Finite Group DatabasebetaFinite Group Databasebeta| order | 12300 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#11717961⋊C5) |
| unambiguous semidirect product | (C3⋊#11717961)⋊C5 |
| Fitting, socle, derived | C3×C5×C41 ⋊ C4×C5 |
| Hall | C3×C5×C41⋊C5 ⋊ C4 |
| Hall | C5×C41⋊C5 ⋊ Dic3 |
| Hall | C3×C41 ⋊ C5×AGL(1,𝔽5) |
| Sylow | C41 ⋊ C5×ΓL(1,𝔽16) |
| Sylow | C3 ⋊ #11717961⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C41 | C4×C5 |
| prime radical | C3×C5⋊(D41⋊C5) | C2 |
| cosocle | C3×D205 | C2×C5 |
| LCS | G ▷ C3×C5×C41 | |
| UCS | C1 | |
52 conjugacy classes of subgroups.
Polycyclic presentation:
2,5,2,3,5,41|a^2=c,d^a=d^2,e^a=e^2,f^a=f^32,f^b=f^10,e^c=e^4,f^c=f^4075 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12300, 85);