Abstract group 256.26490: $D_4:(C_2\times C_{16})$

• Label: 256.26490
• Order: \( 2^{8} \)
• Exponent: \( 2^{4} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $128$
• Rank: $4$

Group information

Description:	$D_4:(C_2\times C_{16})$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Automorphism group:	$C_2^4:C_3.C_2^4.C_2^4$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \)
Composition factors:	$C_2$ x 8
Nilpotency class:	$2$
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	16
Elements	1	15	48	64	128	256
Conjugacy classes	1	9	26	36	64	136
Divisions	1	9	17	9	8	44
Autjugacy classes	1	4	7	5	2	19

Dimension	1	2	4	8	16
Irr. complex chars.	128	0	8	0	0	136
Irr. rational chars.	16	8	10	9	1	44

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$128$
Rank:	$4$
Inequivalent generating quadruples:	$107520$

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:	$\langle a, b, c, d \mid a^{2}=b^{16}=c^{4}=d^{2}=[a,b]=[a,c]=[b,d]=[c,d]=1, d^{a}=c^{2}d, c^{b}=c^{3} \rangle$
Permutation group:	Degree $24$ $\langle(1,2)(3,7)(4,5)(6,8)(9,10,12,15,13,16,19,22,11,14,17,20,18,21,23,24), (1,3) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(Q_8\times C_{16})$ $\,\rtimes\,$ $C_2$	$Q_8$ $\,\rtimes\,$ $(C_2\times C_{16})$	$(D_4\times C_{16})$ $\,\rtimes\,$ $C_2$	$(D_4:C_2)$ $\,\rtimes\,$ $C_{16}$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_8\times Q_8)$ . $C_4$	$(C_4\times Q_8)$ . $C_8$	$(C_8\times D_4)$ . $C_4$	$(C_4\times D_4)$ . $C_8$	all 49

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

Homology

Abelianization:	$C_{2}^{3} \times C_{16} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 367 subgroups in 285 conjugacy classes, 227 normal (26 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_8$	$G/Z \simeq$ $C_2^4$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^3\times C_{16}$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $D_4:(C_2\times C_{16})$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $D_4:(C_2\times C_{16})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^3\times C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:(C_2\times C_{16})$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 256: $D_4:(C_2\times C_{16})$

Order 128: $D_4\times C_{16}$ x 6, $C_4^2.C_8$ x 3, $C_4^2.C_8$ x 3, $Q_8\times C_{16}$ x 2, $C_4^2.C_2^3$

Order 64: $C_4:C_{16}$ x 10, $C_2^2:C_{16}$ x 6, $C_4\times C_{16}$ x 6, $C_2^2\times C_{16}$ x 6, $C_8\times D_4$ x 6, $C_2\times C_4\times C_8$ x 3, $C_4^2.C_4$ x 3, $C_8\times Q_8$ x 2, $C_4^2.C_2^2$

Order 32: $C_2\times C_{16}$ x 14, $C_4\times C_8$ x 10, $C_2^2:C_8$ x 6, $C_2^2\times C_8$ x 6, $C_4\times D_4$ x 6, $C_4:C_8$ x 6, $C_4^2:C_2$ x 3, $C_2\times C_4^2$ x 3, $C_4\times Q_8$ x 2, $D_4:C_2^2$

Order 16: $C_2\times C_8$ x 17, $C_4^2$ x 10, $C_2^2\times C_4$ x 9, $D_4:C_2$ x 8, $C_{16}$ x 8, $C_4:C_4$ x 6, $C_2^2:C_4$ x 6, $C_2\times D_4$ x 3, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 33, $D_4$ x 12, $C_8$ x 9, $Q_8$ x 4, $C_2^3$ x 3

Order 4: $C_4$ x 17, $C_2^2$ x 13

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $D_4:(C_2\times C_{16})$

Order 128: $Q_8\times C_{16}$, $D_4\times C_{16}$, $C_4^2.C_8$, $C_4^2.C_8$, $C_4^2.C_2^3$

