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Properties

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

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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 \(64\)\(\medspace = 2^{6} \) (generators)
Outer automorphisms:$C_8.C_2^3$, 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 $\langle a, b \mid a^{2}=b^{32}=1 \rangle$ Copy content Toggle raw display
Permutation group:Degree $34$ $\langle(3,34,18,26,10,30,14,22,6,32,16,24,8,28,12,20,4,33,17,25,9,29,13,21,5,31,15,23,7,27,11,19) \!\cdots\! \rangle$ Copy content Toggle raw display
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 generators from the presentation given 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: $P_{ 2 } \simeq$ $C_2\times C_{32}$

Subgroup diagram and profile

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Subgroup information

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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 4A 4A 8B 8B 16A 16A
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