Abstract group 256.26336: $(D_4\times C_2^3):C_4$

• Label: 256.26336
• Order: \( 2^{8} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $4$

Group information

Description:	$(D_4\times C_2^3):C_4$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^4.(C_2^2\times D_4^2)$, 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	8
Elements	1	63	128	64	256
Conjugacy classes	1	17	20	8	46
Divisions	1	17	16	4	38
Autjugacy classes	1	10	10	1	22

Dimension	1	2	4	8
Irr. complex chars.	32	8	4	2	46
Irr. rational chars.	16	16	4	2	38

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$4$
Inequivalent generating quadruples:	$322560$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{4}=b^{2}=c^{2}=d^{4}=e^{4}=[a,b]=[a,d]=[b,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,11,3,9)(2,12,4,10)(5,16,8,13)(6,15,7,14), (1,2)(3,4)(5,6)(7,8)(9,14) \!\cdots\! \rangle$
Transitive group:	16T452	16T468	16T469	32T2247	all 24
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^4.D_4)$ $\,\rtimes\,$ $C_2$	$(C_2^3:D_4)$ $\,\rtimes\,$ $C_4$	$(C_2^3:D_4)$ $\,\rtimes\,$ $C_4$	$(D_4\times C_2^3)$ $\,\rtimes\,$ $C_4$	all 25
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^2\times D_4)$ . $D_4$ (3)	$(D_4:C_2^2)$ . $D_4$ (3)	$(D_4^2:C_2)$ . $C_2$	$C_2^4$ . $(C_2\times D_4)$	all 39

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

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 2223 subgroups in 705 conjugacy classes, 181 normal (51 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^5:C_4$
Commutator:	$G' \simeq$ $C_2^3$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times D_4$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $(D_4\times C_2^3):C_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $(D_4\times C_2^3):C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2^5:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(D_4\times C_2^3):C_4$

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: $(D_4\times C_2^3):C_4$

Order 128: $C_2^4.D_4$ x 4, $C_2^5:C_4$ x 4, $(C_2^2\times D_4):C_4$ x 2, $(C_2^3\times C_4):C_4$ x 2, $D_4^2:C_2$, $\OD_{16}.C_2^3$, $C_2^3.C_2^4$

Order 64: $C_2\wr C_4$ x 12, $C_2^4:C_4$ x 6, $\OD_{16}:C_2^2$ x 5, $\OD_{16}.C_2^2$ x 4, $C_2^3.D_4$ x 4, $D_4:D_4$ x 4, $D_4^2$ x 4, $C_2^3:D_4$ x 4, $C_2^3:D_4$ x 4, $C_2^3.C_2^3$ x 3, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $D_4:C_2^3$, $D_4\times C_2^3$, $C_8.C_2^3$

Order 32: $C_2^2\times D_4$ x 24, $C_2^2\wr C_2$ x 24, $C_4:D_4$ x 14, $C_4\times D_4$ x 12, $C_2^3:C_4$ x 12, $C_2^3:C_4$ x 10, $\OD_{16}:C_2$ x 8, $C_2^2.D_4$ x 8, $D_4:C_2^2$ x 6, $C_2\times \OD_{16}$ x 6, $D_4:C_2^2$ x 4, $\OD_{16}:C_2$ x 4, $C_4^2:C_2$ x 4, $C_4.D_4$ x 2, $C_2^5$ x 2, $C_4:D_4$ x 2, $C_4^2:C_2$ x 2, $C_2^2:Q_8$ x 2, $C_2^3\times C_4$

Order 16: $C_2\times D_4$ x 88, $C_2^2:C_4$ x 32, $C_2^2\times C_4$ x 22, $C_2^4$ x 21, $D_4:C_2$ x 14, $\OD_{16}$ x 10, $C_4:C_4$ x 8, $C_2\times C_8$ x 6, $C_4^2$ x 4, $C_2\times Q_8$ x 2

Order 8: $C_2^3$ x 71, $D_4$ x 57, $C_2\times C_4$ x 42, $C_8$ x 4, $Q_8$ x 3

Order 4: $C_2^2$ x 65, $C_4$ x 16

Order 2: $C_2$ x 17

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $(D_4\times C_2^3):C_4$

Order 128: $(C_2^2\times D_4):C_4$, $C_2^4.D_4$, $(C_2^3\times C_4):C_4$, $C_2^5:C_4$, $D_4^2:C_2$, $\OD_{16}.C_2^3$, $C_2^3.C_2^4$

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

Order 32: $C_2^2\times D_4$ x 14, $C_2^2\wr C_2$ x 13, $C_4:D_4$ x 9, $C_4\times D_4$ x 7, $D_4:C_2^2$ x 5, $C_2^2.D_4$ x 5, $C_2^3:C_4$ x 5, $C_2^3:C_4$ x 3, $C_4^2:C_2$ x 3, $\OD_{16}:C_2$ x 2, $C_2\times \OD_{16}$ x 2, $C_4:D_4$ x 2, $C_4^2:C_2$ x 2, $C_2^2:Q_8$ x 2, $C_4.D_4$, $C_2^5$, $D_4:C_2^2$, $C_2^3\times C_4$, $\OD_{16}:C_2$

Order 16: $C_2\times D_4$ x 36, $C_2^2:C_4$ x 17, $C_2^2\times C_4$ x 13, $C_2^4$ x 9, $C_4:C_4$ x 6, $D_4:C_2$ x 5, $\OD_{16}$ x 3, $C_4^2$ x 3, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2

Order 8: $C_2^3$ x 26, $D_4$ x 23, $C_2\times C_4$ x 22, $Q_8$ x 2, $C_8$

Order 4: $C_2^2$ x 27, $C_4$ x 10

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $(D_4\times C_2^3):C_4$ ($C_1$)