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

Order 32: $C_4\times C_8$ x 4, $C_2\times C_{16}$ x 3, $C_2^2\times C_8$ x 2, $C_4\times Q_8$ x 2, $C_4\times D_4$ x 2, $C_4:C_8$ x 2, $C_2^2:C_8$, $D_4:C_2^2$, $C_4^2:C_2$, $C_2\times C_4^2$

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

Order 8: $C_2\times C_4$ x 10, $C_8$ x 5, $C_2^3$, $Q_8$, $D_4$

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

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $D_4:(C_2\times C_{16})$ ($C_1$)

Order 128: $D_4\times C_{16}$ ($C_2$) x 6, $C_4^2.C_8$ ($C_2$) x 3, $C_4^2.C_8$ ($C_2$) x 3, $Q_8\times C_{16}$ ($C_2$) x 2, $C_4^2.C_2^3$ ($C_2$)

Order 64: $C_4:C_{16}$ ($C_2^2$) x 10, $C_2^2:C_{16}$ ($C_2^2$) x 6, $C_4\times C_{16}$ ($C_2^2$) x 6, $C_2^2\times C_{16}$ ($C_2^2$) x 6, $C_2\times C_4\times C_8$ ($C_2^2$) x 3, $C_8\times D_4$ ($C_2^2$) x 3, $C_8\times D_4$ ($C_4$) x 3, $C_4^2.C_4$ ($C_4$) x 3, $C_4^2.C_2^2$ ($C_4$), $C_8\times Q_8$ ($C_2^2$), $C_8\times Q_8$ ($C_4$)

Order 32: $C_2\times C_{16}$ ($C_2^3$) x 8, $C_2^2:C_8$ ($C_2\times C_4$) x 6, $C_4\times C_8$ ($C_2\times C_4$) x 6, $C_4:C_8$ ($C_2\times C_4$) x 6, $C_4\times C_8$ ($C_2^3$) x 4, $C_2^2\times C_8$ ($C_2^3$) x 3, $C_2^2\times C_8$ ($C_2\times C_4$) x 3, $C_4\times D_4$ ($C_2\times C_4$) x 3, $C_4\times D_4$ ($C_8$) x 3, $C_4^2:C_2$ ($C_8$) x 3, $C_2\times C_4^2$ ($C_2\times C_4$) x 3, $D_4:C_2^2$ ($C_8$), $C_4\times Q_8$ ($C_2\times C_4$), $C_4\times Q_8$ ($C_8$)

Order 16: $D_4:C_2$ ($C_{16}$) x 8, $C_2\times C_8$ ($C_2^2\times C_4$) x 7, $C_4:C_4$ ($C_2\times C_8$) x 6, $C_2^2:C_4$ ($C_2\times C_8$) x 6, $C_4^2$ ($C_2\times C_8$) x 6, $C_2^2\times C_4$ ($C_2\times C_8$) x 6, $C_4^2$ ($C_2^2\times C_4$) x 4, $C_2\times D_4$ ($C_2\times C_8$) x 3, $C_2^2\times C_4$ ($C_2^2\times C_4$) x 3, $C_2\times C_8$ ($C_2^4$), $C_2\times Q_8$ ($C_2\times C_8$)

Order 8: $D_4$ ($C_2\times C_{16}$) x 12, $C_2\times C_4$ ($C_2\times C_{16}$) x 12, $C_2\times C_4$ ($C_2^2\times C_8$) x 11, $Q_8$ ($C_2\times C_{16}$) x 4, $C_2^3$ ($C_2^2\times C_8$) x 3, $C_2\times C_4$ ($C_2^3\times C_4$), $C_8$ ($C_4.C_2^3$), $C_8$ ($D_4:C_2^2$)

Order 4: $C_4$ ($C_2^2\times C_{16}$) x 8, $C_2^2$ ($C_2^2\times C_{16}$) x 6, $C_2^2$ ($C_2^3\times C_8$), $C_4$ ($C_8.C_2^3$), $C_4$ ($C_4^2.C_2^2$)

Order 2: $C_2$ ($\OD_{32}:C_2^2$), $C_2$ ($C_2^3\times C_{16}$), $C_2$ ($C_4^2.C_2^3$)

