Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C33 |
| direct product | C2×C33×D3 |
| unambiguous semidirect product | C3⋊(C22×C33) |
| unambiguous semidirect product | (C2×C3)⋊(C2×C33) |
| Fitting, socle | C2×C34 ⋊ C2 |
| derived | C3 ⋊ C22×C33 |
| Sylow | C34 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C33 | D3 |
| commutator, cosocle, hypocenter | C3 | C22×C33 |
| Frattini | C1 | G |
| Fitting, socle | C2×C34 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C33 | |
436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 14 points:
(3,4,5)(13,14);(1,2)(6,7,8)(12,13,14);(9,10,11)162 conjugacy classes — view the full character table.
| maximal subgroup of | C32×D3×Dic3, C32×#625, C32×#626, … |
| maximal normal subgroup of | C32×D3×Dic3, C32×#625, C32×#626, … |
| maximal quotient of | C33×Dic6, C33×C4×D3, C33×D12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 172);