Finite Group DatabasebetaFinite Group Databasebeta| order | 10686 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C13⋊C3)⋊D137 |
| unambiguous semidirect product | C137⋊(D13⋊C3) |
| unambiguous semidirect product | D1781⋊C3 |
| Fitting, socle, derived, Hall | C13×C137 ⋊ C2×C3 |
| Hall | C13⋊C3×C137 ⋊ C2 |
| Hall | D1781 ⋊ C3 |
| Sylow | C137 ⋊ D13⋊C3 |
| Hall | C13⋊C3 ⋊ D137 |
| Sylow | C13 ⋊ C3×D137 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C13×C137 | C2×C3 |
| LCS | G ▷ C13×C137 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,13,137|c^a=c^12,d^a=d^136,c^b=c^3484 conjugacy classes — view the full character table.
| maximal subgroup of | #20297117⋊C3, #20297118⋊C3, D13⋊C3×D137, … |
| maximal normal subgroup of | #20297117⋊C3, #20297118⋊C3, D13⋊C3×D137, … |
| maximal quotient of | Dic1781⋊C3, C2×(C13⋊C3)⋊D137, (C13⋊C9)⋊D137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10686, 12);