Finite Group DatabasebetaFinite Group Databasebeta| order | 32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C109 · Schur multiplier (H2) = C1 |
| direct product | C5×ΓL(1,𝔽16)×C109 |
| unambiguous semidirect product | C3⋊(C5×AGL(1,𝔽5)×C109) |
| Fitting, socle, Hall | C3×C52×C109 ⋊ C4 |
| derived | C3×C5 ⋊ C4×C5×C109 |
| Hall | C52×C109 ⋊ Dic3 |
| Hall | C3×C109 ⋊ C5×AGL(1,𝔽5) |
| Hall | C3×C52 ⋊ C4×C109 |
| Sylow | C52 ⋊ Dic3×C109 |
| Sylow | C3 ⋊ C5×AGL(1,𝔽5)×C109 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C109 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C4×C5×C109 |
| Frattini | C1 | G |
| Fitting, socle | C3×C52×C109 | C4 |
| prime radical | C3×C5×D5×C109 | C2 |
| cosocle | C3×D5 | C2×C5×C109 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C5×C109 | |
52 conjugacy classes of subgroups.
| transitive degree | μt(G) = 6540 |
4905 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32700, 171);