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

• Label: 256.334
• Order: \( 2^{8} \)
• Exponent: \( 2^{4} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $64$
• Trans deg.: $64$
• Rank: $2$

Group information

Description:	$(C_2^2\times C_8).D_4$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Automorphism group:	$C_4\times D_4^2:C_2^2$, of order \(1024\)\(\medspace = 2^{10} \) (generators)
Outer automorphisms:	$C_2^2\times C_4$, of order \(16\)\(\medspace = 2^{4} \)
Composition factors:	$C_2$ x 8
Nilpotency class:	$5$
Derived length:	$3$

This group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).

Group statistics

Order	1	2	4	8	16
Elements	1	31	64	32	128	256
Conjugacy classes	1	6	9	6	12	34
Divisions	1	6	7	2	2	18
Autjugacy classes	1	5	6	2	2	16

Dimension	1	2	4	32
Irr. complex chars.	16	4	14	0	34
Irr. rational chars.	4	4	9	1	18

Minimal Presentations

Permutation degree:	$64$
Transitive degree:	$64$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	32
Arbitrary	4	8	32

Constructions

Presentation:	${\langle a, b, c, d, e \mid b^{2}=c^{2}=d^{4}=e^{4}=[a,e]=[b,c]=[b,e]=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $64$ $\langle(1,40,8,36,4,38,6,34,2,39,7,35,3,37,5,33)(9,48,16,44,12,46,14,42,10,47,15,43,11,45,13,41) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^3.\OD_{16})$ $\,\rtimes\,$ $C_2$	$(C_2^3.\OD_{16})$ $\,\rtimes\,$ $C_2$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $(C_2\wr C_4)$ (2)	$(C_2^2\times C_8)$ . $D_4$	$(D_4:C_2^2)$ . $C_8$ (2)	$(C_4.C_2^4)$ . $C_4$	all 17

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:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 587 subgroups in 159 conjugacy classes, 23 normal (21 characteristic).

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

Special subgroups

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

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

Order 128: $C_2^3.\OD_{16}$, $(C_2^2\times C_8):C_2^2$, $C_2^3.\OD_{16}$

Order 64: $C_2^3.D_4$, $C_2^3.C_2^3$, $(C_2^2\times C_8):C_2$, $\OD_{32}:C_2$, $C_2^2:C_{16}$, $C_4.C_2^4$, $C_4^2:C_2^2$

Order 32: $D_4:C_2^2$ x 5, $D_4:C_2^2$ x 4, $C_4.C_2^3$ x 2, $C_4\wr C_2$ x 2, $D_4:C_4$, $C_2^3:C_4$, $C_2^2:C_8$, $C_2\times \OD_{16}$, $C_2^2\times C_8$, $C_4^2:C_2$, $\OD_{32}$, $C_2\times C_{16}$, $Q_8:C_4$

Order 16: $D_4:C_2$ x 22, $C_2\times D_4$ x 13, $C_2\times Q_8$ x 5, $C_2^2\times C_4$ x 5, $C_2\times C_8$ x 3, $C_{16}$ x 2, $\OD_{16}$, $C_4:C_4$, $C_2^2:C_4$, $C_4^2$

Order 8: $C_2\times C_4$ x 19, $D_4$ x 16, $Q_8$ x 6, $C_2^3$ x 5, $C_8$ x 2

Order 4: $C_2^2$ x 10, $C_4$ x 7

Order 2: $C_2$ x 6

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 128: $C_2^3.\OD_{16}$, $(C_2^2\times C_8):C_2^2$, $C_2^3.\OD_{16}$

Order 64: $C_2^3.D_4$, $C_2^3.C_2^3$, $(C_2^2\times C_8):C_2$, $\OD_{32}:C_2$, $C_2^2:C_{16}$, $C_4.C_2^4$, $C_4^2:C_2^2$

Order 32: $D_4:C_2^2$ x 4, $D_4:C_2^2$ x 3, $C_4.C_2^3$ x 2, $D_4:C_4$, $C_2^3:C_4$, $C_2^2:C_8$, $C_2\times \OD_{16}$, $C_2^2\times C_8$, $C_4^2:C_2$, $\OD_{32}$, $C_2\times C_{16}$, $C_4\wr C_2$, $Q_8:C_4$

Order 16: $D_4:C_2$ x 12, $C_2\times D_4$ x 8, $C_2\times Q_8$ x 4, $C_2^2\times C_4$ x 4, $C_2\times C_8$ x 3, $C_{16}$ x 2, $\OD_{16}$, $C_4:C_4$, $C_2^2:C_4$, $C_4^2$

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

Order 4: $C_2^2$ x 8, $C_4$ x 6

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 128: $C_2^3.\OD_{16}$ ($C_2$), $(C_2^2\times C_8):C_2^2$ ($C_2$), $C_2^3.\OD_{16}$ ($C_2$)

Order 64: $C_2^3.C_2^3$ ($C_4$), $(C_2^2\times C_8):C_2$ ($C_2^2$), $C_4.C_2^4$ ($C_4$)

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

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

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

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

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

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

Normal subgroups up to automorphism (quotient in parentheses)

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

Order 128: $C_2^3.\OD_{16}$ ($C_2$), $(C_2^2\times C_8):C_2^2$ ($C_2$), $C_2^3.\OD_{16}$ ($C_2$)

Order 64: $C_2^3.C_2^3$ ($C_4$), $(C_2^2\times C_8):C_2$ ($C_2^2$), $C_4.C_2^4$ ($C_4$)

Order 32: $D_4:C_2^2$ ($C_8$), $D_4:C_2^2$ ($C_2\times C_4$), $C_2\times \OD_{16}$ ($D_4$), $C_2^2\times C_8$ ($D_4$)

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

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

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

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

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

Series

Derived series

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

$\rhd$

$C_2\times D_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

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

$\rhd$

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

$\rhd$

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

$\rhd$

$D_4:C_2^2$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_2\times D_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_4$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2^2\times C_4$

$\lhd$

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

$\lhd$

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

Supergroups

Character theory

Complex character table

See the $34 \times 34$ 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.