Finite Group DatabasebetaFinite Group Databasebeta| order | 32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | #24926274⋊C3 |
| Fitting, socle, derived, Hall | C52×C109 ⋊ C3×C4 |
| Hall | #24926274 ⋊ C3 |
| Hall | C52×C109⋊C3 ⋊ C4 |
| Hall | C109⋊C3 ⋊ #1044 |
| Sylow | C109 ⋊ C3×#1044 |
| Sylow | C52 ⋊ #89137⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52×C109 | C3×C4 |
| prime radical | C5⋊(C5⋊(D109⋊C3)) | C2 |
| cosocle | C5⋊D545 | C2×C3 |
| LCS | G ▷ C52×C109 | |
| UCS | C1 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1090 |
Polycyclic presentation:
2,3,2,5^2,109|a^2=c,d^a=d^2,e^a=e^3,f^a=f^76,f^b=f^63,d^c=d^4,e^c=e^4,f^c=f^108255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32700, 124);