Abstract group 256.4741: $C_2^5.D_4$

• Label: 256.4741
• Order: \( 2^{8} \)
• Exponent: \( 2^{3} \)
• 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.: $16$
• Trans deg.: $32$
• Rank: $3$

Group information

Description:	$C_2^5.D_4$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
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	8
Elements	1	47	144	64	256
Conjugacy classes	1	14	18	4	37
Divisions	1	14	12	2	29
Autjugacy classes	1	11	8	1	21

Dimension	1	2	4	8
Irr. complex chars.	16	12	8	1	37
Irr. rational chars.	8	8	12	1	29

Minimal presentations

Permutation degree:	$16$
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:	${\langle a, b, c, d, e \mid a^{4}=b^{2}=c^{4}=d^{2}=e^{4}=[a,b]=[a,d]=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,2,4,8)(3,7,5,6)(10,13,14,15), (2,6)(9,10)(11,15)(12,14)(13,16), (1,3) \!\cdots\! \rangle$
Transitive group:	32T4016	32T4394	32T4401	32T4438	all 7
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^3:Q_8)$ $\,\rtimes\,$ $C_4$	$(C_2^3.D_8)$ $\,\rtimes\,$ $C_2$ (2)	$(D_4:C_2^3)$ $\,\rtimes\,$ $C_4$	$(C_2^3.D_4)$ $\,\rtimes\,$ $C_4$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $D_4$	$(C_2^3\times C_4)$ . $D_4$	$(C_2^3:C_4)$ . $D_4$ (2)	$C_2^4$ . $(C_2\times D_4)$	all 29

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:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 1471 subgroups in 389 conjugacy classes, 53 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:	$\Phi \simeq$ $C_2^3:C_4$	$G/\Phi \simeq$ $C_2^3$
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: $C_2^3.Q_{16}$ x 2, $C_2^3.D_8$ x 2, $C_2^3.C_4^2$, $C_2^3:\OD_{16}$, $C_2^3.C_2^4$

Order 64: $C_2^3.D_4$ x 3, $C_2^3:C_8$ x 3, $C_2^2.C_4^2$ x 3, $C_2^3:Q_8$ x 2, $C_2^3:D_4$ x 2, $C_2^4:C_4$ x 2, $C_2^2:\OD_{16}$, $C_2^3.D_4$, $D_4:C_2^3$, $C_2^3:D_4$

Order 32: $C_2^3:C_4$ x 20, $D_4:C_2^2$ x 6, $C_2.C_4^2$ x 5, $C_2^2\times D_4$ x 4, $C_2^3\times C_4$ x 4, $C_2^2\wr C_2$ x 4, $C_2^2:C_8$ x 3, $C_2^5$, $C_2^2\times Q_8$, $C_2\times \OD_{16}$

Order 16: $C_2^2\times C_4$ x 30, $C_2^2:C_4$ x 22, $C_2\times D_4$ x 21, $D_4:C_2$ x 16, $C_2^4$ x 13, $C_2\times Q_8$ x 3, $\OD_{16}$ x 2, $C_2\times C_8$ x 2

Order 8: $C_2^3$ x 50, $C_2\times C_4$ x 49, $D_4$ x 20, $Q_8$ x 4, $C_8$ x 2

Order 4: $C_2^2$ x 52, $C_4$ x 12

Order 2: $C_2$ x 14

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $C_2^5.D_4$

Order 128: $C_2^3.Q_{16}$, $C_2^3.D_8$, $C_2^3.C_4^2$, $C_2^3:\OD_{16}$, $C_2^3.C_2^4$

Order 64: $C_2^3.D_4$ x 3, $C_2^3:Q_8$ x 2, $C_2^3:D_4$ x 2, $C_2^3:C_8$ x 2, $C_2^2.C_4^2$ x 2, $C_2^4:C_4$ x 2, $C_2^2:\OD_{16}$, $C_2^3.D_4$, $D_4:C_2^3$, $C_2^3:D_4$

Order 32: $C_2^3:C_4$ x 15, $C_2.C_4^2$ x 5, $C_2^2\times D_4$ x 4, $C_2^3\times C_4$ x 4, $D_4:C_2^2$ x 3, $C_2^2:C_8$ x 2, $C_2^5$, $C_2^2\times Q_8$, $C_2\times \OD_{16}$, $C_2^2\wr C_2$

