Finite Group DatabasebetaFinite Group Databasebeta| order | 27900 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D75⋊(D31⋊C3) |
| unambiguous semidirect product | (C3⋊(D31⋊C3))⋊D25 |
| unambiguous semidirect product | (C25⋊(D31⋊C3))⋊D3 |
| unambiguous semidirect product | (D75⋊D31)⋊C3 |
| Fitting, derived | C3×C25×C31 ⋊ C22×C3 |
| Hall | C3×C25×C31⋊C3 ⋊ C22 |
| Hall | C25×C31 ⋊ C2×C3×D3 |
| Hall | C3×C31⋊C3 ⋊ C2×D25 |
| Sylow | C31 ⋊ C3×D3×D25 |
| Sylow | C25 ⋊ D3×D31⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C25×C31 | C22×C3 |
| Frattini | C5 | D15⋊(D31⋊C3) |
| socle | C3×C5×C31 | C2×C3×D5 |
| LCS | G ▷ C3×C25×C31 | |
| UCS | C1 | |
132 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,3^2,5,31,5|e^5=g^4,e^a=e^4g,f^a=f^30,g^a=g^4,d^b=d^2,e^b=e^4g,g^b=g^4,f^c=f^5321 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27900, 151);