Finite Group DatabasebetaFinite Group Databasebeta| order | 27900 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D45⋊D155 |
| Fitting, derived, Hall | C52×C9×C31 ⋊ C22 |
| Hall | C52×C31 ⋊ C2×D9 |
| Hall | C9×C31 ⋊ D52 |
| Hall | C52×C9 ⋊ C2×D31 |
| Sylow | C31 ⋊ D45⋊D5 |
| Sylow | C52 ⋊ D9×D31 |
| Sylow | C9 ⋊ D5×D155 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C52×C9×C31 | C22 |
| Frattini | C3 | D15⋊D155 |
| socle | C3×C52×C31 | C2×D3 |
| LCS | G ▷ C52×C9×C31 | |
| UCS | C1 | |
144 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,3,5^2,31,3|c^3=g^2,d^a=d^4,e^a=e^4,f^a=f^30,c^b=c^2g,d^b=d^4,g^b=g^21872 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27900, 321);