Abstract group 128.2011: $D_4\times D_8$

• Label: 128.2011
• Order: \( 2^{7} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{12} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $12$
• Trans deg.: $32$
• Rank: $4$

Group information

Description:	$D_4\times D_8$
Order:	\(128\)\(\medspace = 2^{7} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^4.C_2^6.C_2^2$, of order \(4096\)\(\medspace = 2^{12} \)
Composition factors:	$C_2$ x 7
Nilpotency class:	$3$
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	59	36	32	128
Conjugacy classes	1	15	9	10	35
Divisions	1	15	9	5	30
Autjugacy classes	1	6	6	3	16

Dimension	1	2	4	8
Irr. complex chars.	16	16	3	0	35
Irr. rational chars.	16	8	5	1	30

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$32$
Rank:	$4$
Inequivalent generating quadruples:	$20160$

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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{8}=d^{4}=[a,b]=[a,c]=[b,d]=[c,d]=1, d^{a}=d^{3}, c^{b}=c^{7} \rangle$
Permutation group:	Degree $12$ $\langle(1,2)(4,6)(5,8)(9,10)(11,12), (1,3)(2,5)(4,7)(6,8), (1,4)(2,6)(3,7)(5,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 14 & 18 \\ 9 & 14 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 15 & 13 \end{array}\right), \left(\begin{array}{rr} 0 & 4 \\ 2 & 0 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/21\Z)$
Transitive group:	32T1192				more information
Direct product:	$D_4$ $\, \times\, $ $D_8$
Semidirect product:	$D_4^2$ $\,\rtimes\,$ $C_2$	$(D_4:D_4)$ $\,\rtimes\,$ $C_2$	$D_4$ $\,\rtimes\,$ $(C_2\times D_4)$	$(C_8:D_4)$ $\,\rtimes\,$ $C_2$	all 23
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $D_4^2$	$C_4^2$ . $C_2^3$	$(C_2\times D_4)$ . $D_4$	$(C_4:C_4)$ . $D_4$	all 19

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

Homology

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

Subgroups

There are 824 subgroups in 310 conjugacy classes, 104 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^2\times D_4$
Commutator:	$G' \simeq$ $C_2\times C_4$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times D_8$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $D_4\times D_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times D_8$

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 128: $D_4\times D_8$

Order 64: $D_4:D_4$ x 4, $C_2^2\times D_8$ x 2, $D_4^2$ x 2, $C_8:D_4$ x 2, $C_4:D_8$ x 2, $C_4:D_8$, $C_4\times D_8$, $C_8\times D_4$

Order 32: $C_2\times D_8$ x 17, $D_4:C_4$ x 6, $C_2^2\times D_4$ x 6, $C_4:D_4$ x 6, $C_2^2\wr C_2$ x 4, $C_4\times D_4$ x 3, $C_2^2:C_8$ x 2, $C_2^2\times C_8$ x 2, $C_4:D_4$ x 2, $C_4\times C_8$, $C_8:C_4$, $C_4:C_8$

Order 16: $C_2\times D_4$ x 39, $D_8$ x 18, $C_2\times C_8$ x 8, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 3, $C_4^2$

Order 8: $D_4$ x 40, $C_2^3$ x 28, $C_2\times C_4$ x 15, $C_8$ x 5

Order 4: $C_2^2$ x 45, $C_4$ x 9

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 128: $D_4\times D_8$

Order 64: $C_2^2\times D_8$, $D_4^2$, $C_4:D_8$, $C_8:D_4$, $C_4:D_8$, $D_4:D_4$, $C_4\times D_8$, $C_8\times D_4$

Order 32: $C_2\times D_8$ x 5, $D_4:C_4$ x 2, $C_2^2\times D_4$ x 2, $C_4:D_4$ x 2, $C_4\times D_4$ x 2, $C_2^2:C_8$, $C_2^2\times C_8$, $C_4:D_4$, $C_4\times C_8$, $C_2^2\wr C_2$, $C_8:C_4$, $C_4:C_8$

Order 16: $C_2\times D_4$ x 10, $D_8$ x 4, $C_2\times C_8$ x 4, $C_2^2:C_4$ x 3, $C_4:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4^2$, $C_2^4$

Order 8: $D_4$ x 9, $C_2\times C_4$ x 8, $C_2^3$ x 6, $C_8$ x 3

Order 4: $C_2^2$ x 11, $C_4$ x 6

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 128: $D_4\times D_8$ ($C_1$)

