Finite Group DatabasebetaFinite Group Databasebeta| order | 27900 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | (C31⋊C3)⋊#64480 |
| unambiguous semidirect product | (C31⋊#64480)⋊C3 |
| Fitting, socle, Hall | C52×C31 ⋊ C3×Dic3 |
| derived | C31×C52⋊C3 ⋊ C3×C4 |
| Hall | C52⋊C3×C31⋊C3 ⋊ C4 |
| Sylow | C31 ⋊ C3×#64480 |
| Sylow | C52 ⋊ C3⋊((C31⋊C3)⋊C4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C31×C52⋊C3 | C3×C4 |
| Fitting, socle | C52×C31 | C3×Dic3 |
| prime radical | C31⋊C3×(C5⋊D5)⋊C3 | C2 |
| cosocle | C31×(C5⋊D5)⋊C3 | C2×C3 |
| LCS | G ▷ C31×C52⋊C3 | |
| UCS | C1 | |
80 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,2,3,31,5^2|a^2=c,d^a=d^2,e^a=e^30,f^a=fg^4,g^a=f^2g^4,e^b=e^5,f^c=f^4,g^c=g^4,f^d=f^3g^2,g^d=fg74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27900, 431);