Finite Group DatabasebetaFinite Group Databasebeta| order | 27900 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C2×C5 |
| unambiguous semidirect product | C22⋊(C52⋊(C31⋊C9)) |
| unambiguous semidirect product | C52⋊(C22⋊(C31⋊C9)) |
| unambiguous semidirect product | C31⋊(C22⋊(C52⋊C9)) |
| derived, Hall | C22×C52×C31 ⋊ C9 |
| Hall | C52×C31 ⋊ C22⋊C9 |
| Hall | C22×C31 ⋊ C52⋊C9 |
| Hall | C22×C52 ⋊ C31⋊C9 |
| Sylow | C31 ⋊ C22⋊(C52⋊C9) |
| Sylow | C52 ⋊ C22⋊(C31⋊C9) |
| Sylow | C22 ⋊ C52⋊(C31⋊C9) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C22⋊(C52⋊(C31⋊C3)) |
| commutator, hypocenter | C22×C52×C31 | C9 |
| Fitting, socle, prime radical, cosocle | C22×C3×C52×C31 | C3 |
| LCS | G ▷ C22×C52×C31 | |
| UCS | C1 ◁ C3 | |
72 conjugacy classes of subgroups.
Polycyclic presentation:
3,2^2,5^2,31,3|a^3=g,b^a=c,c^a=bc,d^a=d^2e^2,e^a=d^4e^2,f^a=f^53108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27900, 322);