Abstract group 1024.dhj: $C_2^6:D_8$

• Label: 1024.dhj
• Order: \( 2^{10} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{14} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $3$

Group information

Description:	$C_2^6:D_8$
Order:	\(1024\)\(\medspace = 2^{10} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^8.C_2\wr C_2^2$, of order \(16384\)\(\medspace = 2^{14} \)
Composition factors:	$C_2$ x 10
Nilpotency class:	$6$
Derived length:	$3$

This group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).

Group statistics

Order	1	2	4	8
Elements	1	175	592	256	1024
Conjugacy classes	1	19	19	4	43
Divisions	1	19	19	2	41
Autjugacy classes	1	10	9	1	21

Dimension	1	2	4	8
Irr. complex chars.	8	10	13	12	43
Irr. rational chars.	8	6	15	12	41

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$3$
Inequivalent generating triples:	$21504$

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:	${\langle a, b, c, d, e, f, g \mid b^{8}=c^{2}=d^{2}=e^{2}=f^{2}=g^{4}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,6,2,5)(3,4)(9,12,10,11)(13,15,14,16), (1,11,7,10,5,16,3,14)(2,12,8,9,6,15,4,13), (1,14,4,16,5,10,7,12)(2,13,3,15,6,9,8,11)\rangle$
Transitive group:	16T1253	16T1265	16T1275		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_2^6$ $\,\rtimes\,$ $D_8$	$(C_2^4:D_4)$ $\,\rtimes\,$ $D_4$	$(C_2^6:D_4)$ $\,\rtimes\,$ $C_2$	$(C_2^5:D_8)$ $\,\rtimes\,$ $C_2$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^4:D_4)$ . $D_4$	$C_2^4$ . $(D_4:D_4)$	$C_2^4$ . $(D_4:D_4)$	$(C_2^5:C_4)$ . $D_4$	all 17

Elements of the group are displayed as permutations of degree 16.

Homology

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

Subgroups

There are 19605 subgroups in 2205 conjugacy classes, 39 normal (23 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^5:D_8$
Commutator:	$G' \simeq$ $C_2^5:C_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^5:C_4$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6:D_8$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^6:D_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2^5:D_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6:D_8$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1024: $C_2^6:D_8$

Order 512: $C_2^5:D_8$ x 2, $C_2^5.D_8$ x 2, $C_2^6:D_4$, $C_2^6:C_8$, $C_2^5.C_2^4$

Order 256: $C_2^5:D_4$ x 3, $C_2^4:D_8$ x 2, $C_2^4.D_8$ x 2, $C_2^5:C_8$ x 2, $C_2^5:D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$, $C_2^4.C_4^2$, $C_2^5:C_2^3$, $C_2^5:D_4$, $C_2^5:D_4$, $C_2^6:C_4$

Order 128: $C_2^4:D_4$ x 11, $C_2^5:C_4$ x 9, $C_2^3.C_2^4$ x 6, $C_2^4:D_4$ x 5, $C_2^4:D_4$ x 4, $C_2^4.D_4$ x 4, $C_2^3.C_4^2$ x 3, $C_2^4.D_4$ x 2, $C_2^3.D_8$ x 2, $C_2^4:C_8$ x 2, $C_2^3:D_8$ x 2, $C_2^4:D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^4.D_4$, $D_4:C_2^4$, $D_4^2:C_2$, $C_2^4:D_4$, $C_2^3\wr C_2$

Order 64: $C_2^3:D_4$ x 38, $C_2^3.D_4$ x 30, $C_2\wr C_2^2$ x 22, $C_2^4:C_4$ x 21, $C_2^3:D_4$ x 15, $D_4:C_2^3$ x 15, $C_2^4:C_4$ x 12, $D_4\times C_2^3$ x 9, $C_2^4:C_4$ x 5, $C_2^3:Q_8$ x 4, $C_2^2.C_4^2$ x 4, $C_2^3:D_4$ x 4, $C_2^3.D_4$ x 3, $D_4:C_2^3$ x 3, $C_2^2.D_8$ x 2, $C_2^3:C_8$ x 2, $C_4^2:C_4$ x 2, $C_4^2:C_4$ x 2, $D_4:D_4$ x 2, $D_4^2$ x 2, $C_2^3.D_4$ x 2, $D_4:D_4$ x 2, $C_2^6$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_4^2:C_2^2$

