Finite Group DatabasebetaFinite Group Databasebeta| order | 12 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×D3 (order 12) · Inn(G) = D3 · Out(G) = C2 |
| holomorph | G ⋊ Aut(G) = #3811 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D6 |
| Borel | B(2,𝔽3) |
| direct product | C2×D3 |
| unambiguous semidirect product | C3⋊C22 |
| unambiguous semidirect product | (C2×C3)⋊C2 |
| Fitting, socle | C2×C3 ⋊ C2 |
| derived, Sylow | C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3 |
| commutator, cosocle, hypocenter | C3 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 5 |
| transitive degree | μt(G) = 6 |
A minimal faithful permutation representation — 2 generators on 5 points:
(1,2)(3,4,5);(4,5)| C1 | C2 | C3 | C4 | C5 | C6 | |
|---|---|---|---|---|---|---|
| size | 1 | 2 | 3 | 1 | 2 | 3 |
| order | 1 | 3 | 2 | 2 | 6 | 2 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | -1 | 1 | 1 | -1 |
| χ3 | 2 | -1 | 0 | 2 | -1 | 0 |
| χ4 | 1 | 1 | 1 | -1 | -1 | -1 |
| χ5 | 1 | 1 | -1 | -1 | -1 | 1 |
| χ6 | 2 | -1 | 0 | -2 | 1 | 0 |
| maximal subgroup of | C4×D3, D12, #67, … |
| maximal normal subgroup of | C4×D3, D12, #67, … |
| maximal quotient of | Dic6, C4×D3, D12, … |
| automorphism group of | Dic3, C2×D3, C22×C3, … |
| outer automorphism group of | C22×C3, #41, C3×Q8, … |
| holomorph of | C2×C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12, 4);