Abstract group 256.2715: $(C_4\times C_8).D_4$

• Label: 256.2715
• Order: \( 2^{8} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{13} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $24$
• Trans deg.: $128$
• Rank: $3$

Group information

Description:	$(C_4\times C_8).D_4$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2.C_2^6.C_2^6$, of order \(8192\)\(\medspace = 2^{13} \)
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	7	88	160	256
Conjugacy classes	1	4	17	24	46
Divisions	1	4	13	13	31
Autjugacy classes	1	4	9	7	21

Dimension	1	2	4	8
Irr. complex chars.	16	20	10	0	46
Irr. rational chars.	8	12	6	5	31

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$128$
Rank:	$3$
Inequivalent generating triples:	$672$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	12	12

Constructions

Presentation:	$\langle a, b, c, d \mid c^{4}=d^{8}=[a,c]=[b,c]=[c,d]=1, a^{2}=c^{3}d^{2}, b^{4}=d^{4}, b^{a}=b^{3}c^{3}d^{2}, d^{a}=c^{2}d, d^{b}=d^{7} \rangle$
Permutation group:	Degree $24$ $\langle(1,2,4,6,5,9,13,8)(3,10,12,7,11,14,16,15)(18,21,22,24), (2,6,9,8)(3,11) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_8:Q_8)$ . $C_4$	$(C_4:C_8)$ . $D_4$ (2)	$(C_4\times C_8)$ . $D_4$ (2)	$(C_8:C_4)$ . $D_4$ (2)	all 38

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 431 subgroups in 196 conjugacy classes, 65 normal (41 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_4:D_4$
Commutator:	$G' \simeq$ $C_4^2$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4^2$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $(C_4\times C_8).D_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $(C_4\times C_8).D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^3.C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_4\times C_8).D_4$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 256: $(C_4\times C_8).D_4$

Order 128: $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_8.C_4^2$, $C_4^2.D_4$, $C_8:\OD_{16}$

Order 64: $Q_8:C_8$ x 4, $C_4^2.C_2^2$ x 4, $C_4\times \OD_{16}$ x 3, $C_8:C_8$ x 2, $C_4^2.C_2^2$ x 2, $C_4.Q_{16}$ x 2, $C_2^2:Q_{16}$ x 2, $C_4\times Q_{16}$ x 2, $C_2\times C_4\times C_8$, $C_4.C_4^2$, $C_8:Q_8$, $C_4:Q_{16}$, $C_4^2.C_4$, $C_2^3.Q_8$

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

Order 16: $C_2\times C_8$ x 13, $C_4:C_4$ x 12, $\OD_{16}$ x 8, $C_4^2$ x 6, $Q_{16}$ x 4, $C_2\times Q_8$ x 4, $C_2^2\times C_4$ x 3, $C_2^2:C_4$ x 2

Order 8: $C_2\times C_4$ x 16, $C_8$ x 13, $Q_8$ x 8, $C_2^3$

Order 4: $C_4$ x 13, $C_2^2$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $(C_4\times C_8).D_4$

Order 128: $C_8.C_4^2$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_8:\OD_{16}$

Order 64: $C_4\times \OD_{16}$ x 2, $C_8:C_8$ x 2, $C_2^2:Q_{16}$ x 2, $C_4\times Q_{16}$ x 2, $C_2\times C_4\times C_8$, $Q_8:C_8$, $C_4.C_4^2$, $C_4^2.C_2^2$, $C_8:Q_8$, $C_4:Q_{16}$, $C_4.Q_{16}$, $C_4^2.C_4$, $C_2^3.Q_8$, $C_4^2.C_2^2$

Order 32: $C_4\times C_8$ x 7, $C_2\times Q_{16}$ x 2, $C_8:C_4$ x 2, $C_2\times \OD_{16}$ x 2, $C_2^2\times C_8$ x 2, $C_4:Q_8$ x 2, $C_2^2:Q_8$ x 2, $C_4\times Q_8$ x 2, $C_8:C_4$ x 2, $Q_8:C_4$ x 2, $C_2^2:C_8$, $C_4.Q_8$, $C_4^2:C_2$, $C_2\times C_4^2$, $C_8.C_4$, $C_4:C_8$