Order 1: $C_1$ ($D_4:(C_2\times C_{16})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $D_4:(C_2\times C_{16})$ ($C_1$)

Order 128: $Q_8\times C_{16}$ ($C_2$), $D_4\times C_{16}$ ($C_2$), $C_4^2.C_8$ ($C_2$), $C_4^2.C_8$ ($C_2$), $C_4^2.C_2^3$ ($C_2$)

Order 64: $C_4:C_{16}$ ($C_2^2$) x 3, $C_2\times C_4\times C_8$ ($C_2^2$), $C_2^2:C_{16}$ ($C_2^2$), $C_4\times C_{16}$ ($C_2^2$), $C_4^2.C_2^2$ ($C_4$), $C_2^2\times C_{16}$ ($C_2^2$), $C_8\times Q_8$ ($C_2^2$), $C_8\times Q_8$ ($C_4$), $C_8\times D_4$ ($C_2^2$), $C_8\times D_4$ ($C_4$), $C_4^2.C_4$ ($C_4$)

Order 32: $C_4\times C_8$ ($C_2^3$) x 2, $C_4\times C_8$ ($C_2\times C_4$) x 2, $C_2\times C_{16}$ ($C_2^3$) x 2, $C_4:C_8$ ($C_2\times C_4$) x 2, $C_2^2:C_8$ ($C_2\times C_4$), $D_4:C_2^2$ ($C_8$), $C_2^2\times C_8$ ($C_2^3$), $C_2^2\times C_8$ ($C_2\times C_4$), $C_4\times Q_8$ ($C_2\times C_4$), $C_4\times Q_8$ ($C_8$), $C_4\times D_4$ ($C_2\times C_4$), $C_4\times D_4$ ($C_8$), $C_4^2:C_2$ ($C_8$), $C_2\times C_4^2$ ($C_2\times C_4$)

Order 16: $C_2\times C_8$ ($C_2^2\times C_4$) x 3, $C_4:C_4$ ($C_2\times C_8$) x 2, $C_4^2$ ($C_2\times C_8$) x 2, $C_4^2$ ($C_2^2\times C_4$) x 2, $C_2^2\times C_4$ ($C_2\times C_8$) x 2, $C_2\times C_8$ ($C_2^4$), $C_2^2:C_4$ ($C_2\times C_8$), $D_4:C_2$ ($C_{16}$), $C_2\times Q_8$ ($C_2\times C_8$), $C_2\times D_4$ ($C_2\times C_8$), $C_2^2\times C_4$ ($C_2^2\times C_4$)

Order 8: $C_2\times C_4$ ($C_2^2\times C_8$) x 5, $C_2^3$ ($C_2^2\times C_8$), $Q_8$ ($C_2\times C_{16}$), $D_4$ ($C_2\times C_{16}$), $C_2\times C_4$ ($C_2^3\times C_4$), $C_2\times C_4$ ($C_2\times C_{16}$), $C_8$ ($C_4.C_2^3$), $C_8$ ($D_4:C_2^2$)

Order 4: $C_4$ ($C_2^2\times C_{16}$) x 2, $C_2^2$ ($C_2^3\times C_8$), $C_2^2$ ($C_2^2\times C_{16}$), $C_4$ ($C_8.C_2^3$), $C_4$ ($C_4^2.C_2^2$)

Order 2: $C_2$ ($\OD_{32}:C_2^2$), $C_2$ ($C_2^3\times C_{16}$), $C_2$ ($C_4^2.C_2^3$)

Order 1: $C_1$ ($D_4:(C_2\times C_{16})$)

Series

Derived series

$D_4:(C_2\times C_{16})$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$D_4:(C_2\times C_{16})$

$\rhd$

$C_4^2.C_8$

$\rhd$

$C_2\times C_4\times C_8$

$\rhd$

$C_2^2\times C_8$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$D_4:(C_2\times C_{16})$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_8$

$\lhd$

$D_4:(C_2\times C_{16})$

Supergroups

Character theory

Complex character table

See the $136 \times 136$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $44 \times 44$ rational character table.