Abstract group 128.159: $C_2\times C_{64}$

• Label: 128.159
• Order: \( 2^{7} \)
• Exponent: \( 2^{6} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{7} \)
• Perm deg.: $66$
• Trans deg.: $128$
• Rank: $2$

Group information

Description:	$C_2\times C_{64}$
Order:	\(128\)\(\medspace = 2^{7} \)
Exponent:	\(64\)\(\medspace = 2^{6} \)
Automorphism group:	$\OD_{32}:C_2^2$, of order \(128\)\(\medspace = 2^{7} \)
Composition factors:	$C_2$ x 7
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.

Group statistics

Order	1	2	4	8	16	32	64
Elements	1	3	4	8	16	32	64	128
Conjugacy classes	1	3	4	8	16	32	64	128
Divisions	1	3	2	2	2	2	2	14
Autjugacy classes	1	2	2	2	2	2	1	12

Dimension	1	2	4	8	16	32
Irr. complex chars.	128	0	0	0	0	0	128
Irr. rational chars.	4	2	2	2	2	2	14

Minimal presentations

Permutation degree:	$66$
Transitive degree:	$128$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	2	3	33

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{2}=b^{64}=1 \rangle$
Permutation group:	Degree $66$ $\langle(3,66,34,50,18,58,26,42,10,62,30,46,14,54,22,38,6,64,32,48,16,56,24,40,8,60,28,44,12,52,20,36,4,65,33,49,17,57,25,41,9,61,29,45,13,53,21,37,5,63,31,47,15,55,23,39,7,59,27,43,11,51,19,35) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 125 & 0 \\ 0 & 158 \end{array}\right), \left(\begin{array}{rr} 43 & 0 \\ 0 & 9 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_2$ $\, \times\, $ $C_{64}$
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:	$C_{32}$ . $C_4$	$C_{16}$ . $C_8$	$C_8$ . $C_{16}$	$C_4$ . $C_{32}$	all 14
Aut. group:	$\Aut(C_{256})$

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

Homology

Primary decomposition:	$C_{2} \times C_{64}$
Schur multiplier:	$C_{2}$
Commutator length:	$0$

Subgroups

There are 20 subgroups, all normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{64}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2\times C_{64}$
Frattini:	$\Phi \simeq$ $C_{32}$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{64}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times C_{64}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_{32}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_{64}$

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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Classes of subgroups up to conjugation

Order 128: $C_2\times C_{64}$

Order 64: $C_{64}$ x 2, $C_2\times C_{32}$

Order 32: $C_{32}$ x 2, $C_2\times C_{16}$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 128: $C_2\times C_{64}$

Order 64: $C_2\times C_{32}$, $C_{64}$

Order 32: $C_{32}$ x 2, $C_2\times C_{16}$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

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

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 128: $C_2\times C_{64}$ ($C_1$)

Order 64: $C_{64}$ ($C_2$) x 2, $C_2\times C_{32}$ ($C_2$)

Order 32: $C_2\times C_{16}$ ($C_4$), $C_{32}$ ($C_2^2$), $C_{32}$ ($C_4$)

Order 16: $C_2\times C_8$ ($C_8$), $C_{16}$ ($C_2\times C_4$), $C_{16}$ ($C_8$)

Order 8: $C_2\times C_4$ ($C_{16}$), $C_8$ ($C_2\times C_8$), $C_8$ ($C_{16}$)

Order 4: $C_2^2$ ($C_{32}$), $C_4$ ($C_2\times C_{16}$), $C_4$ ($C_{32}$)

Order 2: $C_2$ ($C_{64}$) x 2, $C_2$ ($C_2\times C_{32}$)

Order 1: $C_1$ ($C_2\times C_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 128: $C_2\times C_{64}$ ($C_1$)

Order 64: $C_2\times C_{32}$ ($C_2$), $C_{64}$ ($C_2$)

Order 32: $C_2\times C_{16}$ ($C_4$), $C_{32}$ ($C_2^2$), $C_{32}$ ($C_4$)

Order 16: $C_2\times C_8$ ($C_8$), $C_{16}$ ($C_2\times C_4$), $C_{16}$ ($C_8$)

Order 8: $C_2\times C_4$ ($C_{16}$), $C_8$ ($C_2\times C_8$), $C_8$ ($C_{16}$)

Order 4: $C_2^2$ ($C_{32}$), $C_4$ ($C_2\times C_{16}$), $C_4$ ($C_{32}$)

Order 2: $C_2$ ($C_2\times C_{32}$), $C_2$ ($C_{64}$)

Order 1: $C_1$ ($C_2\times C_{64}$)

Series

Derived series

$C_2\times C_{64}$

$\rhd$

$C_1$

Chief series

$C_2\times C_{64}$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2\times C_{64}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{64}$

Supergroups

This group is a maximal subgroup of 46 larger groups in the database.

This group is a maximal quotient of 32 larger groups in the database.

Character theory

Complex character table

See the $128 \times 128$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	4A	4B	8A	8B	16A	16B	32A	32B	64A	64B
	Size	1	1	1	1	2	2	4	4	8	8	16	16	32	32
	2 P	1A	1A	1A	1A	2C	2C	4B	4B	8B	8B	16B	16B	32B	32B
128.159.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
128.159.1b		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$
128.159.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$
128.159.1d		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
128.159.1e		$2$	$-2$	$2$	$-2$	$2$	$-2$	$2$	$2$	$2$	$2$	$2$	$-2$	$0$	$0$
128.159.1f		$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$0$	$0$
128.159.1g		$4$	$-4$	$4$	$-4$	$4$	$-4$	$4$	$4$	$-4$	$-4$	$0$	$0$	$0$	$0$
128.159.1h		$4$	$4$	$4$	$4$	$4$	$4$	$4$	$4$	$-4$	$-4$	$0$	$0$	$0$	$0$
128.159.1i		$8$	$-8$	$8$	$-8$	$8$	$-8$	$-8$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$
128.159.1j		$8$	$8$	$8$	$8$	$8$	$8$	$-8$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$
128.159.1k		$16$	$-16$	$16$	$-16$	$-16$	$16$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.159.1l		$16$	$16$	$16$	$16$	$-16$	$-16$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.159.1m		$32$	$-32$	$-32$	$32$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
128.159.1n		$32$	$32$	$-32$	$-32$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$