Abstract group 128.56: $C_4^2:C_8$

• Label: 128.56
• Order: \( 2^{7} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $24$
• Trans deg.: $32$
• Rank: $2$

Group information

Description:	$C_4^2:C_8$
Order:	\(128\)\(\medspace = 2^{7} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^7:D_4$, of order \(1024\)\(\medspace = 2^{10} \)
Composition factors:	$C_2$ x 7
Nilpotency class:	$4$
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	56	64	128
Conjugacy classes	1	5	12	8	26
Divisions	1	5	7	2	15
Autjugacy classes	1	5	5	1	12

Dimension	1	2	4	8
Irr. complex chars.	16	4	6	0	26
Irr. rational chars.	4	4	5	2	15

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$32$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	5	10	12

Constructions

Presentation:	$\langle a, b, c \mid a^{8}=b^{4}=c^{4}=[b,c]=1, b^{a}=b^{3}c^{2}, c^{a}=b^{3}c^{3} \rangle$
Permutation group:	Degree $24$ $\langle(1,2,4,8,5,9,12,16)(3,6,10,14,11,7,13,15)(17,18,20,22,19,21,23,24), (2,6,9,7) \!\cdots\! \rangle$
Transitive group:	32T1579				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_4^2$ $\,\rtimes\,$ $C_8$ (2)				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^2.D_4)$ . $C_4$	$(C_2^2\times C_4)$ . $D_4$ (2)	$C_2^2$ . $(C_4.D_4)$	$(C_4^2:C_4)$ . $C_2$	all 16

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 144 subgroups in 59 conjugacy classes, 22 normal (16 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

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 128: $C_4^2:C_8$

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

Order 32: $C_2^2:C_8$ x 2, $C_2^2.D_4$ x 2, $C_2.C_4^2$ x 2, $C_2\times C_4^2$

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

Order 8: $C_2\times C_4$ x 15, $C_8$ x 2, $C_2^3$

Order 4: $C_4$ x 7, $C_2^2$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 128: $C_4^2:C_8$

Order 64: $C_4^2:C_4$, $C_2^2.\OD_{16}$

Order 32: $C_2^2.D_4$ x 2, $C_2^2:C_8$, $C_2\times C_4^2$, $C_2.C_4^2$

Order 16: $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$ x 2, $C_2\times C_8$

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

Order 4: $C_2^2$ x 5, $C_4$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 128: $C_4^2:C_8$ ($C_1$)

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

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

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

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

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

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

Order 1: $C_1$ ($C_4^2:C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 128: $C_4^2:C_8$ ($C_1$)

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

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

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

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

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

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

Order 1: $C_1$ ($C_4^2:C_8$)

Series

Derived series

$C_4^2:C_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_1$

Chief series

$C_4^2:C_8$

$\rhd$

$C_4^2:C_4$

$\rhd$

$C_2^2.D_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4^2:C_8$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2.D_4$

$\lhd$

$C_4^2:C_8$

Supergroups

This group is a maximal subgroup of 56 larger groups in the database.

This group is a maximal quotient of 14 larger groups in the database.

Character theory

Complex character table

See the $26 \times 26$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	8A	8B
	Size	1	1	1	1	2	2	4	4	8	8	8	8	16	32	32
	2 P	1A	1A	1A	1A	1A	1A	2A	2A	2E	2E	2C	2C	2B	4E	4F
	Schur
128.56.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
128.56.1b		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$
128.56.1c		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$
128.56.1d		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
128.56.1e		$2$	$2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$-2$	$-2$	$-2$	$-2$	$2$	$0$	$0$
128.56.1f		$2$	$2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$-2$	$2$	$2$	$2$	$-2$	$0$	$0$
128.56.1g		$4$	$4$	$-4$	$-4$	$-4$	$4$	$0$	$0$	$0$	$-4$	$-4$	$4$	$0$	$0$	$0$
128.56.1h		$4$	$4$	$-4$	$-4$	$-4$	$4$	$0$	$0$	$0$	$4$	$4$	$-4$	$0$	$0$	$0$
128.56.2a		$2$	$2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
128.56.2b		$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$
128.56.2c		$4$	$4$	$-4$	$-4$	$-4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.56.4a		$4$	$4$	$4$	$4$	$-4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.56.4b	2	$4$	$4$	$-4$	$-4$	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.56.4c		$8$	$-8$	$8$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.56.4d		$8$	$-8$	$-8$	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$