Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C9×C3⋊D3 |
| 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×C9 | C3⋊D3 |
| commutator, hypocenter | C32 | C22×C9 |
| Frattini | C3 | C2×C3×C3⋊D3 |
| Fitting | C2×C32×C9 | C2 |
| socle | C2×C33 | C2×C3 |
| prime radical | C2×C3×C3⋊D3 | C3 |
| cosocle | C33 | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C9 | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2)(4,5)(7,8)(9,13,17,11,16,12,14,10,15);(1,2)(3,4,5);(6,7,8)108 conjugacy classes — view the full character table.
| maximal subgroup of | C9×#624, C9×#626, C3⋊D3×Dic9, … |
| maximal normal subgroup of | C9×#624, C9×#626, C3⋊D3×Dic9, … |
| maximal quotient of | C9×#634, C4×C9×C3⋊D3, C9×C4⋊(C3⋊D3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 143);