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Properties

Label 12.2
Order \( 2^{2} \cdot 3 \)
Exponent \( 2^{2} \cdot 3 \)
Abelian yes
$\card{\operatorname{Aut}(G)}$ \( 2^{2} \)
Perm deg. $7$
Trans deg. $12$
Rank $1$

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Group information

Description:$C_{12}$
Order: \(12\)\(\medspace = 2^{2} \cdot 3 \)
Exponent: \(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:$C_2^2$, of order \(4\)\(\medspace = 2^{2} \) (generators)
Outer automorphisms:$C_2^2$, of order \(4\)\(\medspace = 2^{2} \)
Composition factors:$C_2$ x 2, $C_3$
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 12
Elements 1 1 2 2 2 4 12
Conjugacy classes   1 1 2 2 2 4 12
Divisions 1 1 1 1 1 1 6
Autjugacy classes 1 1 1 1 1 1 6

Dimension 1 2 4
Irr. complex chars.   12 0 0 12
Irr. rational chars. 2 3 1 6

Minimal Presentations

Permutation degree:$7$
Transitive degree:$12$
Rank: $1$
Inequivalent generators: $1$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 1 2 4
Arbitrary 1 2 4

Constructions

Presentation: $\langle a \mid a^{12}=1 \rangle$ Copy content Toggle raw display
Permutation group: $\langle(1,4,2,3), (5,7,6), (1,2)(3,4)\rangle$ Copy content Toggle raw display
Matrix group:$\left\langle \left(\begin{array}{rrrr} 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & -1 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\Z)$
$\left\langle \left(\begin{array}{rr} 4 & 3 \\ 4 & 4 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{5})$
Transitive group: 12T1 more information
Direct product: $C_4$ $\, \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: $C_6$ . $C_2$ $C_2$ . $C_6$ more information
Aut. group: $\Aut(C_{13})$ $\Aut(C_{26})$

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

Homology

Primary decomposition: $C_{4} \times C_{3}$
Schur multiplier: $C_1$
Commutator length: $0$

Subgroups

There are 6 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_{12}$ $G/Z \simeq$ $C_1$
Commutator: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_{12}$
Frattini: $\Phi \simeq$ $C_2$ $G/\Phi \simeq$ $C_6$
Fitting: $\operatorname{Fit} \simeq$ $C_{12}$ $G/\operatorname{Fit} \simeq$ $C_1$
Radical: $R \simeq$ $C_{12}$ $G/R \simeq$ $C_1$
Socle: $\operatorname{soc} \simeq$ $C_6$ $G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_4$
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 $C_{12}$ $\rhd$ $C_1$
Chief series $C_{12}$ $\rhd$ $C_6$ $\rhd$ $C_3$ $\rhd$ $C_1$
Lower central series $C_{12}$ $\rhd$ $C_1$
Upper central series $C_1$ $\lhd$ $C_{12}$

Supergroups

This group is a maximal subgroup of 254 larger groups in the database.

This group is a maximal quotient of 266 larger groups in the database.

Character theory

Complex character table

1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 12A1 12A-1 12A5 12A-5
Size 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 3A-1 3A1 2A 2A 3A1 3A-1 6A-1 6A1 6A1 6A-1
3 P 1A 2A 1A 1A 4A-1 4A1 2A 2A 4A1 4A-1 4A1 4A-1
Type
12.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1
12.2.1b R 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1
12.2.1c1 C 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
12.2.1c2 C 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
12.2.1d1 C 1 −1 1 1 −i i −1 −1 i −i i −i
12.2.1d2 C 1 −1 1 1 i −i −1 −1 −i i −i i
12.2.1e1 C 1 1 ζ3−1 ζ3 −1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
12.2.1e2 C 1 1 ζ3 ζ3−1 −1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1
12.2.1f1 C 1 −1 −ζ122 ζ124 −ζ123 ζ123 −ζ124 ζ122 −ζ12 ζ125 −ζ125 ζ12
12.2.1f2 C 1 −1 ζ124 −ζ122 ζ123 −ζ123 ζ122 −ζ124 ζ125 −ζ12 ζ12 −ζ125
12.2.1f3 C 1 −1 −ζ122 ζ124 ζ123 −ζ123 −ζ124 ζ122 ζ12 −ζ125 ζ125 −ζ12
12.2.1f4 C 1 −1 ζ124 −ζ122 −ζ123 ζ123 ζ122 −ζ124 −ζ125 ζ12 −ζ12 ζ125

Rational character table

1A 2A 3A 4A 6A 12A
Size 1 1 2 2 2 4
2 P 1A 1A 3A 2A 3A 6A
3 P 1A 2A 1A 4A 2A 4A
12.2.1a 1 1 1 1 1 1
12.2.1b 1 1 1 −1 1 −1
12.2.1c 2 2 −1 2 −1 −1
12.2.1d 2 −2 2 0 −2 0
12.2.1e 2 2 −1 −2 −1 1
12.2.1f 4 −4 −2 0 2 0