Finite Group DatabasebetaFinite Group Databasebeta| order | 27900 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C31⋊C3)⋊(#1043⋊C3) |
| unambiguous semidirect product | ((C31⋊C3)⋊#1043)⋊C3 |
| unambiguous semidirect product | #22604861⋊C3 |
| unambiguous semidirect product | #22604863⋊C3 |
| Fitting, socle, derived, Hall | C52×C31 ⋊ C32×C4 |
| Hall | C52⋊C3×C31⋊C3 ⋊ C4 |
| Hall | C31⋊#1043 ⋊ C32 |
| Sylow | C31 ⋊ C3×#1043⋊C3 |
| Sylow | C52 ⋊ C3×(C31⋊C3)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52×C31 | C32×C4 |
| prime radical | C31⋊C3×(C5⋊D5)⋊C3 | C2 |
| cosocle | C31×C5⋊D5 | C2×C32 |
| LCS | G ▷ C52×C31 | |
| UCS | C1 | |
96 conjugacy classes of subgroups.
Polycyclic presentation:
2,3^2,2,5^2,31|a^2=d,e^a=e^2,f^a=f^2,g^a=g^30,g^b=g^5,e^c=ef,f^c=e^2f^3,e^d=e^4,f^d=f^492 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27900, 433);