Abstract group 128.180: $C_2^3.C_4^2$

• Label: 128.180
• Order: \( 2^{7} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\Aut(G)}$: \( 2^{13} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \)
• Perm deg.: $18$
• Trans deg.: $128$
• Rank: $3$

Group information

Description:	$C_2^3.C_4^2$
Order:	\(128\)\(\medspace = 2^{7} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^4.C_2^4\wr C_2$, of order \(8192\)\(\medspace = 2^{13} \)
Composition factors:	$C_2$ x 7
Nilpotency class:	$2$
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	24	96	128
Conjugacy classes	1	7	24	48	80
Divisions	1	7	12	16	36
Autjugacy classes	1	3	4	2	10

Dimension	1	2	4
Irr. complex chars.	64	16	0	80
Irr. rational chars.	8	12	16	36

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$128$
Rank:	$3$
Inequivalent generating triples:	$84$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	7	9

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{8}=c^{8}=[a,b]=[a,c]=1, c^{b}=c^{5} \rangle$
Permutation group:	Degree $18$ $\langle(1,2,4,6)(3,5,7,8)(9,10)(11,12,13,16,14,15,17,18), (1,3,2,5,4,7,6,8)(12,15) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 3 & 11 \\ 4 & 9 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 9 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 7 & 9 \\ 4 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
	$\left\langle \left(\begin{array}{rr} 7 & 6 \\ 12 & 7 \end{array}\right), \left(\begin{array}{rr} 17 & 8 \\ 8 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 21 \\ 6 & 7 \end{array}\right), \left(\begin{array}{rr} 9 & 2 \\ 20 & 5 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_8:C_8)$
Semidirect product:	$(C_2\times C_8)$ $\,\rtimes\,$ $C_8$ (8)	$C_8$ $\,\rtimes\,$ $(C_2\times C_8)$ (8)			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^2$ . $C_2^3$	$C_2^3$ . $C_4^2$	$(C_4\times C_8)$ . $C_4$ (6)	$C_4$ . $(C_4\times C_8)$ (2)	all 23

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

Homology

Abelianization:	$C_{2} \times C_{4} \times C_{8} $
Schur multiplier:	$C_{2}^{2} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 140 subgroups in 124 conjugacy classes, 108 normal (12 characteristic).

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

Special subgroups

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

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 128: $C_2^3.C_4^2$

Order 64: $C_8:C_8$ x 4, $C_2\times C_4\times C_8$ x 3

Order 32: $C_4\times C_8$ x 12, $C_2^2\times C_8$ x 6, $C_2\times C_4^2$

Order 16: $C_2\times C_8$ x 28, $C_4^2$ x 4, $C_2^2\times C_4$ x 3

Order 8: $C_2\times C_4$ x 18, $C_8$ x 16, $C_2^3$

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 128: $C_2^3.C_4^2$

Order 64: $C_2\times C_4\times C_8$ x 2, $C_8:C_8$

Order 32: $C_4\times C_8$ x 4, $C_2^2\times C_8$ x 2, $C_2\times C_4^2$

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

Order 8: $C_2\times C_4$ x 6, $C_8$ x 2, $C_2^3$

Order 4: $C_4$ x 4, $C_2^2$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 128: $C_2^3.C_4^2$ ($C_1$)

Order 64: $C_8:C_8$ ($C_2$) x 4, $C_2\times C_4\times C_8$ ($C_2$) x 3

Order 32: $C_2^2\times C_8$ ($C_4$) x 6, $C_4\times C_8$ ($C_2^2$) x 6, $C_4\times C_8$ ($C_4$) x 6, $C_2\times C_4^2$ ($C_2^2$)

Order 16: $C_2\times C_8$ ($C_2\times C_4$) x 12, $C_2\times C_8$ ($C_8$) x 8, $C_4^2$ ($C_2\times C_4$) x 3, $C_2^2\times C_4$ ($C_2\times C_4$) x 3, $C_4^2$ ($C_2^3$)

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

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

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

Order 1: $C_1$ ($C_2^3.C_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 128: $C_2^3.C_4^2$ ($C_1$)

Order 64: $C_2\times C_4\times C_8$ ($C_2$) x 2, $C_8:C_8$ ($C_2$)

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

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

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

Order 4: $C_2^2$ ($C_8:C_4$), $C_2^2$ ($C_4\times C_8$), $C_2^2$ ($C_2\times C_4^2$), $C_4$ ($C_8:C_4$), $C_4$ ($C_2\times \OD_{16}$), $C_4$ ($C_2^2\times C_8$), $C_4$ ($C_4\times C_8$)

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

Order 1: $C_1$ ($C_2^3.C_4^2$)

Series

Derived series

$C_2^3.C_4^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2^3.C_4^2$

$\rhd$

$C_8:C_8$

$\rhd$

$C_4\times C_8$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3.C_4^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4^2$

$\lhd$

$C_2^3.C_4^2$

Supergroups

This group is a maximal subgroup of 176 larger groups in the database.

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

Character theory

Complex character table

See the $80 \times 80$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $36 \times 36$ rational character table.