Finite Group DatabasebetaFinite Group Databasebeta| order | 27900 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C3⋊#22604802 |
| derived | C3×C52×C31 ⋊ C3×C4 |
| Hall | C3×C52⋊(C31⋊C3) ⋊ C4 |
| Hall | C52×C31 ⋊ C3×Dic3 |
| Sylow | C31 ⋊ C3⋊(#1039⋊C3) |
| Sylow | C52 ⋊ C3⋊((C31⋊C3)⋊C4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#17987869 |
| commutator, hypocenter | C3×C52×C31 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C52×C31 | C2×C3 |
| prime radical | C2×C3×C52⋊(C31⋊C3) | C2 |
| LCS | G ▷ C3×C52×C31 | |
| UCS | C1 ◁ C2 | |
80 conjugacy classes of subgroups.
Polycyclic presentation:
2,3^2,5^2,31,2|a^2=g,c^a=c^2,d^a=d^4,e^a=e^4,f^a=f^30,d^b=de,e^b=d^2e^3,f^b=f^5792 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27900, 388);