Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×D3×C9 (order 216) · Inn(G) = D3 · Out(G) = C22×C9 |
| homology | Abelianization (H1) = C4×C27 · Schur multiplier (H2) = C1 |
| direct product | Dic3×C27 |
| unambiguous semidirect product | C3⋊(C4×C27) |
| derived | C3 ⋊ C4×C27 |
| Sylow | C3×C27 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C27 | D3 |
| commutator, hypocenter | C3 | C4×C27 |
| Frattini | C2×C9 | C3×D3 |
| Fitting | C2×C3×C27 | C2 |
| socle, prime radical | C2×C32 | C2×C9 |
| cosocle | C2×C3×C9 | C2×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C27 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 34 points:
(2,3)(4,5,8,6,9,14,11,16,21,7,10,15,12,17,22,19,24,27,13,18,23,20,25,28,26,29,30)(31,34,33,32);(1,2,3)162 conjugacy classes — view the full character table.
| maximal subgroup of | #10619539, Dic3×D27, (C3⋊D27)⋊C4, … |
| maximal normal subgroup of | #10619539, Dic3×D27, (C3⋊D27)⋊C4, … |
| maximal quotient of | C3⋊C8×C27, C2×Dic3×C27, C27×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 11);