Abstract group 256.5716: $C_2^4.(C_2\times D_4)$

• Label: 256.5716
• Order: \( 2^{8} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• 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:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^6.D_4^2$, of order \(4096\)\(\medspace = 2^{12} \)
Composition factors:	$C_2$ x 8
Nilpotency class:	$4$
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
Elements	1	31	224	256
Conjugacy classes	1	9	24	34
Divisions	1	9	17	27
Autjugacy classes	1	5	10	16

Dimension	1	2	4	8	16
Irr. complex chars.	16	12	4	2	0	34
Irr. rational chars.	8	16	0	2	1	27

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$32$
Rank:	$3$
Inequivalent generating triples:	$1344$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{4}=b^{2}=c^{2}=d^{4}=e^{4}=[a,b]=[b,e]=[c,d]= \!\cdots\! \rangle}$
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:	32T6555	32T6557	32T7453	32T7789	all 7
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^4.D_4)$ $\,\rtimes\,$ $C_2$	$(C_2^4.C_2^3)$ $\,\rtimes\,$ $C_2$	$(C_4^2.C_2^2)$ $\,\rtimes\,$ $C_4$	$(C_4^2.C_2^2)$ $\,\rtimes\,$ $C_4$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4:D_4)$ . $D_4$	$(C_2^4.D_4)$ . $C_2$	$(C_2^3:C_4)$ . $D_4$	$(C_4^2:C_2)$ . $D_4$ (3)	all 27

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 1015 subgroups in 357 conjugacy classes, 69 normal (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^4.C_2^3$
Commutator:	$G' \simeq$ $C_2^2\times C_4$	$G/G' \simeq$ $C_2^2\times C_4$
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:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2^4.C_2^3$
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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

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

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

Order 64: $C_4^2:C_4$ x 6, $C_2^4:C_4$ x 4, $C_4^2:C_2^2$ x 4, $C_2^2.C_4^2$ x 4, $C_4^2:C_4$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2.C_2^2$

Order 32: $C_4^2:C_2$ x 13, $C_4\times D_4$ x 11, $C_2^2.D_4$ x 10, $C_4^2:C_2$ x 10, $C_2^3:C_4$ x 10, $C_2^3:C_4$ x 8, $C_4^2:C_2$ x 8, $C_2^2:Q_8$ x 8, $C_4\times Q_8$ x 7, $C_4:Q_8$ x 2, $C_4:D_4$ x 2, $C_4:D_4$ x 2, $C_2^2\wr C_2$ x 2, $C_2\times C_4^2$ x 2, $D_4:C_2^2$, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_4.Q_8$

Order 16: $C_2^2:C_4$ x 35, $C_4:C_4$ x 19, $C_2^2\times C_4$ x 14, $C_4^2$ x 13, $C_2\times D_4$ x 12, $C_2\times Q_8$ x 8, $C_2^4$ x 2, $D_4:C_2$ x 2

Order 8: $C_2\times C_4$ x 34, $C_2^3$ x 12, $D_4$ x 11, $Q_8$ x 7

Order 4: $C_2^2$ x 19, $C_4$ x 17

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

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

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

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

Order 32: $C_4^2:C_2$ x 6, $C_4^2:C_2$ x 6, $C_4\times D_4$ x 5, $C_4\times Q_8$ x 4, $C_2^2.D_4$ x 3, $C_2^2:Q_8$ x 3, $C_2^3:C_4$ x 3, $C_2^3:C_4$ x 2, $C_4:Q_8$ x 2, $C_4^2:C_2$ x 2, $C_2\times C_4^2$ x 2, $D_4:C_2^2$, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_4:D_4$, $C_4.Q_8$, $C_4:D_4$, $C_2^2\wr C_2$

Order 16: $C_2^2:C_4$ x 11, $C_4:C_4$ x 9, $C_4^2$ x 8, $C_2^2\times C_4$ x 7, $C_2\times D_4$ x 6, $C_2\times Q_8$ x 5, $C_2^4$, $D_4:C_2$

Order 8: $C_2\times C_4$ x 17, $C_2^3$ x 5, $D_4$ x 5, $Q_8$ x 4

Order 4: $C_4$ x 10, $C_2^2$ x 8

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

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

Order 64: $C_2^4:C_4$ ($C_2^2$) x 4, $C_4^2.C_2^2$ ($C_4$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2.C_2^2$ ($C_4$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2.C_2^2$ ($C_4$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2.C_2^2$ ($C_4$)

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

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

Order 8: $C_2^3$ ($C_2^2\wr C_2$) x 4, $C_2\times C_4$ ($C_2^3:C_4$) x 2, $C_2^3$ ($C_2^3:C_4$)

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

Order 2: $C_2$ ($C_2^4.C_2^3$)

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

Order 64: $C_2^4:C_4$ ($C_2^2$), $C_4^2.C_2^2$ ($C_4$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2.C_2^2$ ($C_4$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2.C_2^2$ ($C_4$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2.C_2^2$ ($C_4$)

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

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

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

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

Order 2: $C_2$ ($C_2^4.C_2^3$)

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

Series

Derived series

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

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$C_4^2.C_2^3$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$C_2^2\times D_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^3$

$\lhd$

$C_4^2:C_2^2$

$\lhd$

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

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

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 $27 \times 27$ rational character table.