Finite Group DatabasebetaFinite Group Databasebeta| order | 1350 |
| isoclinism | isoclinic class of C3⋊(C3⋊D75) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C33 |
| unambiguous semidirect product | C3⋊(C3⋊D75) |
| unambiguous semidirect product | C25⋊(C3⋊(C3⋊D3)) |
| Fitting, derived, Hall | C33×C25 ⋊ C2 |
| Sylow | C33 ⋊ D25 |
| Sylow | C25 ⋊ C3⋊(C3⋊D3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C33×C25 | C2 |
| Frattini | C5 | C3⋊(C3⋊D15) |
| socle | C33×C5 | D5 |
| LCS | G ▷ C33×C25 | |
| UCS | C1 | |
168 conjugacy classes of subgroups.
| transitive degree | μt(G) = 675 |
Polycyclic presentation:
2,3^3,5^2|e^5=f^4,b^a=b^2,c^a=c^2,d^a=d^2,e^a=e^4f,f^a=f^4339 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1350, 38);