Finite Group DatabasebetaFinite Group Databasebeta| order | 12300 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×#11717961 |
| Fitting, socle, Hall | C3×C52×C41 ⋊ C4 |
| derived | C5×C41 ⋊ C3×C4×C5 |
| Hall | C52×C41 ⋊ C3×C4 |
| Hall | C3×C41 ⋊ C5×AGL(1,𝔽5) |
| Hall | C3×C52 ⋊ #4223 |
| Sylow | C41 ⋊ C3×C5×AGL(1,𝔽5) |
| Sylow | C52 ⋊ C3×#4223 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | #11717961 |
| commutator, hypocenter | C5×C41 | C3×C4×C5 |
| Frattini | C1 | G |
| Fitting, socle | C3×C52×C41 | C4 |
| prime radical | C3×C5×D205 | C2 |
| cosocle | D205 | C2×C3×C5 |
| LCS | G ▷ C5×C41 | |
| UCS | C1 ◁ C3×C5 | |
52 conjugacy classes of subgroups.
825 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12300, 126);