Finite Group DatabasebetaFinite Group Databasebeta| order | 32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #24926249⋊C3 |
| Fitting, derived, Hall | C25×C109 ⋊ C3×C4 |
| Hall | #24926249 ⋊ C3 |
| Hall | C25×C109⋊C3 ⋊ C4 |
| Hall | C109⋊C3 ⋊ #1035 |
| Sylow | C109 ⋊ C3×#1035 |
| Sylow | C25 ⋊ Dic109⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C25×C109 | C3×C4 |
| Frattini | C5 | #14546128⋊C3 |
| socle | C5×C109 | C3×AGL(1,𝔽5) |
| prime radical | D25×C109⋊C3 | C2 |
| cosocle | D25×C109 | C2×C3 |
| LCS | G ▷ C25×C109 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2725 |
Polycyclic presentation:
2,3,2,5,109,5|a^2=c,d^5=f,d^a=d^2f,e^a=e^108,f^a=f^2,e^b=e^63,d^c=d^4f^4,f^c=f^4282 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32700, 17);