Abstract group 64.216: $C_4^2:C_2^2$

• Label: 64.216
• Order: \( 2^{6} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{11} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $12$
• Trans deg.: $16$
• Rank: $4$

Group information

Description:	$C_4^2:C_2^2$
Order:	\(64\)\(\medspace = 2^{6} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^5.D_4^2$, of order \(2048\)\(\medspace = 2^{11} \)
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	31	32	64
Conjugacy classes	1	11	10	22
Divisions	1	11	10	22
Autjugacy classes	1	5	4	10

Dimension	1	2	4
Irr. complex chars.	16	4	2	22
Irr. rational chars.	16	4	2	22

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$16$
Rank:	$4$
Inequivalent generating quadruples:	$2520$

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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{4}=d^{4}=[a,b]=[a,d]=[c,d]=1, c^{a}=cd^{2}, c^{b}=c^{3}d^{2}, d^{b}=d^{3} \rangle$
Permutation group:	Degree $12$ $\langle(1,2)(3,5)(4,6)(7,8)(10,12), (1,3,5,2)(4,6,8,7), (1,4)(2,6)(3,7)(5,8)(9,10)(11,12), (1,4)(2,7)(3,6)(5,8), (1,5)(2,3)(4,8)(6,7), (9,11)(10,12)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrrrr} 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{5}(\F_{2})$
	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 8 & 15 \end{array}\right), \left(\begin{array}{rr} 9 & 8 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 7 & 2 \\ 3 & 9 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 4 & 13 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 3 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
Transitive group:	16T73	32T62	32T63	32T269	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_4^2$ $\,\rtimes\,$ $C_2^2$	$(C_4:D_4)$ $\,\rtimes\,$ $C_2$	$(C_4:D_4)$ $\,\rtimes\,$ $C_2$	$(C_2\times C_4)$ $\,\rtimes\,$ $D_4$	all 14
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $C_2^2$	$C_2^3$ . $C_2^3$ (3)	$C_2^2$ . $C_2^4$	$C_4$ . $(C_2\times D_4)$ (2)	all 11

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

Homology

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

Subgroups

There are 305 subgroups in 167 conjugacy classes, 81 normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^4$
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:	$\operatorname{Fit} \simeq$ $C_4^2:C_2^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_4^2:C_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^2$

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

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

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

Order 32: $C_4:D_4$ x 4, $C_2^2\wr C_2$ x 4, $C_4:D_4$ x 2, $C_4^2:C_2$ x 2, $D_4:C_2^2$, $C_2^2\times D_4$, $C_4^2:C_2$

Order 16: $C_2\times D_4$ x 19, $C_2^2:C_4$ x 10, $D_4:C_2$ x 4, $C_2^2\times C_4$ x 3, $C_4:C_4$ x 2, $C_4^2$ x 2, $C_2^4$ x 2, $C_2\times Q_8$

Order 8: $D_4$ x 22, $C_2\times C_4$ x 16, $C_2^3$ x 15, $Q_8$ x 2

Order 4: $C_2^2$ x 31, $C_4$ x 10

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

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

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

Order 16: $C_2\times D_4$ x 6, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2^4$, $D_4:C_2$, $C_2\times Q_8$

Order 8: $C_2\times C_4$ x 6, $C_2^3$ x 5, $D_4$ x 5, $Q_8$

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

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

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

Order 16: $C_2\times D_4$ ($C_2^2$) x 15, $C_2^2:C_4$ ($C_2^2$) x 10, $C_2^2\times C_4$ ($C_2^2$) x 3, $C_4:C_4$ ($C_2^2$) x 2, $C_4^2$ ($C_2^2$) x 2, $C_2^4$ ($C_2^2$) x 2, $C_2\times Q_8$ ($C_2^2$)

Order 8: $C_2\times C_4$ ($C_2^3$) x 8, $C_2^3$ ($C_2^3$) x 7, $C_2\times C_4$ ($D_4$) x 4

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

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

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

Normal subgroups up to automorphism (quotient in parentheses)

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

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

Order 16: $C_2\times D_4$ ($C_2^2$) x 5, $C_2^2:C_4$ ($C_2^2$) x 2, $C_2^2\times C_4$ ($C_2^2$) x 2, $C_4:C_4$ ($C_2^2$), $C_4^2$ ($C_2^2$), $C_2^4$ ($C_2^2$), $C_2\times Q_8$ ($C_2^2$)

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

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

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

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

Series

Derived series

$C_4^2:C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_4^2:C_2^2$

$\rhd$

$C_4^2:C_2$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4^2:C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_4^2:C_2^2$

Supergroups

This group is a maximal subgroup of 139 larger groups in the database.

This group is a maximal quotient of 187 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 $22 \times 22$ rational character table. Alternatively, you may search for characters of this group with desired properties.