Abstract group 256.13754: $C_2\times C_4^2.D_4$

• Label: 256.13754
• Order: \( 2^{8} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{18} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{14} \)
• Perm deg.: $26$
• Trans deg.: $128$
• Rank: $4$

Group information

Description:	$C_2\times C_4^2.D_4$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^7.C_2^6.C_2^5$, of order \(262144\)\(\medspace = 2^{18} \)
Composition factors:	$C_2$ x 8
Nilpotency class:	$3$
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	48	192	256
Conjugacy classes	1	11	28	48	88
Divisions	1	11	20	16	48
Autjugacy classes	1	5	6	2	14

Dimension	1	2	4	8
Irr. complex chars.	32	56	0	0	88
Irr. rational chars.	16	16	8	8	48

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$128$
Rank:	$4$
Inequivalent generating quadruples:	$5040$

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:	$\langle a, b, c, d \mid a^{2}=c^{4}=d^{8}=[a,b]=[a,c]=[a,d]=[c,d]=1, b^{4}=d^{4}, c^{b}=c^{3}d^{4}, d^{b}=c^{2}d^{3} \rangle$
Permutation group:	Degree $26$ $\langle(1,2,4,5)(3,6,7,8)(9,10)(11,12,16,19,14,18,15,24)(13,20,23,25,21,26,22,17) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 6 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 3 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 8 \\ 8 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
	$\left\langle \left(\begin{array}{rr} 35 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 18 \\ 24 & 25 \end{array}\right), \left(\begin{array}{rr} 13 & 24 \\ 24 & 13 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 33 \\ 24 & 23 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/48\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_4^2.D_4)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^4.Q_8)$ . $C_2$	$(C_2^3\times C_4)$ . $Q_8$	$(C_2\times C_4^2)$ . $Q_8$	$C_2^4$ . $(C_2\times Q_8)$	all 52

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

Homology

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

Subgroups

There are 671 subgroups in 435 conjugacy classes, 247 normal (23 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_4$	$G/Z \simeq$ $C_2\times D_4$
Commutator:	$G' \simeq$ $C_2\times C_4$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2\times C_4$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_4^2.D_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times C_4^2.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^2.D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4^2.D_4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

No diagram available: subgroups only stored up to automorphism

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 256: $C_2\times C_4^2.D_4$

Order 128: $C_4^2.D_4$ x 8, $C_2^4.Q_8$ x 4, $C_4^2.C_2^3$ x 2, $C_2^2\times C_4\times C_8$

Order 64: $C_4.C_4^2$ x 16, $C_2\times C_4\times C_8$ x 12, $C_4:\OD_{16}$ x 12, $C_2^2\times \OD_{16}$ x 4, $C_4^2.C_4$ x 4, $C_2^3\times C_8$ x 2, $C_2^2\times C_4^2$

Order 32: $C_2\times \OD_{16}$ x 28, $C_2^2\times C_8$ x 24, $C_4\times C_8$ x 16, $C_4:C_8$ x 16, $C_2\times C_4^2$ x 12, $C_2^3\times C_4$ x 3

Order 16: $C_2\times C_8$ x 48, $C_2^2\times C_4$ x 34, $\OD_{16}$ x 24, $C_4^2$ x 16, $C_2^4$

Order 8: $C_2\times C_4$ x 64, $C_8$ x 16, $C_2^3$ x 11

Order 4: $C_2^2$ x 23, $C_4$ x 20

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $C_2\times C_4^2.D_4$

Order 128: $C_4^2.D_4$, $C_2^4.Q_8$, $C_4^2.C_2^3$, $C_2^2\times C_4\times C_8$

Order 64: $C_2\times C_4\times C_8$ x 4, $C_4:\OD_{16}$ x 2, $C_2^2\times \OD_{16}$, $C_2^3\times C_8$, $C_4.C_4^2$, $C_2^2\times C_4^2$, $C_4^2.C_4$

Order 32: $C_2\times C_4^2$ x 7, $C_2^2\times C_8$ x 5, $C_2\times \OD_{16}$ x 3, $C_2^3\times C_4$ x 2, $C_4\times C_8$ x 2, $C_4:C_8$

Order 16: $C_2^2\times C_4$ x 12, $C_2\times C_8$ x 6, $C_4^2$ x 6, $\OD_{16}$ x 2, $C_2^4$

Order 8: $C_2\times C_4$ x 16, $C_2^3$ x 5, $C_8$ x 2

Order 4: $C_2^2$ x 8, $C_4$ x 6

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $C_2\times C_4^2.D_4$ ($C_1$)

Order 128: $C_4^2.D_4$ ($C_2$) x 8, $C_2^4.Q_8$ ($C_2$) x 4, $C_4^2.C_2^3$ ($C_2$) x 2, $C_2^2\times C_4\times C_8$ ($C_2$)

Order 64: $C_4.C_4^2$ ($C_2^2$) x 16, $C_2\times C_4\times C_8$ ($C_4$) x 8, $C_4:\OD_{16}$ ($C_2^2$) x 8, $C_2\times C_4\times C_8$ ($C_2^2$) x 4, $C_2^2\times \OD_{16}$ ($C_2^2$) x 4, $C_2^3\times C_8$ ($C_2^2$) x 2, $C_2^2\times C_4^2$ ($C_2^2$)

Order 32: $C_4\times C_8$ ($C_2\times C_4$) x 16, $C_2\times \OD_{16}$ ($C_2^3$) x 8, $C_2^2\times C_8$ ($C_2\times C_4$) x 8, $C_2^2\times C_8$ ($C_2^3$) x 4, $C_2\times C_4^2$ ($D_4$) x 4, $C_2\times C_4^2$ ($C_2\times C_4$) x 4, $C_2^3\times C_4$ ($Q_8$) x 2, $C_2\times C_4^2$ ($C_2^3$) x 2, $C_2\times C_4^2$ ($Q_8$) x 2, $C_2^3\times C_4$ ($C_2^3$)

