Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C3 |
| direct product | C5×(C3⋊Dic3)⋊C3 |
| unambiguous semidirect product | (C3⋊Dic3)⋊(C3×C5) |
| unambiguous semidirect product | (C5×C3⋊Dic3)⋊C3 |
| derived | C32 ⋊ C3×C4×C5 |
| Hall | C5×H(3,𝔽3) ⋊ C4 |
| Sylow | H(3,𝔽3) ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C3×C4×C5 |
| Frattini | C2×C3 | C3×C5×D3 |
| Fitting, prime radical | C2×C5×H(3,𝔽3) | C2 |
| socle | C2×C3×C5 | C3×D3 |
| cosocle | C2×C32 | C2×C3×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C5 | |
50 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 18);