Abstract group $C_2^3:D_4$

• Label: 64.202
• Order: \( 2^{6} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{12} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3 \)
• Perm deg.: $10$
• Trans deg.: $16$
• Rank: $4$

Group information

Description:	$C_2^3:D_4$
Order:	\(64\)\(\medspace = 2^{6} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^8.(C_2\times S_4)$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \)
Outer automorphisms:	$C_2^6:S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \)
Composition factors:	$C_2$ x 6
Nilpotency class:	$2$
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
Elements	1	39	24	64
Conjugacy classes	1	21	6	28
Divisions	1	21	6	28
Autjugacy classes	1	4	1	6

Dimension	1	2
Irr. complex chars.	16	12	28
Irr. rational chars.	16	12	28

Minimal Presentations

Permutation degree:	$10$
Transitive degree:	$16$
Rank:	$4$
Inequivalent generating quadruples:	$420$

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:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{4}=[a,b]=[a,c]=[a,d]=[b,c]=[b,d]=[c,d]=[c,e]=[d,e]=1, e^{a}=de, e^{b}=e^{3} \rangle$
Permutation group:	Degree $10$ $\langle(1,2)(3,4)(8,10), (1,3)(7,8)(9,10), (2,4)(5,6), (1,3)(2,4)(5,6), (1,3)(2,4)(7,9)(8,10), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{5}(\Z)$
	$\left\langle \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{5}(\F_{2})$
	$\left\langle \left(\begin{array}{rr} 3 & 0 \\ 2 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 3 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 7 & 5 \\ 2 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/8\Z)$
Transitive group:	16T105	32T109	32T275		more information
Direct product:	$C_2$ $\, \times\, $ $(C_2^2\wr C_2)$
Semidirect product:	$C_2^3$ $\,\rtimes\,$ $D_4$	$C_2^5$ $\,\rtimes\,$ $C_2$	$C_2^4$ $\,\rtimes\,$ $C_2^2$ (2)	$C_2^3$ $\,\rtimes\,$ $C_2^3$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $C_2^2$	$C_2^3$ . $C_2^3$ (2)	$C_2^2$ . $C_2^4$	$C_2^2$ . $(C_2\times D_4)$	all 5
Aut. group:	$\Aut(C_2^2:C_8)$	$\Aut(C_4:C_8)$	$\Aut(C_2^2:C_{12})$	$\Aut(C_4:C_{12})$

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{8}\Z)$.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 569 subgroups in 331 conjugacy classes, 105 normal (6 characteristic).

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

Special subgroups

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

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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Classes of subgroups up to conjugation

Order 64: $C_2^3:D_4$

Order 32: $C_2^2\wr C_2$ x 8, $C_2^2\times D_4$ x 3, $C_2^3:C_4$ x 3, $C_2^5$

Order 16: $C_2\times D_4$ x 24, $C_2^4$ x 20, $C_2^2:C_4$ x 12, $C_2^2\times C_4$ x 3

Order 8: $C_2^3$ x 95, $D_4$ x 24, $C_2\times C_4$ x 12

Order 4: $C_2^2$ x 97, $C_4$ x 6

Order 2: $C_2$ x 21

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 64: $C_2^3:D_4$

Order 32: $C_2^5$, $C_2^2\times D_4$, $C_2^2\wr C_2$, $C_2^3:C_4$

Order 16: $C_2^4$ x 4, $C_2\times D_4$ x 2, $C_2^2:C_4$, $C_2^2\times C_4$

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

Order 4: $C_2^2$ x 8, $C_4$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 64: $C_2^3:D_4$ ($C_1$)

Order 32: $C_2^2\wr C_2$ ($C_2$) x 8, $C_2^2\times D_4$ ($C_2$) x 3, $C_2^3:C_4$ ($C_2$) x 3, $C_2^5$ ($C_2$)

Order 16: $C_2^2:C_4$ ($C_2^2$) x 12, $C_2\times D_4$ ($C_2^2$) x 12, $C_2^4$ ($C_2^2$) x 8, $C_2^2\times C_4$ ($C_2^2$) x 3

Order 8: $C_2^3$ ($D_4$) x 12, $C_2^3$ ($C_2^3$) x 9, $C_2\times C_4$ ($C_2^3$) x 6

Order 4: $C_2^2$ ($C_2\times D_4$) x 18, $C_2^2$ ($C_2^4$)

Order 2: $C_2$ ($C_2^2\wr C_2$) x 4, $C_2$ ($C_2^2\times D_4$) x 3

Order 1: $C_1$ ($C_2^3:D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 64: $C_2^3:D_4$ ($C_1$)

Order 32: $C_2^5$ ($C_2$), $C_2^2\times D_4$ ($C_2$), $C_2^2\wr C_2$ ($C_2$), $C_2^3:C_4$ ($C_2$)

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

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

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

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

Order 1: $C_1$ ($C_2^3:D_4$)

Series

Derived series

$C_2^3:D_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^3:D_4$

$\rhd$

$C_2^2\times D_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3:D_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^3:D_4$

Supergroups

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

This group is a maximal quotient of 141 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

See the $28 \times 28$ rational character table. Alternatively, you may search for characters of this group with desired properties.