Finite Group DatabasebetaFinite Group Databasebeta| order | 27900 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2 |
| Fitting, socle, derived | C3×C5×C31 ⋊ C22×C3×C5 |
| Hall | C3×C5×(C31⋊C3)⋊C5 ⋊ C22 |
| Hall | C5×ΓL(1,𝔽32) ⋊ C2×C3×D3 |
| Hall | C3×C31⋊C3 ⋊ C2×C5×D5 |
| Sylow | C31 ⋊ C3×C5×D3×D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C5×C31 | C22×C3×C5 |
| LCS | G ▷ C3×C5×C31 | |
| UCS | C1 | |
212 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,3,5,3,5,31|f^a=f^4,g^a=g^30,e^b=e^2,f^b=f^4,g^c=g^5,g^d=g^16189 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27900, 426);