Abstract group 256.52114: $C_4:D_4^2$

• Label: 256.52114
• Order: \( 2^{8} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{18} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{13} \)
• Perm deg.: $12$
• Trans deg.: $64$
• Rank: $5$

Group information

Description:	$C_4:D_4^2$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^9.C_2^6.C_2^3$, of order \(262144\)\(\medspace = 2^{18} \)
Composition factors:	$C_2$ x 8
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	111	144	256
Conjugacy classes	1	31	32	64
Divisions	1	31	32	64
Autjugacy classes	1	9	7	17

Dimension	1	2	4
Irr. complex chars.	32	24	8	64
Irr. rational chars.	32	24	8	64

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$64$
Rank:	$5$
Inequivalent generating 5-tuples:	$1249920$

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:	${\langle a, b, c, d, e \mid a^{2}=c^{4}=d^{4}=e^{4}=[a,d]=[b,c]=[b,e]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(1,2)(3,4)(6,8)(10,11), (3,4)(5,6,7,8)(10,11), (5,7)(6,8)(9,10,12,11), (1,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 30 \\ 30 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 36 \\ 12 & 43 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right), \left(\begin{array}{rr} 49 & 42 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 56 & 45 \\ 45 & 26 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 20 & 41 \end{array}\right), \left(\begin{array}{rr} 41 & 0 \\ 0 & 41 \end{array}\right), \left(\begin{array}{rr} 56 & 45 \\ 25 & 16 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/60\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_4$ $\,\rtimes\,$ $D_4^2$	$C_2^2$ $\,\rtimes\,$ $D_4^2$	$(C_4:D_4)$ $\,\rtimes\,$ $D_4$	$(C_4:D_4)$ $\,\rtimes\,$ $D_4$	all 35
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $C_2^3$ (2)	$C_2^4$ . $C_2^4$ (4)	$C_2^3$ . $C_2^5$	$C_2$ . $(D_4^2:C_2)$	all 22

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{60}\Z)$.

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$1$

Subgroups

There are 6831 subgroups in 2535 conjugacy classes, 559 normal (21 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2^5$
Commutator:	$G' \simeq$ $C_2^3$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^5$
Fitting:	$\operatorname{Fit} \simeq$ $C_4:D_4^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_4:D_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4:D_4^2$

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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 256: $C_4:D_4^2$

Order 128: $C_2\times D_4^2$ x 5, $C_2.D_4^2$ x 4, $C_2.D_4^2$ x 4, $C_2.D_4^2$ x 4, $C_2.D_4^2$ x 4, $C_2^4:D_4$ x 4, $C_4^2:D_4$ x 2, $C_4^2:D_4$ x 2, $C_4^2:C_2^3$, $C_2.D_4^2$

Order 64: $D_4^2$ x 40, $C_2^3:D_4$ x 24, $C_2^3:D_4$ x 24, $C_2^3:D_4$ x 22, $C_4^2:C_2^2$ x 20, $D_4\times C_2^3$ x 14, $C_2^3.D_4$ x 12, $C_2^3.C_2^3$ x 10, $C_4^2:C_2^2$ x 10, $C_2^3.D_4$ x 8, $C_2^3.C_2^3$ x 4, $C_2^4:C_4$ x 4, $C_2^2:C_4^2$ x 4, $C_2^3.C_2^3$ x 2, $C_2^3.D_4$ x 2, $C_4:C_4^2$ x 2, $C_2^2\times C_4^2$

Order 32: $C_2^2\times D_4$ x 190, $C_4:D_4$ x 88, $C_2^2\wr C_2$ x 88, $C_4:D_4$ x 52, $C_2^3:C_4$ x 52, $C_4\times D_4$ x 40, $C_2\times C_4^2$ x 12, $C_2.C_4^2$ x 12, $C_2^3\times C_4$ x 11, $C_2^2.D_4$ x 10, $C_2^5$ x 8

Order 16: $C_2\times D_4$ x 520, $C_2^2:C_4$ x 100, $C_2^4$ x 97, $C_2^2\times C_4$ x 84, $C_4:C_4$ x 24, $C_4^2$ x 18

