Finite Group DatabasebetaFinite Group Databasebeta| order | 27900 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C3⋊D3)⋊(C5⋊(D31⋊C5)) |
| unambiguous semidirect product | ((C3⋊D3)⋊D155)⋊C5 |
| Fitting, socle, derived | C32×C5×C31 ⋊ C22×C5 |
| Hall | C32×C5×ΓL(1,𝔽32) ⋊ C22 |
| Hall | C5×ΓL(1,𝔽32) ⋊ D32 |
| Hall | C32×C31 ⋊ C2×C5×D5 |
| Sylow | C31 ⋊ C5×(C3⋊D3)⋊D5 |
| Sylow | C32 ⋊ C2×C5⋊(D31⋊C5) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C5×C31 | C22×C5 |
| LCS | G ▷ C32×C5×C31 | |
| UCS | C1 | |
224 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,5,3^2,5,31|e^a=e^2,f^a=f^4,g^a=g^30,d^b=d^2,e^b=e^2,g^c=g^16195 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27900, 470);