Finite Group DatabasebetaFinite Group Databasebeta| order | 27900 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (ΓL(1,𝔽32)⋊C9)⋊Dic5 |
| unambiguous semidirect product | (ΓL(1,𝔽32)⋊Dic5)⋊C9 |
| unambiguous semidirect product | ((C31⋊C9)⋊Dic5)⋊C5 |
| derived | C5×C31 ⋊ C4×C5×C9 |
| Hall | C5×ΓL(1,𝔽32)⋊C9 ⋊ C4 |
| Hall | ΓL(1,𝔽32)⋊Dic5 ⋊ C9 |
| Hall | C5×ΓL(1,𝔽32) ⋊ C4×C9 |
| Hall | C31⋊C9 ⋊ C5×Dic5 |
| Sylow | C31 ⋊ C5×C9×Dic5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | C5⋊AGL(1,𝔽31) |
| commutator, hypocenter | C5×C31 | C4×C5×C9 |
| Fitting, socle, cosocle | C2×C3×C5×C31 | C2×C3×C5 |
| prime radical | C2×C3×C5×ΓL(1,𝔽32) | C2×C3 |
| LCS | G ▷ C5×C31 | |
| UCS | C1 ◁ C2×C3 | |
96 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,5^2,31,2,3|a^2=f,b^3=g,d^a=d^4,e^a=e^30,e^b=e^5,e^c=e^16390 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27900, 88);