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Properties

Label 96.2
Order \( 2^{5} \cdot 3 \)
Exponent \( 2^{5} \cdot 3 \)
Abelian yes
$\card{\operatorname{Aut}(G)}$ \( 2^{5} \)
Perm deg. $35$
Trans deg. $96$
Rank $1$

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Group information

Description:$C_{96}$
Order: \(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent: \(96\)\(\medspace = 2^{5} \cdot 3 \)
Automorphism group:$C_2^2\times C_8$, of order \(32\)\(\medspace = 2^{5} \) (generators)
Outer automorphisms:$C_2^2\times C_8$, of order \(32\)\(\medspace = 2^{5} \)
Composition factors:$C_2$ x 5, $C_3$
Nilpotency class:$1$
Derived length:$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Group statistics

Order 1 2 3 4 6 8 12 16 24 32 48 96
Elements 1 1 2 2 2 4 4 8 8 16 16 32 96
Conjugacy classes   1 1 2 2 2 4 4 8 8 16 16 32 96
Divisions 1 1 1 1 1 1 1 1 1 1 1 1 12
Autjugacy classes 1 1 1 1 1 1 1 1 1 1 1 1 12

Dimension 1 2 4 8 16 32
Irr. complex chars.   96 0 0 0 0 0 96
Irr. rational chars. 2 3 2 2 2 1 12

Minimal Presentations

Permutation degree:$35$
Transitive degree:$96$
Rank: $1$
Inequivalent generators: $1$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 1 2 32
Arbitrary 1 2 18

Constructions

Presentation: $\langle a \mid a^{96}=1 \rangle$ Copy content Toggle raw display
Permutation group:Degree $35$ $\langle(1,32,16,24,8,28,12,20,4,30,14,22,6,26,10,18,2,31,15,23,7,27,11,19,3,29,13,21,5,25,9,17) \!\cdots\! \rangle$ Copy content Toggle raw display
Matrix group:$\left\langle \left(\begin{array}{rr} 12 & 3 \\ 1 & 12 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{17})$
Direct product: $C_{32}$ $\, \times\, $ $C_3$
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: $C_{48}$ . $C_2$ $C_{24}$ . $C_4$ $C_{16}$ . $C_6$ $C_{12}$ . $C_8$ all 8
Aut. group: $\Aut(C_{97})$ $\Aut(C_{194})$

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

Homology

Primary decomposition: $C_{32} \times C_{3}$
Schur multiplier: $C_1$
Commutator length: $0$

Subgroups

There are 12 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_{96}$ $G/Z \simeq$ $C_1$
Commutator: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_{96}$
Frattini: $\Phi \simeq$ $C_{16}$ $G/\Phi \simeq$ $C_6$
Fitting: $\operatorname{Fit} \simeq$ $C_{96}$ $G/\operatorname{Fit} \simeq$ $C_1$
Radical: $R \simeq$ $C_{96}$ $G/R \simeq$ $C_1$
Socle: $\operatorname{soc} \simeq$ $C_6$ $G/\operatorname{soc} \simeq$ $C_{16}$
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_{32}$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_{96}$ $\rhd$ $C_1$
Chief series $C_{96}$ $\rhd$ $C_{48}$ $\rhd$ $C_{24}$ $\rhd$ $C_{12}$ $\rhd$ $C_6$ $\rhd$ $C_3$ $\rhd$ $C_1$
Lower central series $C_{96}$ $\rhd$ $C_1$
Upper central series $C_1$ $\lhd$ $C_{96}$

Supergroups

This group is a maximal subgroup of 87 larger groups in the database.

This group is a maximal quotient of 75 larger groups in the database.

Character theory

Complex character table

See the $96 \times 96$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

1A 2A 3A 4A 6A 8A 12A 16A 24A 32A 48A 96A
Size 1 1 2 2 2 4 4 8 8 16 16 32
2 P 1A 1A 3A 2A 3A 4A 6A 8A 12A 16A 24A 48A
3 P 1A 2A 1A 4A 2A 8A 4A 16A 8A 32A 16A 32A
96.2.1a 1 1 1 1 1 1 1 1 1 1 1 1
96.2.1b 1 1 1 1 1 1 1 1 1 −1 1 −1
96.2.1c 2 2 −1 2 −1 2 −1 2 −1 2 −1 −1
96.2.1d 2 2 2 2 2 2 2 −2 2 0 −2 0
96.2.1e 2 2 −1 2 −1 2 −1 2 −1 −2 −1 1
96.2.1f 4 4 4 4 4 −4 4 0 −4 0 0 0
96.2.1g 4 4 −2 4 −2 4 −2 −4 −2 0 2 0
96.2.1h 8 8 8 −8 8 0 −8 0 0 0 0 0
96.2.1i 8 8 −4 8 −4 −8 −4 0 4 0 0 0
96.2.1j 16 −16 16 0 −16 0 0 0 0 0 0 0
96.2.1k 16 16 −8 −16 −8 0 8 0 0 0 0 0
96.2.1l 32 −32 −16 0 16 0 0 0 0 0 0 0