Finite Group DatabasebetaFinite Group Databasebeta| order | 54 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = D3×GL(2,𝔽3) (order 288) · Inn(G) = D3 · Out(G) = GL(2,𝔽3) |
| holomorph | G ⋊ Aut(G) = D32×AGL(2,𝔽3) |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C32×D3 |
| unambiguous semidirect product | C3⋊(C2×C32) |
| Fitting, socle, Sylow | C33 ⋊ C2 |
| derived | C3 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C33 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C32 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 9 points:
(4,5,6)(8,9);(1,2,3)(7,8,9)27 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D32, D3×C3⋊D3, C2×C32×D3, … |
| maximal normal subgroup of | C3×D32, D3×C3⋊D3, C2×C32×D3, … |
| maximal quotient of | C32×Dic3, C2×C32×D3, C32×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(54, 12);