Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of AGL(1,ℤ/9) |
| automorphism | Aut(G) = C2×C3×AGL(1,ℤ/9) (order 324) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×D9⋊C9 |
| unambiguous semidirect product | D9⋊(C2×C9) |
| unambiguous semidirect product | (C2×D9)⋊C9 |
| unambiguous semidirect product | (C9⋊C9)⋊C22 |
| unambiguous semidirect product | (C2×C9⋊C9)⋊C2 |
| Fitting | C2×C9⋊C9 ⋊ C2 |
| derived | C9 ⋊ C22×C9 |
| Sylow | C9⋊C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | AGL(1,ℤ/9) |
| commutator, hypocenter | C9 | C22×C9 |
| Frattini | C32 | C2×C3×D3 |
| Fitting | C2×C9⋊C9 | C2 |
| socle | C2×C32 | C3×D3 |
| prime radical | C2×C3×D9 | C3 |
| cosocle | C3×C9 | C22×C3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C3 | |
53 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 20 points:
(4,8,11,6,7,10)(5,9)(12,16,20,14,19,15,17,13,18);(1,2)(3,11,6,9,7,8,5,4,10)60 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D9⋊C9, C4⋊(D9⋊C9), #611⋊C9, … |
| maximal normal subgroup of | C4×D9⋊C9, C4⋊(D9⋊C9), #611⋊C9, … |
| maximal quotient of | Dic18⋊C9, C4×D9⋊C9, C4⋊(D9⋊C9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 64);