Order 16: $C_2\times C_8$ ($C_2^2\times C_4$) x 8, $C_4^2$ ($C_4:C_4$) x 8, $C_2^2\times C_4$ ($C_4:C_4$) x 8, $C_2^2\times C_4$ ($D_4:C_2$) x 8, $C_2^2\times C_4$ ($C_2\times Q_8$) x 5, $C_4^2$ ($C_2\times D_4$) x 4, $C_4^2$ ($C_2^2\times C_4$) x 4, $C_2^2\times C_4$ ($C_2\times D_4$) x 2, $C_2^2\times C_4$ ($C_2^2\times C_4$) x 2, $C_2^4$ ($C_2\times Q_8$), $C_2^2\times C_4$ ($C_2^4$)

Order 8: $C_2\times C_4$ ($C_2^2.D_4$) x 10, $C_2\times C_4$ ($C_4^2:C_2$) x 8, $C_2\times C_4$ ($C_4^2:C_2$) x 8, $C_2\times C_4$ ($C_8.C_4$) x 8, $C_2^3$ ($C_4.Q_8$) x 4, $C_2\times C_4$ ($D_4:C_2^2$) x 4, $C_2\times C_4$ ($C_4.Q_8$) x 4, $C_2^3$ ($C_2^2.D_4$) x 2, $C_2^3$ ($C_2^2\times Q_8$), $C_2\times C_4$ ($C_2^2\times D_4$), $C_2\times C_4$ ($C_2^3\times C_4$)

Order 4: $C_4$ ($C_2^3.Q_8$) x 8, $C_2^2$ ($C_4^2:C_4$) x 4, $C_2^2$ ($C_2^3.Q_8$) x 4, $C_4$ ($C_4^2:C_4$) x 4, $C_2^2$ ($C_4^2.C_2^2$) x 2, $C_4$ ($C_4^2:C_2^2$) x 2, $C_4$ ($C_4^2:C_2^2$) x 2, $C_2^2$ ($C_2^3.D_4$)

Order 2: $C_2$ ($C_4^2.D_4$) x 4, $C_2$ ($C_2^4.Q_8$) x 2, $C_2$ ($C_2^3.C_2^4$)

Order 1: $C_1$ ($C_2\times C_4^2.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $C_2\times C_4^2.D_4$ ($C_1$)

Order 128: $C_4^2.D_4$ ($C_2$), $C_2^4.Q_8$ ($C_2$), $C_4^2.C_2^3$ ($C_2$), $C_2^2\times C_4\times C_8$ ($C_2$)

Order 64: $C_2\times C_4\times C_8$ ($C_4$) x 3, $C_2\times C_4\times C_8$ ($C_2^2$), $C_2^2\times \OD_{16}$ ($C_2^2$), $C_2^3\times C_8$ ($C_2^2$), $C_4.C_4^2$ ($C_2^2$), $C_2^2\times C_4^2$ ($C_2^2$), $C_4:\OD_{16}$ ($C_2^2$)

Order 32: $C_2\times C_4^2$ ($C_2\times C_4$) x 3, $C_2^2\times C_8$ ($C_2\times C_4$) x 2, $C_4\times C_8$ ($C_2\times C_4$) x 2, $C_2\times C_4^2$ ($D_4$) x 2, $C_2^3\times C_4$ ($C_2^3$), $C_2^3\times C_4$ ($Q_8$), $C_2\times \OD_{16}$ ($C_2^3$), $C_2^2\times C_8$ ($C_2^3$), $C_2\times C_4^2$ ($C_2^3$), $C_2\times C_4^2$ ($Q_8$)

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

Order 8: $C_2\times C_4$ ($C_2^2.D_4$) x 4, $C_2^3$ ($C_2^2.D_4$) x 2, $C_2\times C_4$ ($C_4^2:C_2$) x 2, $C_2\times C_4$ ($C_4^2:C_2$) x 2, $C_2^3$ ($C_2^2\times Q_8$), $C_2^3$ ($C_4.Q_8$), $C_2\times C_4$ ($D_4:C_2^2$), $C_2\times C_4$ ($C_2^2\times D_4$), $C_2\times C_4$ ($C_2^3\times C_4$), $C_2\times C_4$ ($C_4.Q_8$), $C_2\times C_4$ ($C_8.C_4$)

Order 4: $C_2^2$ ($C_4^2:C_4$) x 2, $C_4$ ($C_4^2:C_4$) x 2, $C_2^2$ ($C_4^2.C_2^2$), $C_2^2$ ($C_2^3.D_4$), $C_2^2$ ($C_2^3.Q_8$), $C_4$ ($C_4^2:C_2^2$), $C_4$ ($C_4^2:C_2^2$), $C_4$ ($C_2^3.Q_8$)

Order 2: $C_2$ ($C_4^2.D_4$), $C_2$ ($C_2^4.Q_8$), $C_2$ ($C_2^3.C_2^4$)

Order 1: $C_1$ ($C_2\times C_4^2.D_4$)

Series

Derived series

$C_2\times C_4^2.D_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_1$

Chief series

$C_2\times C_4^2.D_4$

$\rhd$

$C_4^2.D_4$

$\rhd$

$C_2\times C_4\times C_8$

$\rhd$

$C_2^2\times C_8$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2\times C_4^2.D_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_2^2\times C_4^2$

$\lhd$

$C_2\times C_4^2.D_4$

Supergroups

This group is a maximal subgroup of 20 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $88 \times 88$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $48 \times 48$ rational character table.