Finite Group DatabasebetaFinite Group Databasebeta| order | 24 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = D3×D4 (order 48) · Inn(G) = D3 · Out(G) = D4 |
| holomorph | G ⋊ Aut(G) = #11865254 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×Dic3 |
| unambiguous semidirect product | (C2×C3)⋊C4 |
| derived, Sylow | C3 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3 |
| commutator, hypocenter | C3 | C2×C4 |
| Frattini | C2 | C2×D3 |
| Fitting, socle, prime radical | C22×C3 | C2 |
| cosocle | C2×C3 | C22 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C22 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 9 points:
(1,2)(4,5)(6,7,8,9);(1,2)(3,4,5)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 2 | 2 | 3 | 3 | 1 | 1 | 2 | 2 | 3 | 3 |
| order | 1 | 2 | 3 | 6 | 4 | 4 | 2 | 2 | 6 | 6 | 4 | 4 |
| 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 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | ||||
| χ4 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | ||||
| χ5 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 |
| χ6 | 2 | -2 | -1 | 1 | 0 | 0 | 2 | -2 | -1 | 1 | 0 | 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 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | ||||
| χ10 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | ||||
| χ11 | 2 | 2 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | 0 | 0 |
| χ12 | 2 | -2 | -1 | 1 | 0 | 0 | -2 | 2 | 1 | -1 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| maximal subgroup of | C4×Dic3, Dic3⋊C4, C4⋊Dic3, … |
| maximal normal subgroup of | C4×Dic3, Dic3⋊C4, C4⋊Dic3, … |
| maximal quotient of | C2×C3⋊C8, #208, C4×Dic3, … |
| outer automorphism group of | C3×C24⋊C5 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24, 7);