Finite Group DatabasebetaFinite Group Databasebeta| order | 54 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = AGL(2,𝔽3) (order 432) · Inn(G) = C32 · Out(G) = GL(2,𝔽3) |
| holomorph | G ⋊ Aut(G) = C2×#72210405 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
| direct product | C2×H(3,𝔽3) |
| unambiguous semidirect product | (C2×C32)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32 |
| commutator, Frattini, cosocle | C3 | C2×C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ G | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,2)(4,6,9)(7,11,10);(3,4,7)(5,6,10)(8,9,11)22 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(54, 10);