Finite Group DatabasebetaFinite Group Databasebeta| order | 27900 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3×C5 |
| unambiguous semidirect product | (C31⋊C3)⋊(#1039⋊C3) |
| unambiguous semidirect product | ((C31⋊C3)⋊#1039)⋊C3 |
| unambiguous semidirect product | #22604802⋊C3 |
| unambiguous semidirect product | #22604804⋊C3 |
| derived, Hall | C52×C31 ⋊ C32×C4 |
| Hall | C52⋊C3×C31⋊C3 ⋊ C4 |
| Hall | C31⋊#1039 ⋊ C32 |
| Sylow | C31 ⋊ C3×#1039⋊C3 |
| Sylow | C52 ⋊ C3×(C31⋊C3)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | (C31⋊C3)⋊((C5⋊D5)⋊C3) |
| commutator, hypocenter | C52×C31 | C32×C4 |
| Fitting, socle, cosocle | C2×C52×C31 | C2×C32 |
| prime radical | C2×C52⋊C3×C31⋊C3 | C2 |
| LCS | G ▷ C52×C31 | |
| UCS | C1 ◁ C2 | |
96 conjugacy classes of subgroups.
Polycyclic presentation:
2,3^2,5^2,31,2|a^2=g,d^a=d^4,e^a=e^4,f^a=f^30,f^b=f^5,d^c=de,e^c=d^2e^3170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27900, 350);