Order 8: $C_2^3$ x 271, $D_4$ x 240, $C_2\times C_4$ x 124

Order 4: $C_2^2$ x 195, $C_4$ x 32

Order 2: $C_2$ x 31

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $C_4:D_4^2$

Order 128: $C_2\times D_4^2$ x 2, $C_4^2:C_2^3$, $C_2.D_4^2$, $C_4^2:D_4$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2^4:D_4$, $C_2.D_4^2$, $C_4^2:D_4$

Order 64: $C_4^2:C_2^2$ x 6, $C_2^3:D_4$ x 5, $D_4\times C_2^3$ x 4, $C_2^3:D_4$ x 4, $C_2^3:D_4$ x 3, $C_2^3.C_2^3$ x 3, $C_2^3.D_4$ x 3, $C_4^2:C_2^2$ x 3, $D_4^2$ x 2, $C_2^3.D_4$, $C_2^3.C_2^3$, $C_2^3.C_2^3$, $C_2^3.D_4$, $C_2^4:C_4$, $C_4:C_4^2$, $C_2^2:C_4^2$, $C_2^2\times C_4^2$

Order 32: $C_2^2\times D_4$ x 22, $C_2^3:C_4$ x 10, $C_4:D_4$ x 5, $C_4:D_4$ x 5, $C_2\times C_4^2$ x 5, $C_2^3\times C_4$ x 4, $C_2^2\wr C_2$ x 4, $C_2^2.D_4$ x 4, $C_4\times D_4$ x 3, $C_2.C_4^2$ x 3, $C_2^5$ x 2

Order 16: $C_2\times D_4$ x 37, $C_2^2\times C_4$ x 18, $C_2^4$ x 14, $C_2^2:C_4$ x 10, $C_4:C_4$ x 4, $C_4^2$ x 4

Order 8: $C_2^3$ x 33, $C_2\times C_4$ x 25, $D_4$ x 11

Order 4: $C_2^2$ x 29, $C_4$ x 7

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $C_4:D_4^2$ ($C_1$)

Order 128: $C_2\times D_4^2$ ($C_2$) x 5, $C_2.D_4^2$ ($C_2$) x 4, $C_2.D_4^2$ ($C_2$) x 4, $C_2.D_4^2$ ($C_2$) x 4, $C_2.D_4^2$ ($C_2$) x 4, $C_2^4:D_4$ ($C_2$) x 4, $C_4^2:D_4$ ($C_2$) x 2, $C_4^2:D_4$ ($C_2$) x 2, $C_4^2:C_2^3$ ($C_2$), $C_2.D_4^2$ ($C_2$)

Order 64: $C_2^3:D_4$ ($C_2^2$) x 24, $C_2^3:D_4$ ($C_2^2$) x 24, $C_2^3:D_4$ ($C_2^2$) x 22, $D_4\times C_2^3$ ($C_2^2$) x 14, $C_2^3.D_4$ ($C_2^2$) x 12, $C_4^2:C_2^2$ ($C_2^2$) x 12, $C_2^3.C_2^3$ ($C_2^2$) x 10, $C_4^2:C_2^2$ ($C_2^2$) x 10, $C_2^3.D_4$ ($C_2^2$) x 8, $C_2^3.C_2^3$ ($C_2^2$) x 4, $C_2^4:C_4$ ($C_2^2$) x 4, $C_2^2:C_4^2$ ($C_2^2$) x 4, $C_2^3.C_2^3$ ($C_2^2$) x 2, $C_2^3.D_4$ ($C_2^2$) x 2, $C_4:C_4^2$ ($C_2^2$) x 2, $C_2^2\times C_4^2$ ($C_2^2$)

Order 32: $C_2^2\times D_4$ ($C_2^3$) x 54, $C_2^3:C_4$ ($C_2^3$) x 52, $C_4:D_4$ ($D_4$) x 16, $C_2.C_4^2$ ($C_2^3$) x 12, $C_2^3\times C_4$ ($C_2^3$) x 11, $C_2^2.D_4$ ($C_2^3$) x 10, $C_2^5$ ($C_2^3$) x 8, $C_4:D_4$ ($D_4$) x 8, $C_2\times C_4^2$ ($C_2^3$) x 8