Order 32: $C_2^2\times D_4$ x 116, $C_2^2\wr C_2$ x 96, $D_4:C_2^2$ x 76, $C_2^3:C_4$ x 72, $C_2^3:C_4$ x 40, $D_4:C_2^2$ x 38, $C_2^5$ x 15, $C_2.C_4^2$ x 15, $C_2^3\times C_4$ x 9, $C_4:D_4$ x 9, $C_2^2.D_4$ x 6, $C_4\times D_4$ x 4, $D_4:C_4$ x 2, $C_2^2:C_8$ x 2, $C_2\times D_8$ x 2, $C_4:D_4$ x 2, $C_4^2:C_2$ x 2, $C_2^2\times Q_8$, $C_2^2:Q_8$, $C_4^2:C_2$

Order 16: $C_2\times D_4$ x 362, $C_2^4$ x 106, $D_4:C_2$ x 102, $C_2^2:C_4$ x 98, $C_2^2\times C_4$ x 62, $C_2\times Q_8$ x 7, $C_4:C_4$ x 5, $D_8$ x 4, $C_2\times C_8$ x 2, $C_4^2$ x 2

Order 8: $C_2^3$ x 225, $D_4$ x 145, $C_2\times C_4$ x 104, $Q_8$ x 9, $C_8$ x 2

Order 4: $C_2^2$ x 129, $C_4$ x 19

Order 2: $C_2$ x 19

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1024: $C_2^6:D_8$

Order 512: $C_2^5:D_8$, $C_2^6:D_4$, $C_2^5.D_8$, $C_2^6:C_8$, $C_2^5.C_2^4$

Order 256: $C_2^5:D_4$ x 2, $C_2^5.D_4$, $C_2^4:D_8$, $C_2^4.D_8$, $C_2^5:C_8$, $C_2^4.C_4^2$, $C_2^5:C_2^3$, $C_2^5:D_4$, $C_2^5.D_4$, $C_2^5:D_4$, $C_2^5:D_4$, $C_2^6:C_4$

Order 128: $C_2^5:C_4$ x 6, $C_2^4:D_4$ x 5, $C_2^3.C_2^4$ x 3, $C_2^4:D_4$ x 2, $C_2^3.C_4^2$ x 2, $C_2^4:D_4$ x 2, $C_2^4:D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$, $C_2^3.D_8$, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^4:C_8$, $C_2^3:D_8$, $D_4:C_2^4$, $D_4^2:C_2$, $C_2^4:D_4$, $C_2^3\wr C_2$

Order 64: $C_2^3:D_4$ x 15, $C_2^3.D_4$ x 10, $C_2^4:C_4$ x 9, $C_2^4:C_4$ x 7, $C_2^3:D_4$ x 6, $D_4\times C_2^3$ x 6, $D_4:C_2^3$ x 5, $C_2^4:C_4$ x 4, $C_2^3:D_4$ x 3, $C_2\wr C_2^2$ x 3, $C_2^3:Q_8$ x 2, $C_2^3.D_4$ x 2, $D_4:C_2^3$ x 2, $C_2^2.C_4^2$ x 2, $C_2^2.D_8$, $C_2^3:C_8$, $C_4^2:C_4$, $C_4^2:C_4$, $C_2^6$, $D_4:D_4$, $D_4^2$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_2^3.D_4$, $D_4:D_4$

Order 32: $C_2^2\times D_4$ x 39, $C_2^3:C_4$ x 30, $C_2^2\wr C_2$ x 22, $D_4:C_2^2$ x 12, $C_2^3:C_4$ x 9, $C_2^5$ x 9, $D_4:C_2^2$ x 9, $C_2^3\times C_4$ x 6, $C_2.C_4^2$ x 6, $C_4:D_4$ x 5, $C_2^2.D_4$ x 4, $C_4\times D_4$ x 2, $D_4:C_4$, $C_2^2:C_8$, $C_2^2\times Q_8$, $C_2\times D_8$, $C_4:D_4$, $C_4^2:C_2$, $C_2^2:Q_8$, $C_4^2:C_2$

Order 16: $C_2\times D_4$ x 77, $C_2^4$ x 45, $C_2^2:C_4$ x 31, $C_2^2\times C_4$ x 29, $D_4:C_2$ x 10, $C_4:C_4$ x 3, $C_2\times Q_8$ x 3, $D_8$, $C_2\times C_8$, $C_4^2$

