Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of ((C3⋊D3)⋊C3)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×((C3⋊D3)⋊C3)⋊C2 |
| unambiguous semidirect product | ((C3⋊D3)⋊C3)⋊(C2×C5) |
| unambiguous semidirect product | #272⋊(C2×C5) |
| Fitting, Hall | C5×H(3,𝔽3) ⋊ C22 |
| derived, Sylow | H(3,𝔽3) ⋊ C22×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C22×C5 |
| Frattini | C3 | C5×D32 |
| Fitting | C5×H(3,𝔽3) | C22 |
| socle | C3×C5 | D32 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C5 | |
78 conjugacy classes of subgroups.
| transitive degree | μt(G) = 45 |
55 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 51);