Finite Group DatabasebetaFinite Group Databasebeta| order | 108 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×C3×D3 (order 72) · Inn(G) = D3 · Out(G) = C22×C3 |
| holomorph | G ⋊ Aut(G) = AGL(1,ℤ/9)×#3811 |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×Dic3 |
| unambiguous semidirect product | C3⋊(C4×C9) |
| derived | C3 ⋊ C4×C9 |
| Sylow | C3×C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | D3 |
| commutator, hypocenter | C3 | C4×C9 |
| Frattini | C2×C3 | C3×D3 |
| Fitting | C2×C3×C9 | C2 |
| socle, prime radical, cosocle | C2×C32 | C2×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C9 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3)(4,5,7,6,8,10,9,11,12)(13,14,15,16);(1,2,3)54 conjugacy classes — view the full character table.
| maximal subgroup of | #6189, Dic3×D9, (C3⋊D9)⋊C4, … |
| maximal normal subgroup of | #6189, Dic3×D9, (C3⋊D9)⋊C4, … |
| maximal quotient of | C9×C3⋊C8, C2×C9×Dic3, C9×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(108, 7);