Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×D3×#272 (order 1296) · Inn(G) = D3 · Out(G) = C22×#272 |
| homology | Abelianization (H1) = C22×C3×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3×D3×C9 |
| unambiguous semidirect product | C3⋊(C22×C3×C9) |
| unambiguous semidirect product | (C2×C3)⋊(C2×C3×C9) |
| Fitting | C2×C32×C9 ⋊ C2 |
| derived | C3 ⋊ C22×C3×C9 |
| Sylow | C32×C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C9 | D3 |
| commutator, hypocenter | C3 | C22×C3×C9 |
| Frattini | C3 | C2×C32×D3 |
| Fitting | C2×C32×C9 | C2 |
| socle | C2×C33 | C2×C3 |
| prime radical | C2×C32×D3 | C3 |
| cosocle | C32 | C22×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C3×C9 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 17 points:
(4,5)(9,13,17,11,16,12,14,10,15);(1,2)(3,4,5)(6,7,8)162 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×Dic9, C3×#6194, C3×#6195, … |
| maximal normal subgroup of | C3×D3×Dic9, C3×#6194, C3×#6195, … |
| maximal quotient of | C3×C9×Dic6, C3×C4×D3×C9, C3×C9×D12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 137);