Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | Aut(G) = C2×C3×((C3⋊D3)⋊C3)⋊C2 (order 648) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×(C3⋊D3)⋊C9 |
| unambiguous semidirect product | (C2×C3⋊D3)⋊C9 |
| Fitting | C2×C32⋊C9 ⋊ C2 |
| derived | C32 ⋊ C22×C9 |
| Sylow | C32⋊C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C22×C9 |
| Frattini | C32 | C2×C3×D3 |
| Fitting | C2×C32⋊C9 | C2 |
| socle | C2×C32 | C3×D3 |
| prime radical | C2×C3×C3⋊D3 | C3 |
| cosocle | C33 | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C3 | |
67 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 20 points:
(4,9,11,6,7,10)(5,8)(12,16,20,14,19,15,17,13,18);(1,2)(3,4,6)(5,7,10)(8,11,9)60 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊C9)⋊C4, #10619535, #10619536, … |
| maximal normal subgroup of | ((C3⋊D3)⋊C9)⋊C4, #10619535, #10619536, … |
| maximal quotient of | #634⋊C9, C4×(C3⋊D3)⋊C9, C4⋊((C3⋊D3)⋊C9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 62);