Abstract group 256.6076: $C_2^5.D_4$

• Label: 256.6076
• Order: \( 2^{8} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{12} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $12$
• Trans deg.: $32$
• Rank: $3$

Group information

Description:	$C_2^5.D_4$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^7.C_2^5$, 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	63	192	256
Conjugacy classes	1	16	20	37
Divisions	1	16	14	31
Autjugacy classes	1	13	7	21

Dimension	1	2	4	8
Irr. complex chars.	16	12	8	1	37
Irr. rational chars.	8	12	10	1	31

Minimal presentations

Permutation degree:	$12$
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	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{4}=d^{4}=e^{4}=[a,b]=[a,c]=[a,d]=[b,c]=[b,d]=[b,e]=[d,e]=1, e^{a}=be, d^{c}=d^{3}e^{2}, e^{c}=de \rangle$
Permutation group:	Degree $12$ $\langle(1,2,5,6)(3,7,8,4)(9,10,11,12), (1,3)(2,4)(5,6,7,8)(10,12), (1,4)(2,3)(5,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
Transitive group:	32T4050	32T4105	32T4110	32T4370	all 11
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^4.D_4)$ $\,\rtimes\,$ $C_2$ (2)	$(C_2^4.D_4)$ $\,\rtimes\,$ $C_2$ (2)	$(C_2^3:D_4)$ $\,\rtimes\,$ $C_4$	$(C_2^3:C_4)$ $\,\rtimes\,$ $D_4$ (4)	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $D_4$ (2)	$(C_2^2\times D_4)$ . $D_4$ (2)	$(C_2.D_4^2)$ . $C_2$	$C_2^4$ . $(C_2\times D_4)$ (2)	all 27

Elements of the group are displayed as permutations of degree 12.

Homology

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

Subgroups

There are 2047 subgroups in 520 conjugacy classes, 65 normal (41 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^4:C_4$
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:	$\operatorname{Fit} \simeq$ $C_2^5.D_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^5.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^4:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.D_4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 256: $C_2^5.D_4$

Order 128: $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^5:C_4$, $C_2^5:C_4$, $C_2.D_4^2$

Order 64: $C_2^4:C_4$ x 10, $C_4^2:C_4$ x 4, $C_2^3:D_4$ x 2, $C_2^3.C_2^3$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3:D_4$ x 2, $C_2^4:C_4$, $C_2^2:C_4^2$, $D_4\times C_2^3$, $C_2^3:D_4$, $C_2^4:C_4$

Order 32: $C_2^3:C_4$ x 19, $C_2^2\times D_4$ x 18, $C_2^3:C_4$ x 16, $C_2^2\wr C_2$ x 8, $C_4:D_4$ x 6, $C_2^5$ x 2, $C_2^3\times C_4$ x 2, $C_4:D_4$ x 2, $C_2\times C_4^2$ x 2, $C_2^2.D_4$, $C_2.C_4^2$

Order 16: $C_2\times D_4$ x 66, $C_2^2:C_4$ x 32, $C_2^4$ x 23, $C_2^2\times C_4$ x 22, $C_4^2$ x 3, $C_4:C_4$

Order 8: $C_2^3$ x 78, $C_2\times C_4$ x 44, $D_4$ x 38

Order 4: $C_2^2$ x 70, $C_4$ x 14

Order 2: $C_2$ x 16

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $C_2^5.D_4$

Order 128: $C_2^4.D_4$, $C_2^5:C_4$, $C_2^5:C_4$, $C_2^4.D_4$, $C_2.D_4^2$

Order 64: $C_2^4:C_4$ x 5, $C_2^3:D_4$ x 2, $C_2^3.C_2^3$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3:D_4$ x 2, $C_2^4:C_4$, $C_2^2:C_4^2$, $C_4^2:C_4$, $D_4\times C_2^3$, $C_2^3:D_4$, $C_2^4:C_4$

