Finite Group DatabasebetaFinite Group Databasebeta| order | 12300 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C25 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(Dic41⋊C25) |
| unambiguous semidirect product | Dic123⋊C25 |
| derived, Hall | C3×C41 ⋊ C4×C25 |
| Hall | C3×C41⋊C25 ⋊ C4 |
| Hall | C41⋊C25 ⋊ Dic3 |
| Hall | Dic123 ⋊ C25 |
| Sylow | C41 ⋊ Dic3×C25 |
| Sylow | C3 ⋊ Dic41⋊C25 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C5 | C3⋊(D41⋊C5) |
| commutator, hypocenter | C3×C41 | C4×C25 |
| Fitting, socle, prime radical, cosocle | C2×C3×C5×C41 | C2×C5 |
| LCS | G ▷ C3×C41 | |
| UCS | C1 ◁ C2×C5 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2,5,3,41,2,5|a^2=e,b^5=f,c^a=c^2,d^a=d^40,d^b=d^37270 conjugacy classes — view the full character table.
| maximal subgroup of | C9⋊(Dic41⋊C25), C3×C3⋊(Dic41⋊C25), C3⋊(C3⋊(Dic41⋊C25)) |
| maximal normal subgroup of | C3×C3⋊(Dic41⋊C25) |
| maximal quotient of | C9⋊(Dic41⋊C25), C3×C3⋊(Dic41⋊C25), C3⋊(C3⋊(Dic41⋊C25)) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12300, 4);