Order 16: $C_2^2\times C_4$ x 20, $C_2\times D_4$ x 13, $C_2^2:C_4$ x 12, $C_2^4$ x 11, $D_4:C_2$ x 4, $C_2\times Q_8$ x 2, $\OD_{16}$, $C_2\times C_8$

Order 8: $C_2^3$ x 36, $C_2\times C_4$ x 26, $D_4$ x 7, $Q_8$ x 2, $C_8$

Order 4: $C_2^2$ x 36, $C_4$ x 8

Order 2: $C_2$ x 11

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 128: $C_2^3.Q_{16}$ ($C_2$) x 2, $C_2^3.D_8$ ($C_2$) x 2, $C_2^3.C_4^2$ ($C_2$), $C_2^3:\OD_{16}$ ($C_2$), $C_2^3.C_2^4$ ($C_2$)

Order 64: $C_2^3:C_8$ ($C_2^2$) x 2, $C_2^2.C_4^2$ ($C_2^2$) x 2, $C_2^3:Q_8$ ($C_2^2$), $C_2^3:Q_8$ ($C_4$), $C_2^3:D_4$ ($C_2^2$), $C_2^3:D_4$ ($C_4$), $C_2^3.D_4$ ($C_4$), $D_4:C_2^3$ ($C_4$), $C_2^4:C_4$ ($C_2^2$)

Order 32: $C_2^3:C_4$ ($D_4$) x 2, $C_2.C_4^2$ ($C_2\times C_4$) x 2, $C_2^5$ ($D_4$), $C_2^2\times Q_8$ ($C_2\times C_4$), $C_2^2\times D_4$ ($C_2\times C_4$), $C_2^3\times C_4$ ($D_4$), $C_2^3\times C_4$ ($C_2\times C_4$), $C_2^3:C_4$ ($C_2^3$), $C_2^3:C_4$ ($C_2\times C_4$)

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

Order 8: $C_2^3$ ($C_2^3:C_4$) x 2, $C_2^3$ ($C_4\wr C_2$) x 2, $C_2^3$ ($Q_{16}:C_2$), $C_2^3$ ($D_8:C_2$), $C_2^3$ ($C_2^3:C_4$)

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

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

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: $C_2^3.Q_{16}$ ($C_2$), $C_2^3.D_8$ ($C_2$), $C_2^3.C_4^2$ ($C_2$), $C_2^3:\OD_{16}$ ($C_2$), $C_2^3.C_2^4$ ($C_2$)

Order 64: $C_2^3:Q_8$ ($C_2^2$), $C_2^3:Q_8$ ($C_4$), $C_2^3:D_4$ ($C_2^2$), $C_2^3:D_4$ ($C_4$), $C_2^3.D_4$ ($C_4$), $C_2^3:C_8$ ($C_2^2$), $D_4:C_2^3$ ($C_4$), $C_2^2.C_4^2$ ($C_2^2$), $C_2^4:C_4$ ($C_2^2$)

Order 32: $C_2^3:C_4$ ($D_4$) x 2, $C_2.C_4^2$ ($C_2\times C_4$) x 2, $C_2^5$ ($D_4$), $C_2^2\times Q_8$ ($C_2\times C_4$), $C_2^2\times D_4$ ($C_2\times C_4$), $C_2^3\times C_4$ ($D_4$), $C_2^3\times C_4$ ($C_2\times C_4$), $C_2^3:C_4$ ($C_2^3$), $C_2^3:C_4$ ($C_2\times C_4$)

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

Order 8: $C_2^3$ ($C_2^3:C_4$) x 2, $C_2^3$ ($Q_{16}:C_2$), $C_2^3$ ($D_8:C_2$), $C_2^3$ ($C_2^3:C_4$), $C_2^3$ ($C_4\wr C_2$)

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

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

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

$C_2^5.D_4$

$\rhd$

$C_2^3.C_2^4$

$\rhd$

$C_2^3:Q_8$

$\rhd$

$C_2^3:C_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^5.D_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^4$

$\lhd$

$C_2^4:C_4$

$\lhd$

$C_2^5.D_4$

Supergroups

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 $29 \times 29$ rational character table.