Finite Group DatabasebetaFinite Group Databasebeta| order | 32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×AGL(1,𝔽5)×C109⋊C3 |
| unambiguous semidirect product | (AGL(1,𝔽5)×C109)⋊(C3×C5) |
| unambiguous semidirect product | (C5×AGL(1,𝔽5)×C109)⋊C3 |
| Fitting, socle, Hall | C52×C109 ⋊ C3×C4 |
| derived | C5×C109 ⋊ C3×C4×C5 |
| Hall | C5×AGL(1,𝔽5)×C109 ⋊ C3 |
| Hall | C52×C109⋊C3 ⋊ C4 |
| Sylow | C109 ⋊ C3×C5×AGL(1,𝔽5) |
| Sylow | C52 ⋊ C4×C109⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | AGL(1,𝔽5)×C109⋊C3 |
| commutator, hypocenter | C5×C109 | C3×C4×C5 |
| Frattini | C1 | G |
| Fitting, socle | C52×C109 | C3×C4 |
| prime radical | C5×D5×C109⋊C3 | C2 |
| cosocle | D5×C109 | C2×C3×C5 |
| LCS | G ▷ C5×C109 | |
| UCS | C1 ◁ C5 | |
52 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2180 |
975 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32700, 110);