Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| isoclinism | isoclinic class of C3⋊(C3⋊D15) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C33 |
| direct product | C3×C3⋊(C3⋊D15) |
| unambiguous semidirect product | C3⋊(C3×C3⋊D15) |
| unambiguous semidirect product | C5⋊(C3×C3⋊(C3⋊D3)) |
| Fitting, socle, Hall | C34×C5 ⋊ C2 |
| derived | C33×C5 ⋊ C2×C3 |
| Sylow | C34 ⋊ D5 |
| Sylow | C5 ⋊ C3×C3⋊(C3⋊D3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊(C3⋊D15) |
| commutator, cosocle, hypocenter | C33×C5 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C34×C5 | C2 |
| LCS | G ▷ C33×C5 | |
| UCS | C1 ◁ C3 | |
380 conjugacy classes of subgroups.
| transitive degree | μt(G) = 270 |
207 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D5×C3⋊(C3⋊D3), C3×D3×C3⋊D15, C3×C3⋊D3×D15, … |
| maximal normal subgroup of | C3×D5×C3⋊(C3⋊D3), C3×D3×C3⋊D15, C3×C3⋊D3×D15, … |
| maximal quotient of | C3×C3⋊(C3⋊Dic15), C2×C3×C3⋊(C3⋊D15) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 110);