Abstract group $(C_2^2\times D_4):C_8$

• Label: 256.327
• Order: \( 2^{8} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{11} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $32$
• Trans deg.: $32$
• Rank: $2$

Group information

Description:	$(C_2^2\times D_4):C_8$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	Group of order \(2048\)\(\medspace = 2^{11} \) (generators)
Outer automorphisms:	$C_2^4$, of order \(16\)\(\medspace = 2^{4} \)
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
Elements	1	31	96	128	256
Conjugacy classes	1	8	11	8	28
Divisions	1	8	7	2	18
Autjugacy classes	1	6	5	1	13

Dimension	1	2	4	8	16
Irr. complex chars.	16	4	6	2	0	28
Irr. rational chars.	4	4	9	0	1	18

Minimal Presentations

Permutation degree:	$32$
Transitive degree:	$32$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	16
Arbitrary	8	16	16

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{8}=c^{2}=d^{2}=e^{4}=[a,c]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $32$ $\langle(1,17)(2,18)(3,20,4,19)(5,21)(6,22)(7,24,8,23)(9,28,12,26,10,27,11,25)(13,32,16,30,14,31,15,29) \!\cdots\! \rangle$
Transitive group:	32T7558	32T8271	32T8350	32T8365	all 5
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^2\times Q_8)$ $\,\rtimes\,$ $C_8$	$(C_2^2\times D_4)$ $\,\rtimes\,$ $C_8$	$(C_2^3.\OD_{16})$ $\,\rtimes\,$ $C_2$ (2)		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_4:C_2^3)$ . $C_4$	$(C_2^3.D_4)$ . $C_4$	$(C_2^3:C_4)$ . $D_4$ (2)	$C_2$ . $(C_2^4:C_8)$	all 15

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 707 subgroups in 178 conjugacy classes, 23 normal (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^4:C_8$
Commutator:	$G' \simeq$ $C_2^2\times C_4$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_2^3.D_4$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $(C_2^2\times D_4):C_8$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $(C_2^2\times D_4):C_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2^4:C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_2^2\times D_4):C_8$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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

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Classes of subgroups up to conjugation

Order 256: $(C_2^2\times D_4):C_8$

Order 128: $C_2^3.\OD_{16}$ x 2, $C_2^4.C_2^3$

Order 64: $C_2^4:C_4$ x 2, $C_2^3.D_4$ x 2, $C_2^3:C_8$ x 2, $D_4:C_2^3$

Order 32: $D_4:C_2^2$ x 6, $C_2^3:C_4$ x 4, $C_2^3:C_4$ x 3, $C_2^2:C_8$ x 2, $C_2^2\times D_4$ x 2, $C_2^3\times C_4$ x 2, $C_2.C_4^2$ x 2, $C_2^2\times Q_8$

Order 16: $D_4:C_2$ x 16, $C_2^2\times C_4$ x 16, $C_2\times D_4$ x 11, $C_2^2:C_4$ x 4, $C_2\times Q_8$ x 3, $C_2\times C_8$ x 2, $C_2^4$ x 2

Order 8: $C_2\times C_4$ x 27, $C_2^3$ x 12, $D_4$ x 12, $Q_8$ x 4, $C_8$ x 2

Order 4: $C_2^2$ x 18, $C_4$ x 7

Order 2: $C_2$ x 8

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $(C_2^2\times D_4):C_8$

Order 128: $C_2^4.C_2^3$, $C_2^3.\OD_{16}$

Order 64: $C_2^3.D_4$ x 2, $C_2^4:C_4$, $C_2^3:C_8$, $D_4:C_2^3$

Order 32: $D_4:C_2^2$ x 3, $C_2^2\times D_4$ x 2, $C_2^3\times C_4$ x 2, $C_2^3:C_4$ x 2, $C_2.C_4^2$ x 2, $C_2^3:C_4$, $C_2^2:C_8$, $C_2^2\times Q_8$

Order 16: $C_2^2\times C_4$ x 11, $C_2\times D_4$ x 6, $D_4:C_2$ x 4, $C_2^2:C_4$ x 2, $C_2^4$ x 2, $C_2\times Q_8$ x 2, $C_2\times C_8$

Order 8: $C_2\times C_4$ x 14, $C_2^3$ x 9, $D_4$ x 4, $Q_8$ x 2, $C_8$

Order 4: $C_2^2$ x 12, $C_4$ x 5

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $(C_2^2\times D_4):C_8$ ($C_1$)

Order 128: $C_2^3.\OD_{16}$ ($C_2$) x 2, $C_2^4.C_2^3$ ($C_2$)

Order 64: $C_2^3.D_4$ ($C_2^2$), $C_2^3.D_4$ ($C_4$), $D_4:C_2^3$ ($C_4$)

Order 32: $C_2^3:C_4$ ($D_4$) x 2, $C_2^2\times Q_8$ ($C_8$), $C_2^2\times D_4$ ($C_8$), $C_2^3\times C_4$ ($C_2\times C_4$)

Order 16: $C_2^4$ ($C_2^2:C_4$), $C_2^2\times C_4$ ($\OD_{16}$), $C_2^2\times C_4$ ($C_2\times C_8$)

Order 8: $C_2^3$ ($\OD_{16}:C_2$), $C_2^3$ ($C_2^3:C_4$), $C_2^3$ ($C_2^2:C_8$)

Order 4: $C_2^2$ ($C_2\wr C_4$) x 2, $C_2^2$ ($C_2^3:C_8$)

Order 2: $C_2$ ($C_2^4:C_8$)

Order 1: $C_1$ ($(C_2^2\times D_4):C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $(C_2^2\times D_4):C_8$ ($C_1$)

Order 128: $C_2^4.C_2^3$ ($C_2$), $C_2^3.\OD_{16}$ ($C_2$)

Order 64: $C_2^3.D_4$ ($C_2^2$), $C_2^3.D_4$ ($C_4$), $D_4:C_2^3$ ($C_4$)

Order 32: $C_2^2\times Q_8$ ($C_8$), $C_2^2\times D_4$ ($C_8$), $C_2^3\times C_4$ ($C_2\times C_4$), $C_2^3:C_4$ ($D_4$)

Order 16: $C_2^4$ ($C_2^2:C_4$), $C_2^2\times C_4$ ($\OD_{16}$), $C_2^2\times C_4$ ($C_2\times C_8$)

Order 8: $C_2^3$ ($\OD_{16}:C_2$), $C_2^3$ ($C_2^3:C_4$), $C_2^3$ ($C_2^2:C_8$)

Order 4: $C_2^2$ ($C_2^3:C_8$), $C_2^2$ ($C_2\wr C_4$)

Order 2: $C_2$ ($C_2^4:C_8$)

Order 1: $C_1$ ($(C_2^2\times D_4):C_8$)

Series

Derived series

$(C_2^2\times D_4):C_8$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_1$

Chief series

$(C_2^2\times D_4):C_8$

$\rhd$

$C_2^3.\OD_{16}$

$\rhd$

$C_2^3.D_4$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times D_4):C_8$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^4$

$\lhd$

$C_2^3.D_4$

$\lhd$

$(C_2^2\times D_4):C_8$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 10 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.