Abstract group 256.5426: $C_{16}.D_8$

• Label: 256.5426
• Order: \( 2^{8} \)
• Exponent: \( 2^{4} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{11} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $32$
• Trans deg.: $32$
• Rank: $3$

Group information

Description:	$C_{16}.D_8$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Automorphism group:	$C_4^2.C_2^3.C_2^4$, of order \(2048\)\(\medspace = 2^{11} \)
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	16
Elements	1	43	52	32	128	256
Conjugacy classes	1	4	6	8	18	37
Divisions	1	4	6	5	6	22
Autjugacy classes	1	4	5	5	6	21

Dimension	1	2	4	8	16
Irr. complex chars.	8	18	11	0	0	37
Irr. rational chars.	8	4	6	2	2	22

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$32$
Rank:	$3$
Inequivalent generating triples:	$2688$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	16
Arbitrary	4	8	16

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=c^{16}=[a,c]=1, b^{8}=c^{8}, b^{a}=b^{7}c^{8}, c^{b}=c^{7} \rangle$
Permutation group:	Degree $32$ $\langle(3,4)(5,7)(6,8)(9,13)(10,14)(11,16)(12,15)(17,24)(18,23)(19,22)(20,21)(25,26) \!\cdots\! \rangle$
Transitive group:	32T7468				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_8.D_8)$ $\,\rtimes\,$ $C_2$	$(C_8.D_8)$ $\,\rtimes\,$ $C_2$	$(C_8.D_8)$ $\,\rtimes\,$ $C_2$	$(C_{16}.D_4)$ $\,\rtimes\,$ $C_2$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_8$ . $D_8$	$Q_{16}$ . $D_8$	$C_{16}$ . $D_8$ (2)	$(D_8:D_4)$ . $C_2$	all 23

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 415 subgroups in 112 conjugacy classes, 37 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_8:D_8$
Commutator:	$G' \simeq$ $C_4\times C_8$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4\times C_8$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_{16}.D_8$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{16}.D_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_8:D_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{16}.D_8$

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

Normal subgroups up to automorphism

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Subgroup information Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 256: $C_{16}.D_8$

Order 128: $C_{16}:D_4$, $C_8.D_8$, $D_8:D_4$, $C_{16}.D_4$, $C_8.D_8$, $C_8.D_8$, $C_{16}.C_8$

Order 64: $C_8.C_8$ x 3, $C_2\times \SD_{32}$ x 2, $C_4.D_8$, $\OD_{32}:C_2$, $\OD_{32}:C_2$, $C_4\times C_{16}$, $Q_{32}:C_2$, $D_{16}:C_2$, $\OD_{32}:C_2$, $C_4:Q_{16}$, $C_4:D_8$, $D_8:C_4$

Order 32: $\SD_{32}$ x 6, $\OD_{32}$ x 4, $C_2\times C_{16}$ x 3, $C_2\times Q_{16}$ x 2, $C_2\times D_8$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $C_4:Q_8$, $C_4:D_4$, $C_4\times C_8$, $Q_{32}$, $D_{16}$, $C_8.C_4$, $C_4\wr C_2$

Order 16: $C_{16}$ x 6, $Q_{16}$ x 5, $D_8$ x 5, $C_2\times C_8$ x 3, $\OD_{16}$ x 2, $C_2\times D_4$ x 2, $\SD_{16}$, $C_4:C_4$, $C_4^2$, $D_4:C_2$, $C_2\times Q_8$

Order 8: $D_4$ x 6, $C_8$ x 5, $C_2\times C_4$ x 4, $Q_8$ x 3, $C_2^3$

Order 4: $C_4$ x 6, $C_2^2$ x 5

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $C_{16}.D_8$

Order 128: $C_{16}:D_4$, $C_8.D_8$, $D_8:D_4$, $C_{16}.D_4$, $C_8.D_8$, $C_8.D_8$, $C_{16}.C_8$

Order 64: $C_8.C_8$ x 3, $C_2\times \SD_{32}$ x 2, $C_4.D_8$, $\OD_{32}:C_2$, $\OD_{32}:C_2$, $C_4\times C_{16}$, $Q_{32}:C_2$, $D_{16}:C_2$, $\OD_{32}:C_2$, $C_4:Q_{16}$, $C_4:D_8$, $D_8:C_4$

