Finite Group DatabasebetaFinite Group Databasebeta| order | 5166 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊D861 |
| unambiguous semidirect product | C7⋊(C3⋊D123) |
| unambiguous semidirect product | C41⋊(C3⋊D21) |
| Fitting, socle, derived, Hall | C32×C7×C41 ⋊ C2 |
| Hall | C32×C41 ⋊ D7 |
| Hall | C7×C41 ⋊ C3⋊D3 |
| Hall | C32×C7 ⋊ D41 |
| Sylow | C41 ⋊ C3⋊D21 |
| Sylow | C32 ⋊ D287 |
| Sylow | C7 ⋊ C3⋊D123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C7×C41 | C2 |
| LCS | G ▷ C32×C7×C41 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
Polycyclic presentation:
2,3^2,7,41|b^a=b^2,c^a=c^2,d^a=d^6,e^a=e^401293 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3⋊(C3⋊(C7⋊(D41⋊C5))), C5×C3⋊D861, C3⋊D4305, … |
| maximal normal subgroup of | C3⋊(C3⋊(C7⋊(D41⋊C5))), C5×C3⋊D861, C7×C3⋊D861 |
| maximal quotient of | C5×C3⋊D861, C3⋊D4305, C3⋊D6027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5166, 32);