Abstract group 256.5742: $C_2^4.(C_2\times D_4)$

• Label: 256.5742
• Order: \( 2^{8} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $32$
• Trans deg.: $32$
• Rank: $3$

Group information

Description:	$C_2^4.(C_2\times D_4)$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^6:C_2^4$, of order \(1024\)\(\medspace = 2^{10} \)
Composition factors:	$C_2$ x 8
Nilpotency class:	$4$
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
Elements	1	39	184	32	256
Conjugacy classes	1	10	21	2	34
Divisions	1	10	16	1	28
Autjugacy classes	1	10	15	1	27

Dimension	1	2	4	8	16
Irr. complex chars.	16	12	4	2	0	34
Irr. rational chars.	8	16	2	1	1	28

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$32$
Rank:	$3$
Inequivalent generating triples:	$5376$

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:	${\langle a, b, c, d, e \mid a^{4}=b^{2}=c^{2}=d^{4}=e^{4}=[a,c]=[c,d]=[d,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $32$ $\langle(1,19,3,17)(2,20,4,18)(5,23,7,21)(6,24,8,22)(9,28,12,25)(10,27,11,26)(13,32,16,29) \!\cdots\! \rangle$
Transitive group:	32T6591	32T6603	32T6604	32T6606	all 19
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4^2:D_4)$ $\,\rtimes\,$ $C_2$	$(C_2^4.D_4)$ $\,\rtimes\,$ $C_2$	$(C_2^4.D_4)$ $\,\rtimes\,$ $C_2$	$(C_2^4.D_4)$ $\,\rtimes\,$ $C_2$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4:D_4)$ . $D_4$	$(C_2^2\times D_4)$ . $D_4$	$(D_4:C_2^2)$ . $D_4$	$(C_2^2.D_4)$ . $D_4$ (3)	all 39

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

Homology

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

Subgroups

There are 1159 subgroups in 391 conjugacy classes, 69 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

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

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.

Classes of subgroups up to conjugation

Order 256: $C_2^4.(C_2\times D_4)$

Order 128: $C_2^4.D_4$, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^4.D_4$, $C_4^2:D_4$, $C_2^4.D_4$, $C_4^2.C_2^3$

Order 64: $C_2^4:C_4$ x 3, $C_2^3.D_4$ x 3, $C_4^2:C_2^2$ x 2, $C_2^2.C_4^2$ x 2, $C_4^2:C_2^2$ x 2, $\OD_{16}:C_2^2$, $C_2^3.D_4$, $C_4^2:C_4$, $C_4^2:C_4$, $C_2\wr C_4$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $D_4:D_4$, $D_4:D_4$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_4^2:C_2^2$, $C_2^3.D_4$, $C_4^2:C_2^2$, $C_4^2:C_4$, $C_4^2:C_2^2$

Order 32: $C_2^2.D_4$ x 18, $C_4:D_4$ x 12, $C_4\times D_4$ x 11, $C_2^2:Q_8$ x 8, $C_2^3:C_4$ x 8, $C_2^3:C_4$ x 6, $C_4^2:C_2$ x 6, $C_4^2:C_2$ x 6, $C_2^2\wr C_2$ x 6, $C_4^2:C_2$ x 5, $C_4.Q_8$ x 3, $\OD_{16}:C_2$ x 2, $D_4:C_2^2$ x 2, $C_2^2\times D_4$ x 2, $C_4:D_4$ x 2, $C_2^2.D_4$ x 2, $C_2.C_4^2$ x 2, $C_4\wr C_2$ x 2, $C_2^3\times C_4$, $C_2\times \OD_{16}$, $C_4:Q_8$, $C_4\times Q_8$

Order 16: $C_2^2:C_4$ x 35, $C_4:C_4$ x 20, $C_2\times D_4$ x 20, $C_2^2\times C_4$ x 20, $C_4^2$ x 8, $D_4:C_2$ x 4, $C_2^4$ x 3, $C_2\times Q_8$ x 3, $\OD_{16}$ x 2, $C_2\times C_8$