Order 32: $\SD_{32}$ x 5, $\OD_{32}$ x 4, $C_2\times C_{16}$ x 3, $C_2\times Q_{16}$ x 2, $C_2\times D_8$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $C_4:Q_8$, $C_4:D_4$, $C_4\times C_8$, $Q_{32}$, $D_{16}$, $C_8.C_4$, $C_4\wr C_2$

Order 16: $C_{16}$ x 6, $Q_{16}$ x 5, $D_8$ x 5, $C_2\times C_8$ x 3, $\OD_{16}$ x 2, $C_2\times D_4$ x 2, $\SD_{16}$, $C_4:C_4$, $C_4^2$, $D_4:C_2$, $C_2\times Q_8$

Order 8: $D_4$ x 5, $C_8$ x 5, $C_2\times C_4$ x 4, $Q_8$ x 3, $C_2^3$

Order 4: $C_4$ x 5, $C_2^2$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $C_{16}.D_8$ ($C_1$)

Order 128: $C_{16}:D_4$ ($C_2$), $C_8.D_8$ ($C_2$), $D_8:D_4$ ($C_2$), $C_{16}.D_4$ ($C_2$), $C_8.D_8$ ($C_2$), $C_8.D_8$ ($C_2$), $C_{16}.C_8$ ($C_2$)

Order 64: $C_8.C_8$ ($C_2^2$) x 3, $C_4\times C_{16}$ ($C_2^2$), $C_4:Q_{16}$ ($C_2^2$), $C_4:D_8$ ($C_2^2$), $D_8:C_4$ ($C_2^2$)

Order 32: $C_2\times C_{16}$ ($D_4$) x 2, $D_8:C_2$ ($D_4$), $C_4\times C_8$ ($C_2^3$), $C_8.C_4$ ($D_4$)

Order 16: $C_2\times C_8$ ($C_2\times D_4$) x 2, $C_{16}$ ($D_8$) x 2, $Q_{16}$ ($D_8$), $D_8$ ($D_8$), $C_4^2$ ($D_4:C_2$)

Order 8: $C_8$ ($C_2\times D_8$) x 2, $C_2\times C_4$ ($C_4:D_4$), $C_8$ ($D_8:C_2$), $C_8$ ($D_8:C_2$)

Order 4: $C_2^2$ ($C_8.D_4$), $C_4$ ($C_8:D_4$), $C_4$ ($C_4:D_8$)

Order 2: $C_2$ ($C_8:D_8$)

Order 1: $C_1$ ($C_{16}.D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $C_{16}.D_8$ ($C_1$)

Order 128: $C_{16}:D_4$ ($C_2$), $C_8.D_8$ ($C_2$), $D_8:D_4$ ($C_2$), $C_{16}.D_4$ ($C_2$), $C_8.D_8$ ($C_2$), $C_8.D_8$ ($C_2$), $C_{16}.C_8$ ($C_2$)

Order 64: $C_8.C_8$ ($C_2^2$) x 3, $C_4\times C_{16}$ ($C_2^2$), $C_4:Q_{16}$ ($C_2^2$), $C_4:D_8$ ($C_2^2$), $D_8:C_4$ ($C_2^2$)

Order 32: $C_2\times C_{16}$ ($D_4$) x 2, $D_8:C_2$ ($D_4$), $C_4\times C_8$ ($C_2^3$), $C_8.C_4$ ($D_4$)

Order 16: $C_2\times C_8$ ($C_2\times D_4$) x 2, $C_{16}$ ($D_8$) x 2, $Q_{16}$ ($D_8$), $D_8$ ($D_8$), $C_4^2$ ($D_4:C_2$)

Order 8: $C_8$ ($C_2\times D_8$) x 2, $C_2\times C_4$ ($C_4:D_4$), $C_8$ ($D_8:C_2$), $C_8$ ($D_8:C_2$)

Order 4: $C_2^2$ ($C_8.D_4$), $C_4$ ($C_8:D_4$), $C_4$ ($C_4:D_8$)

Order 2: $C_2$ ($C_8:D_8$)

Order 1: $C_1$ ($C_{16}.D_8$)

Series

Derived series

$C_{16}.D_8$

$\rhd$

$C_4\times C_8$

$\rhd$

$C_1$

Chief series

$C_{16}.D_8$

$\rhd$

$C_{16}.C_8$

$\rhd$

$C_4\times C_{16}$

$\rhd$

$C_4\times C_8$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_{16}.D_8$

$\rhd$

$C_4\times C_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_4\times C_8$

$\lhd$

$C_{16}.D_8$

Supergroups

This group is a maximal subgroup of 3 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 $37 \times 37$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.