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) = C3×C4×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×C9×Dic3 |
| unambiguous semidirect product | C3⋊(C3×C4×C9) |
| derived | C3 ⋊ C3×C4×C9 |
| Sylow | C32×C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C9 | D3 |
| commutator, hypocenter | C3 | C3×C4×C9 |
| Frattini | C2×C3 | C32×D3 |
| Fitting | C2×C32×C9 | C2 |
| socle, prime radical | C2×C33 | C2×C3 |
| cosocle | C2×C32 | C2×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C3×C9 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 19 points:
(5,6)(7,8,10,9,11,13,12,14,15)(16,17,18,19);(1,2,3)(4,5,6)162 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#6189, C3×Dic3×D9, C3×(C3⋊D9)⋊C4, … |
| maximal normal subgroup of | C3×#6189, C3×Dic3×D9, C3×(C3⋊D9)⋊C4, … |
| maximal quotient of | C3×C9×C3⋊C8, C2×C3×C9×Dic3, C3×C9×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 91);