_2\times Q_8$ ($C_2^2$) x 7, $\OD_{16}$ ($C_2^2$) x 4, $D_4:C_2$ ($C_2^2$) x 4, $C_2\times C_8$ ($C_2^2$) x 2, $C_2\times D_4$ ($C_2^2$), $C_2^2\times C_4$ ($C_2^2$)

Order 8: $Q_8$ ($C_2^3$) x 6, $C_8$ ($C_2^3$) x 4, $C_2\times C_4$ ($C_2^3$) x 3, $C_2\times C_4$ ($D_4$) x 3, $D_4$ ($C_2^3$) x 2, $C_2^3$ ($D_4$)

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

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

Order 1: $C_1$ ($Q_{16}:C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 64: $Q_{16}:C_2^2$ ($C_1$)

Order 32: $D_4:C_2^2$ ($C_2$), $C_2^2\times Q_8$ ($C_2$), $Q_{16}:C_2$ ($C_2$), $C_2\times Q_{16}$ ($C_2$), $C_2\times \SD_{16}$ ($C_2$), $C_2\times \OD_{16}$ ($C_2$)

Order 16: $C_2\times Q_8$ ($C_2^2$) x 3, $Q_{16}$ ($C_2^2$), $\SD_{16}$ ($C_2^2$), $\OD_{16}$ ($C_2^2$), $C_2\times C_8$ ($C_2^2$), $D_4:C_2$ ($C_2^2$), $C_2\times D_4$ ($C_2^2$), $C_2^2\times C_4$ ($C_2^2$)

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

Order 4: $C_2^2$ ($C_2\times D_4$) x 2, $C_4$ ($C_2\times D_4$) x 2, $C_4$ ($C_2^4$)

Order 2: $C_2$ ($C_2^2\times D_4$), $C_2$ ($Q_{16}:C_2$)

Order 1: $C_1$ ($Q_{16}:C_2^2$)

Series

Derived series

$Q_{16}:C_2^2$

$\rhd$

$C_4$

$\rhd$

$C_1$

Chief series

$Q_{16}:C_2^2$

$\rhd$

$Q_{16}:C_2$

$\rhd$

$\SD_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$Q_{16}:C_2^2$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$Q_{16}:C_2^2$

Supergroups

This group is a maximal subgroup of 131 larger groups in the database.

This group is a maximal quotient of 172 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 $22 \times 22$ rational character table. Alternatively, you may search for characters of this group with desired properties.