Abstract group 64.137: $D_4.D_4$

• Label: 64.137
• Order: \( 2^{6} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $3$

Group information

Description:	$D_4.D_4$
Order:	\(64\)\(\medspace = 2^{6} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$D_4^2:C_2^2$, of order \(256\)\(\medspace = 2^{8} \)
Composition factors:	$C_2$ x 6
Nilpotency class:	$3$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8
Elements	1	11	36	16	64
Conjugacy classes	1	4	9	2	16
Divisions	1	4	9	2	16
Autjugacy classes	1	3	5	1	10

Dimension	1	2	4
Irr. complex chars.	8	6	2	16
Irr. rational chars.	8	6	2	16

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$3$
Inequivalent generating triples:	$336$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=c^{4}=d^{4}=[a,b]=[a,d]=[c,d]=1, b^{2}=d^{2}, c^{a}=c^{3}d^{3}, c^{b}=c^{3}, d^{b}=d^{3} \rangle$
Permutation group:	Degree $16$ $\langle(1,5,2,6)(3,7,4,8)(9,16,10,15)(11,13,12,14), (1,9)(2,10)(3,11)(4,12)(5,13) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 1 & 1 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ 1 & 0 & 2 & 2 \\ 0 & 1 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 1 & 1 & 2 \\ 0 & 1 & 1 & 1 \\ 2 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 2 & 1 \\ 1 & 1 & 2 & 0 \\ 1 & 0 & 1 & 1 \\ 2 & 2 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 2 & 2 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 2 & 2 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 1 & 0 \\ 1 & 2 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	16T137	32T150	32T344		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4:Q_8)$ $\,\rtimes\,$ $C_2$	$(C_4.D_4)$ $\,\rtimes\,$ $C_2$	$(Q_{16}:C_2)$ $\,\rtimes\,$ $C_2$ (2)	$(C_4\wr C_2)$ $\,\rtimes\,$ $C_2$ (2)	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_8$ . $D_4$ (2)	$D_4$ . $D_4$ (2)	$C_4^2$ . $C_2^2$	$\OD_{16}$ . $C_2^2$ (2)	all 12

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

Homology

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

Subgroups

There are 121 subgroups in 71 conjugacy classes, 27 normal (11 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

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.

Classes of subgroups up to conjugation

Order 64: $D_4.D_4$

Order 32: $Q_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $C_4.D_4$, $C_4.C_2^3$, $C_4:Q_8$

Order 16: $D_4:C_2$ x 6, $C_2\times Q_8$ x 4, $Q_{16}$ x 2, $\SD_{16}$ x 2, $\OD_{16}$ x 2, $C_4:C_4$ x 2, $C_4^2$

Order 8: $C_2\times C_4$ x 11, $Q_8$ x 8, $D_4$ x 6, $C_8$ x 2

Order 4: $C_4$ x 9, $C_2^2$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 64: $D_4.D_4$

Order 32: $C_4.D_4$, $C_4.C_2^3$, $Q_{16}:C_2$, $C_4:Q_8$, $C_4\wr C_2$

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

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 64: $D_4.D_4$ ($C_1$)

Order 32: $Q_{16}:C_2$ ($C_2$) x 2, $C_4\wr C_2$ ($C_2$) x 2, $C_4.D_4$ ($C_2$), $C_4.C_2^3$ ($C_2$), $C_4:Q_8$ ($C_2$)

Order 16: $\OD_{16}$ ($C_2^2$) x 2, $D_4:C_2$ ($C_2^2$) x 2, $C_2\times Q_8$ ($C_2^2$) x 2, $C_4^2$ ($C_2^2$)

Order 8: $Q_8$ ($D_4$) x 2, $D_4$ ($D_4$) x 2, $C_2\times C_4$ ($D_4$) x 2, $C_2\times C_4$ ($C_2^3$)

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

Order 2: $C_2$ ($C_2^2\wr C_2$)

Order 1: $C_1$ ($D_4.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 64: $D_4.D_4$ ($C_1$)

Order 32: $C_4.D_4$ ($C_2$), $C_4.C_2^3$ ($C_2$), $Q_{16}:C_2$ ($C_2$), $C_4:Q_8$ ($C_2$), $C_4\wr C_2$ ($C_2$)

Order 16: $C_2\times Q_8$ ($C_2^2$) x 2, $\OD_{16}$ ($C_2^2$), $C_4^2$ ($C_2^2$), $D_4:C_2$ ($C_2^2$)

Order 8: $Q_8$ ($D_4$), $D_4$ ($D_4$), $C_2\times C_4$ ($C_2^3$), $C_2\times C_4$ ($D_4$)

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

Order 2: $C_2$ ($C_2^2\wr C_2$)

Order 1: $C_1$ ($D_4.D_4$)

Series

Derived series

$D_4.D_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_1$

Chief series

$D_4.D_4$

$\rhd$

$C_4:Q_8$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$D_4.D_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

$\lhd$

$D_4.D_4$

Supergroups

This group is a maximal subgroup of 81 larger groups in the database.

This group is a maximal quotient of 99 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

		1A	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B
	Size	1	1	2	4	4	2	2	4	4	4	4	4	4	8	8	8
	2 P	1A	1A	1A	1A	1A	2A	2A	2B	2A	2B	2A	2A	2A	2A	4A	4B
	Schur
64.137.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.137.1b		$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$
64.137.1c		$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
64.137.1d		$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
64.137.1e		$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$
64.137.1f		$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$
64.137.1g		$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$
64.137.1h		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
64.137.2a		$2$	$2$	$-2$	$-2$	$0$	$-2$	$2$	$0$	$0$	$0$	$0$	$2$	$0$	$0$	$0$	$0$
64.137.2b		$2$	$2$	$-2$	$0$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$2$	$0$	$0$	$0$
64.137.2c		$2$	$2$	$-2$	$0$	$2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$-2$	$0$	$0$	$0$
64.137.2d		$2$	$2$	$-2$	$2$	$0$	$-2$	$2$	$0$	$0$	$0$	$0$	$-2$	$0$	$0$	$0$	$0$
64.137.2e		$2$	$2$	$2$	$0$	$0$	$-2$	$-2$	$0$	$-2$	$0$	$2$	$0$	$0$	$0$	$0$	$0$
64.137.2f		$2$	$2$	$2$	$0$	$0$	$-2$	$-2$	$0$	$2$	$0$	$-2$	$0$	$0$	$0$	$0$	$0$
64.137.4a	2	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$-2$	$0$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
64.137.4b	2	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$2$	$0$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$