Finite Group DatabasebetaFinite Group Databasebeta| order | 108 |
| isoclinism | isoclinic class of D3×C3⋊D3 |
| automorphism | Aut(G) = D3×AGL(2,𝔽3) (order 2592) · Inn(G) = G · Out(G) = S4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| direct product | D3×C3⋊D3 |
| unambiguous semidirect product | (C3×D3)⋊D3 |
| unambiguous semidirect product | (C3⋊(C3⋊D3))⋊C2 |
| Fitting, socle, derived, Sylow | C33 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C33 | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 3 generators on 9 points:
(1,2,3)(5,6)(8,9);(2,3)(7,8,9);(4,5,6)18 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#157, D33, C2×D3×C3⋊D3, … |
| maximal normal subgroup of | D3×#157, D33, C2×D3×C3⋊D3, … |
| maximal quotient of | D3×C3⋊Dic3, Dic3×C3⋊D3, #6289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(108, 39);