Order 16: $C_2\times C_8$ x 9, $C_4:C_4$ x 6, $C_4^2$ x 5, $\OD_{16}$ x 3, $C_2^2\times C_4$ x 3, $Q_{16}$ x 2, $C_2\times Q_8$ x 2, $C_2^2:C_4$

Order 8: $C_2\times C_4$ x 12, $C_8$ x 7, $Q_8$ x 3, $C_2^3$

Order 4: $C_4$ x 9, $C_2^2$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $(C_4\times C_8).D_4$ ($C_1$)

Order 128: $C_4^2.D_4$ ($C_2$) x 2, $C_4^2.D_4$ ($C_2$) x 2, $C_8.C_4^2$ ($C_2$), $C_4^2.D_4$ ($C_2$), $C_8:\OD_{16}$ ($C_2$)

Order 64: $C_4\times \OD_{16}$ ($C_2^2$) x 3, $C_4^2.C_2^2$ ($C_2^2$) x 2, $C_4.Q_{16}$ ($C_4$) x 2, $C_2\times C_4\times C_8$ ($C_2^2$), $C_8:Q_8$ ($C_4$), $C_4:Q_{16}$ ($C_4$), $C_4^2.C_4$ ($C_2^2$)

Order 32: $C_4:Q_8$ ($C_2\times C_4$) x 4, $C_8:C_4$ ($D_4$) x 2, $C_2^2\times C_8$ ($D_4$) x 2, $C_4\times C_8$ ($D_4$) x 2, $C_4\times C_8$ ($C_2\times C_4$) x 2, $C_4:C_8$ ($D_4$) x 2, $C_2\times C_4^2$ ($C_2^3$)

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

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

Order 4: $C_4$ ($C_8.D_4$) x 2, $C_4$ ($D_8:C_4$) x 2, $C_2^2$ ($C_2^3.C_2^3$), $C_4$ ($C_4^2:C_2^2$), $C_4$ ($C_4^2:C_2^2$)

Order 2: $C_2$ ($\OD_{16}:D_4$), $C_2$ ($\OD_{16}.D_4$), $C_2$ ($C_4^2.D_4$)

Order 1: $C_1$ ($(C_4\times C_8).D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $(C_4\times C_8).D_4$ ($C_1$)

Order 128: $C_8.C_4^2$ ($C_2$), $C_4^2.D_4$ ($C_2$), $C_4^2.D_4$ ($C_2$), $C_4^2.D_4$ ($C_2$), $C_8:\OD_{16}$ ($C_2$)

Order 64: $C_4\times \OD_{16}$ ($C_2^2$) x 2, $C_2\times C_4\times C_8$ ($C_2^2$), $C_4^2.C_2^2$ ($C_2^2$), $C_8:Q_8$ ($C_4$), $C_4:Q_{16}$ ($C_4$), $C_4.Q_{16}$ ($C_4$), $C_4^2.C_4$ ($C_2^2$)

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

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

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

Order 4: $C_2^2$ ($C_2^3.C_2^3$), $C_4$ ($C_8.D_4$), $C_4$ ($D_8:C_4$), $C_4$ ($C_4^2:C_2^2$), $C_4$ ($C_4^2:C_2^2$)

Order 2: $C_2$ ($\OD_{16}:D_4$), $C_2$ ($\OD_{16}.D_4$), $C_2$ ($C_4^2.D_4$)

Order 1: $C_1$ ($(C_4\times C_8).D_4$)

Series

Derived series

$(C_4\times C_8).D_4$

$\rhd$

$C_4^2$

$\rhd$

$C_1$

Chief series

$(C_4\times C_8).D_4$

$\rhd$

$C_8.C_4^2$

$\rhd$

$C_2\times C_4\times C_8$

$\rhd$

$C_2\times C_4^2$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_4\times C_8).D_4$

$\rhd$

$C_4^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times C_4^2$

$\lhd$

$(C_4\times C_8).D_4$

Character theory

Complex character table

See the $46 \times 46$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $31 \times 31$ rational character table.