Abstract group 64.89: $(C_2^2\times C_8):C_2$

• Label: 64.89
• Order: \( 2^{6} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $20$
• Trans deg.: $32$
• Rank: $3$

Group information

Description:	$(C_2^2\times C_8):C_2$
Order:	\(64\)\(\medspace = 2^{6} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^6:D_4$, of order \(512\)\(\medspace = 2^{9} \)
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), and metabelian.

Group statistics

Order	1	2	4	8
Elements	1	15	16	32	64
Conjugacy classes	1	7	8	12	28
Divisions	1	7	6	4	18
Autjugacy classes	1	4	4	2	11

Dimension	1	2	8
Irr. complex chars.	16	12	0	28
Irr. rational chars.	8	8	2	18

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$32$
Rank:	$3$
Inequivalent generating triples:	$168$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	10

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{8}=[a,c]=[b,c]=[b,d]=[c,d]=1, b^{a}=bd^{4}, d^{a}=cd \rangle$
Permutation group:	Degree $20$ $\langle(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,19,20), (1,3)(2,6)(4,10) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 2 \\ 8 & 13 \end{array}\right), \left(\begin{array}{rr} 9 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 3 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 15 & 4 \\ 0 & 15 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
	$\left\langle \left(\begin{array}{rr} 7 & 9 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 33 \\ 24 & 23 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/48\Z)$
Transitive group:	32T191	32T193			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^2\times C_8)$ $\,\rtimes\,$ $C_2$	$(C_2^2:C_8)$ $\,\rtimes\,$ $C_2$ (4)	$(C_2\times \OD_{16})$ $\,\rtimes\,$ $C_2$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times Q_8)$ . $C_4$	$(C_2\times D_4)$ . $C_4$ (3)	$(C_2\times C_4)$ . $D_4$ (4)	$C_4$ . $(C_2\times D_4)$ (2)	all 15

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{16}\Z)$.

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 121 subgroups in 79 conjugacy classes, 41 normal (15 characteristic).

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

Special subgroups

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

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: $(C_2^2\times C_8):C_2$

Order 32: $C_2^2:C_8$ x 4, $D_4:C_2^2$, $C_2\times \OD_{16}$, $C_2^2\times C_8$

Order 16: $C_2\times C_8$ x 6, $D_4:C_2$ x 4, $C_2\times D_4$ x 3, $C_2^2\times C_4$ x 3, $\OD_{16}$ x 2, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 12, $D_4$ x 6, $C_8$ x 4, $C_2^3$ x 3, $Q_8$ x 2

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

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 64: $(C_2^2\times C_8):C_2$

Order 32: $C_2^2:C_8$, $D_4:C_2^2$, $C_2\times \OD_{16}$, $C_2^2\times C_8$

Order 16: $C_2\times C_8$ x 3, $D_4:C_2$ x 2, $C_2\times D_4$ x 2, $C_2^2\times C_4$ x 2, $\OD_{16}$, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 7, $C_2^3$ x 2, $D_4$ x 2, $C_8$ x 2, $Q_8$

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

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 64: $(C_2^2\times C_8):C_2$ ($C_1$)

Order 32: $C_2^2:C_8$ ($C_2$) x 4, $D_4:C_2^2$ ($C_2$), $C_2\times \OD_{16}$ ($C_2$), $C_2^2\times C_8$ ($C_2$)

Order 16: $C_2\times C_8$ ($C_2^2$) x 4, $C_2\times D_4$ ($C_4$) x 3, $C_2^2\times C_4$ ($C_2^2$) x 3, $C_2\times Q_8$ ($C_4$)

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

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

Order 2: $C_2$ ($\OD_{16}:C_2$) x 2, $C_2$ ($C_2^3:C_4$)

Order 1: $C_1$ ($(C_2^2\times C_8):C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 64: $(C_2^2\times C_8):C_2$ ($C_1$)

Order 32: $C_2^2:C_8$ ($C_2$), $D_4:C_2^2$ ($C_2$), $C_2\times \OD_{16}$ ($C_2$), $C_2^2\times C_8$ ($C_2$)

Order 16: $C_2\times C_8$ ($C_2^2$) x 2, $C_2\times D_4$ ($C_4$) x 2, $C_2^2\times C_4$ ($C_2^2$) x 2, $C_2\times Q_8$ ($C_4$)

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

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

Order 2: $C_2$ ($\OD_{16}:C_2$), $C_2$ ($C_2^3:C_4$)

Order 1: $C_1$ ($(C_2^2\times C_8):C_2$)

Series

Derived series

$(C_2^2\times C_8):C_2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$(C_2^2\times C_8):C_2$

$\rhd$

$C_2^2\times C_8$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times C_8):C_2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$(C_2^2\times C_8):C_2$

Supergroups

This group is a maximal subgroup of 131 larger groups in the database.

This group is a maximal quotient of 101 larger groups in the database.

Character theory

Complex character table

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

Rational character table

See the $18 \times 18$ rational character table.