Finite Group DatabasebetaFinite Group Databasebeta| order | 54 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×SL(3,𝔽3) (order 11232) · Inn(G) = C1 · Out(G) = C2×SL(3,𝔽3) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C33 |
| abelian | C2×C33 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 11 points:
(1,2)(3,4,5);(6,7,8);(9,10,11)54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(54, 15);