Abstract group 64.56: $C_2^2.C_4^2$

• Label: 64.56
• Order: \( 2^{6} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $64$
• Rank: $3$

Group information

Description:	$C_2^2.C_4^2$
Order:	\(64\)\(\medspace = 2^{6} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^9.S_4$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \)
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), and metabelian.

Group statistics

Order	1	2	4
Elements	1	15	48	64
Conjugacy classes	1	15	24	40
Divisions	1	15	12	28
Autjugacy classes	1	4	1	6

Dimension	1	2
Irr. complex chars.	32	8	40
Irr. rational chars.	8	20	28

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$64$
Rank:	$3$
Inequivalent generating triples:	$7$

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

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

Homology

Abelianization:	$C_{2} \times C_{4}^{2} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 225 subgroups in 165 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:	$Z \simeq$ $C_2^4$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2\times C_4^2$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2.C_4^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^2.C_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2.C_4^2$

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

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 64: $C_2^2.C_4^2$

Order 32: $C_2.C_4^2$ x 4, $C_2^3\times C_4$ x 3

Order 16: $C_2^2\times C_4$ x 30, $C_2^4$

Order 8: $C_2\times C_4$ x 48, $C_2^3$ x 15

Order 4: $C_2^2$ x 35, $C_4$ x 12

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 64: $C_2^2.C_4^2$

Order 32: $C_2^3\times C_4$, $C_2.C_4^2$

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

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

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

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 64: $C_2^2.C_4^2$ ($C_1$)

Order 32: $C_2.C_4^2$ ($C_2$) x 4, $C_2^3\times C_4$ ($C_2$) x 3

Order 16: $C_2^2\times C_4$ ($C_4$) x 12, $C_2^2\times C_4$ ($C_2^2$) x 6, $C_2^4$ ($C_2^2$)

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

Order 4: $C_2^2$ ($C_4:C_4$) x 12, $C_2^2$ ($C_2^2:C_4$) x 12, $C_2^2$ ($C_4^2$) x 4, $C_2^2$ ($C_2\times D_4$) x 3, $C_2^2$ ($C_2^2\times C_4$) x 3, $C_2^2$ ($C_2\times Q_8$)

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

Order 1: $C_1$ ($C_2^2.C_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 64: $C_2^2.C_4^2$ ($C_1$)

Order 32: $C_2^3\times C_4$ ($C_2$), $C_2.C_4^2$ ($C_2$)

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

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

Order 4: $C_2^2$ ($C_4:C_4$), $C_2^2$ ($C_2^2:C_4$), $C_2^2$ ($C_4^2$), $C_2^2$ ($C_2\times Q_8$), $C_2^2$ ($C_2\times D_4$), $C_2^2$ ($C_2^2\times C_4$)

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

Order 1: $C_1$ ($C_2^2.C_4^2$)

Series

Derived series

$C_2^2.C_4^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2^2.C_4^2$

$\rhd$

$C_2.C_4^2$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^2.C_4^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^4$

$\lhd$

$C_2^2.C_4^2$

Supergroups

This group is a maximal subgroup of 115 larger groups in the database.

This group is a maximal quotient of 60 larger groups in the database.

Character theory

Complex character table

See the $40 \times 40$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $28 \times 28$ rational character table.