Order 32: $C_2^2\times D_4$ x 15, $C_2^3:C_4$ x 13, $C_2^3:C_4$ x 6, $C_4:D_4$ x 3, $C_2^5$ x 2, $C_2^3\times C_4$ x 2, $C_4:D_4$ x 2, $C_2^2\wr C_2$ x 2, $C_2\times C_4^2$ x 2, $C_2^2.D_4$, $C_2.C_4^2$

Order 16: $C_2\times D_4$ x 38, $C_2^4$ x 19, $C_2^2\times C_4$ x 14, $C_2^2:C_4$ x 13, $C_4^2$ x 2, $C_4:C_4$

Order 8: $C_2^3$ x 54, $C_2\times C_4$ x 24, $D_4$ x 14

Order 4: $C_2^2$ x 48, $C_4$ x 7

Order 2: $C_2$ x 13

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $C_2^5.D_4$ ($C_1$)

Order 128: $C_2^4.D_4$ ($C_2$) x 2, $C_2^4.D_4$ ($C_2$) x 2, $C_2^5:C_4$ ($C_2$), $C_2^5:C_4$ ($C_2$), $C_2.D_4^2$ ($C_2$)

Order 64: $C_2^4:C_4$ ($C_2^2$) x 4, $C_2^3.C_2^3$ ($C_2^2$), $C_2^3.C_2^3$ ($C_4$), $C_2^2:C_4^2$ ($C_4$), $D_4\times C_2^3$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2:C_2^2$ ($C_4$), $C_2^3:D_4$ ($C_4$)

Order 32: $C_2^3:C_4$ ($D_4$) x 4, $C_2^5$ ($D_4$) x 2, $C_2^2\times D_4$ ($D_4$) x 2, $C_2^2\times D_4$ ($C_2\times C_4$) x 2, $C_2\times C_4^2$ ($C_2\times C_4$) x 2, $C_2^2\times D_4$ ($C_2^3$), $C_2^3\times C_4$ ($C_2\times C_4$), $C_2^2.D_4$ ($C_2\times C_4$)

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

Order 8: $C_2^3$ ($C_2^3:C_4$) x 4, $C_2^3$ ($C_4:D_4$) x 2, $C_2\times C_4$ ($C_4\times D_4$) x 2, $C_2^3$ ($C_2^2.D_4$), $C_2^3$ ($C_2^2\wr C_2$), $C_2^3$ ($C_2^3:C_4$)

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

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

Order 1: $C_1$ ($C_2^5.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $C_2^5.D_4$ ($C_1$)

Order 128: $C_2^4.D_4$ ($C_2$), $C_2^5:C_4$ ($C_2$), $C_2^5:C_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_2.D_4^2$ ($C_2$)

Order 64: $C_2^4:C_4$ ($C_2^2$) x 2, $C_2^3.C_2^3$ ($C_2^2$), $C_2^3.C_2^3$ ($C_4$), $C_2^2:C_4^2$ ($C_4$), $D_4\times C_2^3$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2:C_2^2$ ($C_4$), $C_2^3:D_4$ ($C_4$)

Order 32: $C_2^5$ ($D_4$) x 2, $C_2^2\times D_4$ ($D_4$) x 2, $C_2^2\times D_4$ ($C_2\times C_4$) x 2, $C_2\times C_4^2$ ($C_2\times C_4$) x 2, $C_2^3:C_4$ ($D_4$), $C_2^2\times D_4$ ($C_2^3$), $C_2^3\times C_4$ ($C_2\times C_4$), $C_2^2.D_4$ ($C_2\times C_4$)

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

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

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

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

Order 1: $C_1$ ($C_2^5.D_4$)

Series

Derived series

$C_2^5.D_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_1$

Chief series

$C_2^5.D_4$

$\rhd$

$C_2^5:C_4$

$\rhd$

$C_2^4:C_4$

$\rhd$

$C_2^2\times D_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^5.D_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^4$

$\lhd$

$D_4\times C_2^3$

$\lhd$

$C_2^5.D_4$

Supergroups

This group is a maximal subgroup of 12 larger groups in the database.

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

See the $37 \times 37$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $31 \times 31$ rational character table.