Order 16: $C_2\times D_4$ ($C_2\times D_4$) x 48, $C_2^2\times C_4$ ($C_2^4$) x 18, $C_2^2:C_4$ ($C_2\times D_4$) x 16, $C_2^4$ ($C_2^4$) x 13, $C_4:C_4$ ($C_2\times D_4$) x 8, $C_2^2\times C_4$ ($C_2\times D_4$) x 8, $C_4^2$ ($C_2\times D_4$) x 4

Order 8: $C_2\times C_4$ ($C_2^2\times D_4$) x 22, $C_2^3$ ($C_2^2\times D_4$) x 20, $C_2\times C_4$ ($D_4:C_2^2$) x 6, $C_2^3$ ($D_4:C_2^2$) x 2, $C_2^3$ ($C_2^5$)

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

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

Order 1: $C_1$ ($C_4:D_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $C_4:D_4^2$ ($C_1$)

Order 128: $C_2\times D_4^2$ ($C_2$) x 2, $C_4^2:C_2^3$ ($C_2$), $C_2.D_4^2$ ($C_2$), $C_4^2:D_4$ ($C_2$), $C_2.D_4^2$ ($C_2$), $C_2.D_4^2$ ($C_2$), $C_2.D_4^2$ ($C_2$), $C_2^4:D_4$ ($C_2$), $C_2.D_4^2$ ($C_2$), $C_4^2:D_4$ ($C_2$)

Order 64: $C_4^2:C_2^2$ ($C_2^2$) x 5, $C_2^3:D_4$ ($C_2^2$) x 5, $D_4\times C_2^3$ ($C_2^2$) x 4, $C_2^3:D_4$ ($C_2^2$) x 4, $C_2^3:D_4$ ($C_2^2$) x 3, $C_2^3.C_2^3$ ($C_2^2$) x 3, $C_2^3.D_4$ ($C_2^2$) x 3, $C_4^2:C_2^2$ ($C_2^2$) x 3, $C_2^3.D_4$ ($C_2^2$), $C_2^3.C_2^3$ ($C_2^2$), $C_2^3.C_2^3$ ($C_2^2$), $C_2^3.D_4$ ($C_2^2$), $C_2^4:C_4$ ($C_2^2$), $C_4:C_4^2$ ($C_2^2$), $C_2^2:C_4^2$ ($C_2^2$), $C_2^2\times C_4^2$ ($C_2^2$)

Order 32: $C_2^2\times D_4$ ($C_2^3$) x 11, $C_2^3:C_4$ ($C_2^3$) x 10, $C_2^3\times C_4$ ($C_2^3$) x 4, $C_2^2.D_4$ ($C_2^3$) x 4, $C_2\times C_4^2$ ($C_2^3$) x 4, $C_2.C_4^2$ ($C_2^3$) x 3, $C_2^5$ ($C_2^3$) x 2, $C_4:D_4$ ($D_4$), $C_4:D_4$ ($D_4$)

Order 16: $C_2^2\times C_4$ ($C_2^4$) x 7, $C_2^4$ ($C_2^4$) x 4, $C_2\times D_4$ ($C_2\times D_4$) x 4, $C_4:C_4$ ($C_2\times D_4$), $C_2^2:C_4$ ($C_2\times D_4$), $C_4^2$ ($C_2\times D_4$), $C_2^2\times C_4$ ($C_2\times D_4$)

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

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

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

Order 1: $C_1$ ($C_4:D_4^2$)

Series

Derived series

$C_4:D_4^2$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_4:D_4^2$

$\rhd$

$C_4^2:D_4$

$\rhd$

$C_4:C_4^2$

$\rhd$

$C_2^2.D_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4:D_4^2$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_4:D_4^2$

Supergroups

This group is a maximal subgroup of 19 larger groups in the database.

This group is a maximal quotient of 15 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 $64 \times 64$ rational character table. Alternatively, you may search for characters of this group with desired properties.