Order 64: $D_4:D_4$ ($C_2$) x 4, $C_2^2\times D_8$ ($C_2$) x 2, $D_4^2$ ($C_2$) x 2, $C_8:D_4$ ($C_2$) x 2, $C_4:D_8$ ($C_2$) x 2, $C_4:D_8$ ($C_2$), $C_4\times D_8$ ($C_2$), $C_8\times D_4$ ($C_2$)

Order 32: $C_2\times D_8$ ($C_2^2$) x 9, $D_4:C_4$ ($C_2^2$) x 6, $C_2^2\times D_4$ ($C_2^2$) x 4, $C_4:D_4$ ($C_2^2$) x 4, $C_4\times D_4$ ($C_2^2$) x 3, $C_2^2:C_8$ ($C_2^2$) x 2, $C_2^2\times C_8$ ($C_2^2$) x 2, $C_4:D_4$ ($C_2^2$) x 2, $C_4\times C_8$ ($C_2^2$), $C_8:C_4$ ($C_2^2$), $C_4:C_8$ ($C_2^2$)

Order 16: $C_2\times D_4$ ($C_2^3$) x 6, $D_8$ ($D_4$) x 4, $C_2\times C_8$ ($C_2^3$) x 4, $C_4:C_4$ ($C_2^3$) x 2, $C_2^2:C_4$ ($D_4$) x 2, $C_2^2\times C_4$ ($C_2^3$) x 2, $C_4:C_4$ ($D_4$), $C_4^2$ ($C_2^3$), $C_2\times D_4$ ($D_4$)

Order 8: $D_4$ ($D_8$) x 4, $D_4$ ($C_2\times D_4$) x 4, $C_2\times C_4$ ($C_2\times D_4$) x 4, $C_2^3$ ($C_2\times D_4$) x 2, $C_8$ ($C_2\times D_4$) x 2, $C_2\times C_4$ ($C_2^4$)

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

Order 2: $C_2$ ($D_8:C_2^2$), $C_2$ ($C_2^2\times D_8$), $C_2$ ($D_4^2$)

Order 1: $C_1$ ($D_4\times D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 128: $D_4\times D_8$ ($C_1$)

Order 64: $C_2^2\times D_8$ ($C_2$), $D_4^2$ ($C_2$), $C_4:D_8$ ($C_2$), $C_8:D_4$ ($C_2$), $C_4:D_8$ ($C_2$), $D_4:D_4$ ($C_2$), $C_4\times D_8$ ($C_2$), $C_8\times D_4$ ($C_2$)

Order 32: $C_2\times D_8$ ($C_2^2$) x 4, $D_4:C_4$ ($C_2^2$) x 2, $C_4\times D_4$ ($C_2^2$) x 2, $C_2^2:C_8$ ($C_2^2$), $C_2^2\times D_4$ ($C_2^2$), $C_2^2\times C_8$ ($C_2^2$), $C_4:D_4$ ($C_2^2$), $C_4\times C_8$ ($C_2^2$), $C_4:D_4$ ($C_2^2$), $C_8:C_4$ ($C_2^2$), $C_4:C_8$ ($C_2^2$)

Order 16: $C_2\times C_8$ ($C_2^3$) x 3, $C_2\times D_4$ ($C_2^3$) x 2, $D_8$ ($D_4$), $C_4:C_4$ ($C_2^3$), $C_4:C_4$ ($D_4$), $C_2^2:C_4$ ($D_4$), $C_4^2$ ($C_2^3$), $C_2\times D_4$ ($D_4$), $C_2^2\times C_4$ ($C_2^3$)

Order 8: $C_2\times C_4$ ($C_2\times D_4$) x 3, $C_2^3$ ($C_2\times D_4$), $D_4$ ($D_8$), $D_4$ ($C_2\times D_4$), $C_2\times C_4$ ($C_2^4$), $C_8$ ($C_2\times D_4$)

Order 4: $C_2^2$ ($C_2^2\times D_4$), $C_2^2$ ($C_2\times D_8$), $C_4$ ($D_4:C_2^2$), $C_4$ ($C_2^2\times D_4$), $C_4$ ($C_2\times D_8$)

Order 2: $C_2$ ($D_8:C_2^2$), $C_2$ ($C_2^2\times D_8$), $C_2$ ($D_4^2$)

Order 1: $C_1$ ($D_4\times D_8$)

Series

Derived series

$D_4\times D_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_1$

Chief series

$D_4\times D_8$

$\rhd$

$C_4\times D_8$

$\rhd$

$C_4\times C_8$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$D_4\times D_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_4\times D_4$

$\lhd$

$D_4\times D_8$

Supergroups

This group is a maximal subgroup of 105 larger groups in the database.

This group is a maximal quotient of 146 larger groups in the database.

Character theory

Complex character table

See the $35 \times 35$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.