Finite Group DatabasebetaFinite Group Databasebeta| order | 24 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = D3×S4 (order 144) · Inn(G) = D3 · Out(G) = S4 |
| holomorph | G ⋊ Aut(G) = #72210357 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C22×D3 |
| unambiguous semidirect product | C3⋊C23 |
| unambiguous semidirect product | (C2×C3)⋊C22 |
| Fitting, socle | C22×C3 ⋊ C2 |
| derived, Sylow | C3 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3 |
| commutator, cosocle, hypocenter | C3 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C22 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 7 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 3 generators on 7 points:
(1,2)(3,4)(6,7);(3,4)(5,6,7);(1,2)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 |
| order | 1 | 3 | 2 | 2 | 6 | 2 | 2 | 6 | 2 | 2 | 6 | 2 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 |
| χ3 | 2 | -1 | 0 | 2 | -1 | 0 | 2 | -1 | 0 | 2 | -1 | 0 |
| χ4 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 |
| χ5 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 |
| χ6 | 2 | -1 | 0 | -2 | 1 | 0 | 2 | -1 | 0 | -2 | 1 | 0 |
| χ7 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 |
| χ8 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 |
| χ9 | 2 | -1 | 0 | 2 | -1 | 0 | -2 | 1 | 0 | -2 | 1 | 0 |
| χ10 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 |
| χ11 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 |
| χ12 | 2 | -1 | 0 | -2 | 1 | 0 | -2 | 1 | 0 | 2 | -1 | 0 |
| maximal subgroup of | (C2×D3)⋊C4, C2×C4×D3, C2×D12, … |
| maximal normal subgroup of | (C2×D3)⋊C4, C2×C4×D3, C2×D12, … |
| maximal quotient of | C2×Dic6, C2×C4×D3, C2×D12, … |
| automorphism group of | C3⋊C8, C4×D3, #67, … |
| outer automorphism group of | #131, C3×#41, #576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24, 14);