Finite Group DatabasebetaFinite Group Databasebeta| order | 32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#24926249 |
| Fitting, derived, Hall | C3×C25×C109 ⋊ C4 |
| Hall | C25×C109 ⋊ Dic3 |
| Hall | C3×C109 ⋊ #1035 |
| Sylow | C109 ⋊ C3⋊#1035 |
| Hall | C3×C25 ⋊ Dic109 |
| Sylow | C25 ⋊ Dic327 |
| Sylow | C3 ⋊ #24926249 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C3×C25×C109 | C4 |
| Frattini | C5 | C3⋊#14546128 |
| socle | C3×C5×C109 | AGL(1,𝔽5) |
| prime radical, cosocle | C3×D25×C109 | C2 |
| LCS | G ▷ C3×C25×C109 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 8175 |
Polycyclic presentation:
2^2,3,5,109,5|a^2=b,d^5=f,c^a=c^2,d^a=d^2f,e^a=e^108,f^a=f^2,d^b=d^4f^4,f^b=f^42292 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32700, 36);