Finite Group DatabasebetaFinite Group Databasebeta| order | 12300 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D75⋊D41 |
| unambiguous semidirect product | D123⋊D25 |
| unambiguous semidirect product | D1025⋊D3 |
| Fitting, derived, Hall | C3×C25×C41 ⋊ C22 |
| Hall | C25×C41 ⋊ C2×D3 |
| Hall | C3×C41 ⋊ C2×D25 |
| Hall | C3×C25 ⋊ C2×D41 |
| Sylow | C41 ⋊ D3×D25 |
| Sylow | C25 ⋊ D3×D41 |
| Sylow | C3 ⋊ D25×D41 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C25×C41 | C22 |
| Frattini | C5 | D15⋊D41 |
| socle | C3×C5×C41 | C2×D5 |
| LCS | G ▷ C3×C25×C41 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,5,41,5|d^5=f^4,d^a=d^4f,e^a=e^40,f^a=f^4,c^b=c^2,d^b=d^4f,f^b=f^4822 conjugacy classes — view the full character table.
| maximal subgroup of | D225⋊D41, C3×D75⋊D41, D1025⋊(C3⋊D3), … |
| maximal normal subgroup of | C3×D75⋊D41 |
| maximal quotient of | D225⋊D41, C3×D75⋊D41, D1025⋊(C3⋊D3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12300, 40);