Order 8: $C_2\times C_4$ x 37, $D_4$ x 18, $C_2^3$ x 15, $Q_8$ x 3, $C_8$

Order 4: $C_2^2$ x 24, $C_4$ x 16

Order 2: $C_2$ x 10

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $C_2^4.(C_2\times D_4)$

Order 128: $C_2^4.D_4$, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^4.D_4$, $C_4^2:D_4$, $C_2^4.D_4$, $C_4^2.C_2^3$

Order 64: $C_2^4:C_4$ x 3, $C_2^3.D_4$ x 3, $C_4^2:C_2^2$ x 2, $C_2^2.C_4^2$ x 2, $C_4^2:C_2^2$ x 2, $\OD_{16}:C_2^2$, $C_2^3.D_4$, $C_4^2:C_4$, $C_4^2:C_4$, $C_2\wr C_4$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $D_4:D_4$, $D_4:D_4$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_4^2:C_2^2$, $C_2^3.D_4$, $C_4^2:C_2^2$, $C_4^2:C_4$, $C_4^2:C_2^2$

Order 32: $C_2^2.D_4$ x 18, $C_4:D_4$ x 12, $C_4\times D_4$ x 10, $C_2^2:Q_8$ x 8, $C_2^3:C_4$ x 8, $C_4^2:C_2$ x 6, $C_4^2:C_2$ x 6, $C_2^2\wr C_2$ x 6, $C_2^3:C_4$ x 5, $C_4^2:C_2$ x 5, $C_4.Q_8$ x 3, $\OD_{16}:C_2$ x 2, $D_4:C_2^2$ x 2, $C_2^2\times D_4$ x 2, $C_4:D_4$ x 2, $C_2^2.D_4$ x 2, $C_2^3\times C_4$, $C_2\times \OD_{16}$, $C_4:Q_8$, $C_4\times Q_8$, $C_2.C_4^2$, $C_4\wr C_2$

Order 16: $C_2^2:C_4$ x 34, $C_4:C_4$ x 20, $C_2\times D_4$ x 20, $C_2^2\times C_4$ x 19, $C_4^2$ x 7, $D_4:C_2$ x 4, $C_2^4$ x 3, $C_2\times Q_8$ x 3, $\OD_{16}$ x 2, $C_2\times C_8$

Order 8: $C_2\times C_4$ x 34, $D_4$ x 18, $C_2^3$ x 15, $Q_8$ x 3, $C_8$

Order 4: $C_2^2$ x 23, $C_4$ x 15

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $C_2^4.(C_2\times D_4)$ ($C_1$)

Order 128: $C_2^4.D_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_4^2:D_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_4^2.C_2^3$ ($C_2$)

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

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

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

Order 8: $C_2^3$ ($C_2^2\wr C_2$) x 3, $C_2^3$ ($C_2^3:C_4$) x 2, $C_2\times C_4$ ($C_2^2\wr C_2$), $C_2\times C_4$ ($C_2^3:C_4$)

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

Order 2: $C_2$ ($C_2^4.D_4$)

Order 1: $C_1$ ($C_2^4.(C_2\times D_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $C_2^4.(C_2\times D_4)$ ($C_1$)

Order 128: $C_2^4.D_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_4^2:D_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_4^2.C_2^3$ ($C_2$)

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

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

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

Order 8: $C_2^3$ ($C_2^2\wr C_2$) x 3, $C_2^3$ ($C_2^3:C_4$) x 2, $C_2\times C_4$ ($C_2^2\wr C_2$), $C_2\times C_4$ ($C_2^3:C_4$)

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

Order 2: $C_2$ ($C_2^4.D_4$)

Order 1: $C_1$ ($C_2^4.(C_2\times D_4)$)

Series

Derived series

$C_2^4.(C_2\times D_4)$

$\rhd$

$C_2\times D_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2^4.(C_2\times D_4)$

$\rhd$

$C_4^2.C_2^3$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$C_2^2\times D_4$

$\rhd$

$C_2\times D_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^4.(C_2\times D_4)$

$\rhd$

$C_2\times D_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2\times D_4$

$\lhd$

$C_2^4.(C_2\times D_4)$

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 $28 \times 28$ rational character table.