Order 8: $C_2^3$ x 79, $C_2\times C_4$ x 36, $D_4$ x 25, $Q_8$ x 2, $C_8$

Order 4: $C_2^2$ x 48, $C_4$ x 9

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1024: $C_2^6:D_8$ ($C_1$)

Order 512: $C_2^5:D_8$ ($C_2$) x 2, $C_2^5.D_8$ ($C_2$) x 2, $C_2^6:D_4$ ($C_2$), $C_2^6:C_8$ ($C_2$), $C_2^5.C_2^4$ ($C_2$)

Order 256: $C_2^5:D_4$ ($C_2^2$) x 2, $C_2^5:C_8$ ($C_2^2$) x 2, $C_2^4.C_4^2$ ($C_2^2$), $C_2^5:D_4$ ($C_2^2$), $C_2^6:C_4$ ($C_2^2$)

Order 128: $C_2^3.C_4^2$ ($D_4$) x 2, $C_2^4:D_4$ ($D_4$) x 2, $C_2^5:C_4$ ($C_2^3$), $C_2^5:C_4$ ($D_4$), $C_2^4:D_4$ ($D_4$)

Order 64: $C_2^4:C_4$ ($C_2\times D_4$) x 2, $C_2^6$ ($D_8$), $D_4\times C_2^3$ ($C_2\times D_4$), $C_2^4:C_4$ ($D_8$)

Order 32: $C_2^5$ ($C_2\times D_8$), $C_2^5$ ($C_2^2\wr C_2$), $C_2^3\times C_4$ ($D_8:C_2$)

Order 16: $C_2^4$ ($C_2\wr C_2^2$), $C_2^4$ ($D_4:D_4$), $C_2^4$ ($D_4:D_4$)

Order 8: $C_2^3$ ($C_2\wr D_4$) x 2, $C_2^3$ ($C_2^3:D_8$)

Order 4: $C_2^2$ ($C_2^4:D_8$)

Order 2: $C_2$ ($C_2^5:D_8$)

Order 1: $C_1$ ($C_2^6:D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1024: $C_2^6:D_8$ ($C_1$)

Order 512: $C_2^5:D_8$ ($C_2$), $C_2^6:D_4$ ($C_2$), $C_2^5.D_8$ ($C_2$), $C_2^6:C_8$ ($C_2$), $C_2^5.C_2^4$ ($C_2$)

Order 256: $C_2^5:D_4$ ($C_2^2$), $C_2^5:C_8$ ($C_2^2$), $C_2^4.C_4^2$ ($C_2^2$), $C_2^5:D_4$ ($C_2^2$), $C_2^6:C_4$ ($C_2^2$)

Order 128: $C_2^5:C_4$ ($C_2^3$), $C_2^5:C_4$ ($D_4$), $C_2^3.C_4^2$ ($D_4$), $C_2^4:D_4$ ($D_4$), $C_2^4:D_4$ ($D_4$)

Order 64: $C_2^4:C_4$ ($C_2\times D_4$), $C_2^6$ ($D_8$), $D_4\times C_2^3$ ($C_2\times D_4$), $C_2^4:C_4$ ($D_8$)

Order 32: $C_2^5$ ($C_2\times D_8$), $C_2^5$ ($C_2^2\wr C_2$), $C_2^3\times C_4$ ($D_8:C_2$)

Order 16: $C_2^4$ ($C_2\wr C_2^2$), $C_2^4$ ($D_4:D_4$), $C_2^4$ ($D_4:D_4$)

Order 8: $C_2^3$ ($C_2\wr D_4$), $C_2^3$ ($C_2^3:D_8$)

Order 4: $C_2^2$ ($C_2^4:D_8$)

Order 2: $C_2$ ($C_2^5:D_8$)

Order 1: $C_1$ ($C_2^6:D_8$)

Series

Derived series

$C_2^6:D_8$

$\rhd$

$C_2^5:C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_2^6:D_8$

$\rhd$

$C_2^5.C_2^4$

$\rhd$

$C_2^6:C_4$

$\rhd$

$C_2^5:C_4$

$\rhd$

$D_4\times C_2^3$

$\rhd$

$C_2^5$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^6:D_8$

$\rhd$

$C_2^5:C_4$

$\rhd$

$C_2^5$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2^4$

$\lhd$

$C_2^5$

$\lhd$

$C_2^5:C_4$

$\lhd$

$C_2^6:D_8$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 5 larger groups in the database.

Character theory

Complex character table

See the $43 \times 43$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $41 \times 41$ rational character table.