Finite Group DatabasebetaFinite Group Databasebeta| order | 12300 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C25 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#4223⋊C25) |
| unambiguous semidirect product | (C3⋊#4223)⋊C25 |
| derived, Hall | C3×C41 ⋊ C4×C25 |
| Hall | C3×C41⋊C25 ⋊ C4 |
| Hall | C41⋊C25 ⋊ Dic3 |
| Hall | C3⋊#4223 ⋊ C25 |
| Sylow | C41 ⋊ Dic3×C25 |
| Sylow | C3 ⋊ #4223⋊C25 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C5 | C3⋊(#4223⋊C5) |
| commutator, hypocenter | C3×C41 | C4×C25 |
| Fitting, socle | C3×C5×C41 | C4×C5 |
| prime radical, cosocle | C3×C5×D41 | C2×C5 |
| LCS | G ▷ C3×C41 | |
| UCS | C1 ◁ C5 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2,5,2,3,41,5|a^2=c,b^5=f,d^a=d^2,e^a=e^32,e^b=e^10,e^c=e^40180 conjugacy classes — view the full character table.
| maximal subgroup of | C9⋊(#4223⋊C25), C3×C3⋊(#4223⋊C25), C3⋊(C3⋊(#4223⋊C25)) |
| maximal normal subgroup of | C3×C3⋊(#4223⋊C25) |
| maximal quotient of | C9⋊(#4223⋊C25), C3×C3⋊(#4223⋊C25), C3⋊(C3⋊(#4223⋊C25)) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12300, 15);