Order 128: $C_2^4.D_4$ ($C_2$) x 4, $C_2^5:C_4$ ($C_2$) x 4, $(C_2^2\times D_4):C_4$ ($C_2$) x 2, $(C_2^3\times C_4):C_4$ ($C_2$) x 2, $D_4^2:C_2$ ($C_2$), $\OD_{16}.C_2^3$ ($C_2$), $C_2^3.C_2^4$ ($C_2$)

Order 64: $C_2\wr C_4$ ($C_2^2$) x 12, $\OD_{16}:C_2^2$ ($C_2^2$) x 4, $C_2^4:C_4$ ($C_2^2$) x 4, $C_2^3.D_4$ ($C_2^2$) x 4, $\OD_{16}.C_2^2$ ($C_2^2$) x 2, $C_2^3.C_2^3$ ($C_2^2$) x 2, $C_2^3:D_4$ ($C_2^2$) x 2, $C_2^3:D_4$ ($C_4$) x 2, $C_2^3:D_4$ ($C_2^2$) x 2, $C_2^3:D_4$ ($C_4$) x 2, $D_4:C_2^3$ ($C_2^2$), $D_4\times C_2^3$ ($C_4$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2:C_2^2$ ($C_4$), $C_4^2:C_2^2$ ($C_2^2$), $C_4^2:C_2^2$ ($C_4$), $C_4^2:C_2^2$ ($C_4$)

Order 32: $C_2^2\wr C_2$ ($C_2\times C_4$) x 9, $C_2^2\times D_4$ ($C_2\times C_4$) x 6, $\OD_{16}:C_2$ ($C_2^3$) x 4, $C_2^3:C_4$ ($C_2^3$) x 4, $D_4:C_2^2$ ($D_4$) x 4, $C_2^2\times D_4$ ($D_4$) x 4, $C_2^3:C_4$ ($C_2\times C_4$) x 4, $C_2^2.D_4$ ($C_2\times C_4$) x 3, $C_2^2\wr C_2$ ($C_2^3$) x 3, $C_2^5$ ($C_2\times C_4$) x 2, $C_2^2\times D_4$ ($C_2^3$) x 2, $C_4^2:C_2$ ($C_2\times C_4$) x 2, $D_4:C_2^2$ ($C_2^3$), $D_4:C_2^2$ ($C_2\times C_4$), $C_2^3\times C_4$ ($C_2\times C_4$), $C_2^2.D_4$ ($C_2^3$)

Order 16: $C_2\times D_4$ ($C_2^2:C_4$) x 9, $C_2^4$ ($C_2^2\times C_4$) x 5, $C_2\times D_4$ ($C_2\times D_4$) x 5, $C_2^2:C_4$ ($C_2^2\times C_4$) x 4, $C_2^2\times C_4$ ($C_2^2:C_4$) x 4, $C_2^2\times C_4$ ($C_2\times D_4$) x 4, $C_2\times D_4$ ($C_2^2\times C_4$) x 3, $C_2^4$ ($C_2^2:C_4$) x 2, $C_2^4$ ($C_2\times D_4$) x 2, $C_2^2\times C_4$ ($C_2^2\times C_4$) x 2, $C_2\times Q_8$ ($C_2^2:C_4$), $C_2\times Q_8$ ($C_2\times D_4$), $C_2\times D_4$ ($C_2^4$)

Order 8: $C_2^3$ ($C_2^3:C_4$) x 7, $C_2\times C_4$ ($C_2^3:C_4$) x 5, $D_4$ ($C_2^3:C_4$) x 4, $C_2^3$ ($C_2^2\times D_4$), $C_2^3$ ($C_2^3\times C_4$), $C_2\times C_4$ ($C_2^2\times D_4$)

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

Order 2: $C_2$ ($C_2^5:C_4$)

Order 1: $C_1$ ($(D_4\times C_2^3):C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $(D_4\times C_2^3):C_4$ ($C_1$)

Order 128: $(C_2^2\times D_4):C_4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $(C_2^3\times C_4):C_4$ ($C_2$), $C_2^5:C_4$ ($C_2$), $D_4^2:C_2$ ($C_2$), $\OD_{16}.C_2^3$ ($C_2$), $C_2^3.C_2^4$ ($C_2$)

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

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

Order 16: $C_2\times D_4$ ($C_2^2:C_4$) x 6, $C_2^2:C_4$ ($C_2^2\times C_4$) x 3, $C_2^4$ ($C_2^2\times C_4$) x 3, $C_2\times D_4$ ($C_2\times D_4$) x 3, $C_2^2\times C_4$ ($C_2\times D_4$) x 3, $C_2\times D_4$ ($C_2^2\times C_4$) x 2, $C_2^2\times C_4$ ($C_2^2:C_4$) x 2, $C_2^2\times C_4$ ($C_2^2\times C_4$) x 2, $C_2^4$ ($C_2^2:C_4$), $C_2^4$ ($C_2\times D_4$), $C_2\times Q_8$ ($C_2^2:C_4$), $C_2\times Q_8$ ($C_2\times D_4$), $C_2\times D_4$ ($C_2^4$)

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

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

Order 2: $C_2$ ($C_2^5:C_4$)

Order 1: $C_1$ ($(D_4\times C_2^3):C_4$)

Series

Derived series

$(D_4\times C_2^3):C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$(D_4\times C_2^3):C_4$

$\rhd$

$D_4^2:C_2$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$C_2^2\wr C_2$

$\rhd$

$C_2\times D_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(D_4\times C_2^3):C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times D_4$

$\lhd$

$D_4:C_2^3$

$\lhd$

$(D_4\times C_2^3):C_4$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 6 larger groups in the database.

Character theory

Complex character table

See the $46 \times 46$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.