Abstract group 64.50: $C_2\times C_{32}$

• Label: 64.50
• Order: \( 2^{6} \)
• Exponent: \( 2^{5} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{6} \)
• Perm deg.: $34$
• Trans deg.: $64$
• Rank: $2$

Group information

Description:	$C_2\times C_{32}$
Order:	\(64\)\(\medspace = 2^{6} \)
Exponent:	\(32\)\(\medspace = 2^{5} \)
Automorphism group:	$C_8.C_2^3$, of order \(64\)\(\medspace = 2^{6} \)
Composition factors:	$C_2$ x 6
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
Elements	1	3	4	8	16	32	64
Conjugacy classes	1	3	4	8	16	32	64
Divisions	1	3	2	2	2	2	12
Autjugacy classes	1	2	2	2	2	1	10

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

Minimal presentations

Permutation degree:	$34$
Transitive degree:	$64$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	2	3	17

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{2}=b^{32}=1 \rangle$
Permutation group:	Degree $34$ $\langle(3,34,18,26,10,30,14,22,6,32,16,24,8,28,12,20,4,33,17,25,9,29,13,21,5,31,15,23,7,27,11,19) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 77 & 0 \\ 0 & 18 \end{array}\right), \left(\begin{array}{rr} 96 & 0 \\ 0 & 96 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{97})$
Direct product:	$C_2$ $\, \times\, $ $C_{32}$
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:	$C_8$ . $C_8$	$C_{16}$ . $C_4$	$C_4$ . $C_{16}$	$C_{16}$ . $C_2^2$	all 11
Aut. group:	$\Aut(C_{128})$

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

Homology

Primary decomposition:	$C_{2} \times C_{32}$
Schur multiplier:	$C_{2}$
Commutator length:	$0$

Subgroups

There are 17 subgroups, all normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{32}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2\times C_{32}$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{32}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times C_{32}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_{32}$

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 64: $C_2\times C_{32}$

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 64: $C_2\times C_{32}$

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

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 64: $C_2\times C_{32}$ ($C_1$)

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

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

Order 8: $C_2\times C_4$ ($C_8$), $C_8$ ($C_2\times C_4$), $C_8$ ($C_8$)

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

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

Order 1: $C_1$ ($C_2\times C_{32}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 64: $C_2\times C_{32}$ ($C_1$)

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

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

Order 8: $C_2\times C_4$ ($C_8$), $C_8$ ($C_2\times C_4$), $C_8$ ($C_8$)

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

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

Order 1: $C_1$ ($C_2\times C_{32}$)

Series

Derived series

$C_2\times C_{32}$

$\rhd$

$C_1$

Chief series

$C_2\times C_{32}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2\times C_{32}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{32}$

Supergroups

This group is a maximal subgroup of 96 larger groups in the database.

This group is a maximal quotient of 82 larger groups in the database.

Character theory

Complex character table

See the $64 \times 64$ character table. 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
	Size	1	1	1	1	2	2	4	4	8	8	16	16
	2 P	1A	1A	1A	1A	2C	2C	4B	4B	8B	8B	16B	16B
64.50.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
64.50.1b		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$
64.50.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$
64.50.1d		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
64.50.1e		$2$	$-2$	$2$	$-2$	$2$	$-2$	$2$	$2$	$2$	$-2$	$0$	$0$
64.50.1f		$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$0$	$0$
64.50.1g		$4$	$-4$	$4$	$-4$	$4$	$-4$	$-4$	$-4$	$0$	$0$	$0$	$0$
64.50.1h		$4$	$4$	$4$	$4$	$4$	$4$	$-4$	$-4$	$0$	$0$	$0$	$0$
64.50.1i		$8$	$-8$	$8$	$-8$	$-8$	$8$	$0$	$0$	$0$	$0$	$0$	$0$
64.50.1j		$8$	$8$	$8$	$8$	$-8$	$-8$	$0$	$0$	$0$	$0$	$0$	$0$
64.50.1k		$16$	$-16$	$-16$	$16$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
64.50.1l		$16$	$16$	$-16$	$-16$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$