Finite Group DatabasebetaFinite Group Databasebeta| order | 27900 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C25 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C3⋊D3)⋊(D31⋊C25) |
| unambiguous semidirect product | ((C3⋊D3)⋊D31)⋊C25 |
| unambiguous semidirect product | (C3⋊(D31⋊C25))⋊D3 |
| derived, Hall | C32×C31 ⋊ C22×C25 |
| Hall | C32×C31⋊C25 ⋊ C22 |
| Hall | (C3⋊D3)⋊D31 ⋊ C25 |
| Hall | C31⋊C25 ⋊ D32 |
| Sylow | C31 ⋊ D32×C25 |
| Sylow | C32 ⋊ C2×D31⋊C25 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C5 | (C3⋊D3)⋊(D31⋊C5) |
| commutator, hypocenter | C32×C31 | C22×C25 |
| Fitting, socle, cosocle | C32×C5×C31 | C22×C5 |
| prime radical | C5×(C3⋊D3)⋊D31 | C5 |
| LCS | G ▷ C32×C31 | |
| UCS | C1 ◁ C5 | |
132 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,5,3^2,31,5|c^5=g,e^a=e^2,f^a=f^30,d^b=d^2,e^b=e^2,f^c=f^16315 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27900, 135);