Finite Group DatabasebetaFinite Group Databasebeta| order | 108 |
| isoclinism | isoclinic class of C3⋊D9 |
| automorphism | Aut(G) = C2×#72210336 (order 5832) · Inn(G) = C3⋊D9 · Out(G) = C2×#272 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3⋊D9 |
| unambiguous semidirect product | (C2×C3)⋊D9 |
| unambiguous semidirect product | (C2×C9)⋊D3 |
| Fitting | C2×C3×C9 ⋊ C2 |
| derived, Sylow | C3×C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊D9 |
| commutator, cosocle, hypocenter | C3×C9 | C22 |
| Frattini | C3 | C2×C3⋊D3 |
| Fitting | C2×C3×C9 | C2 |
| socle | C2×C32 | D3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C2 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(4,5)(7,9)(8,12)(10,11)(13,14);(1,2)(3,4,5);(6,14,9,12,11,10,8,7,13)30 conjugacy classes — view the full character table.
| maximal subgroup of | (C3⋊D9)⋊C4, #6192, #6195, … |
| maximal normal subgroup of | (C3⋊D9)⋊C4, #6192, #6195, … |
| maximal quotient of | #6226, C4×C3⋊D9, C4⋊(C3⋊D9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(108, 27);