Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C4×C5×3−1+2 |
| unambiguous semidirect product | (C4×C9)⋊(C3×C5) |
| unambiguous semidirect product | (C4×C5×C9)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C5 | C32 |
| commutator | C3 | C32×C4×C5 |
| Frattini, cosocle | C2×C3 | C2×C32×C5 |
| socle | C2×C3×C5 | C2×C32 |
| prime radical | C2×C32×C5 | C2×C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C5 ◁ G | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4)(5,6,7,8,9)(10,11,13,12,15,17,16,14,18);(11,14,15)(13,17,18)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 30);