Finite Group DatabasebetaFinite Group Databasebeta| order | 32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | #89137⋊(C52⋊C3) |
| Fitting, socle, derived, Hall | C52×C109 ⋊ C3×C4 |
| Hall | C52×#89137 ⋊ C3 |
| Hall | C52⋊(C109⋊C3) ⋊ C4 |
| Hall | #89137 ⋊ C52⋊C3 |
| Sylow | C109 ⋊ C4×C52⋊C3 |
| Sylow | C52 ⋊ #89137⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52×C109 | C3×C4 |
| prime radical | D109⋊(C52⋊C3) | C2 |
| cosocle | C52×D109 | C2×C3 |
| LCS | G ▷ C52×C109 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1635 |
Polycyclic presentation:
2,3,2,5^2,109|a^2=c,f^a=f^76,d^b=d^4e^2,e^b=d^2,f^b=f^63,f^c=f^108269 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32700, 82);