Finite Group DatabasebetaFinite Group Databasebeta| order | 36900 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | #20301654⋊C5 |
| Fitting, socle, derived | C32×C5×C41 ⋊ C4×C5 |
| Hall | C32×C5×C41⋊C5 ⋊ C4 |
| Hall | C5×C41⋊C5 ⋊ #157 |
| Hall | C32×C41 ⋊ C5×AGL(1,𝔽5) |
| Sylow | C41 ⋊ C5×#4380 |
| Sylow | C32 ⋊ #11717962⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C5×C41 | C4×C5 |
| prime radical | C3⋊(C3⋊(C5⋊(D41⋊C5))) | C2 |
| cosocle | C3⋊D615 | C2×C5 |
| LCS | G ▷ C32×C5×C41 | |
| UCS | C1 | |
92 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1230 |
Polycyclic presentation:
2,5,2,3^2,5,41|a^2=c,d^a=de,e^a=de^2,f^a=f^2,g^a=g^9,g^b=g^10,d^c=d^2,e^c=e^2,f^c=f^4,g^c=g^40165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36900, 344);