Finite Group DatabasebetaFinite Group Databasebeta| order | 10686 |
| automorphism | Aut(G) order 17439552 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D5343 |
| unambiguous semidirect product | C3⋊D1781 |
| unambiguous semidirect product | C13⋊D411 |
| unambiguous semidirect product | C137⋊D39 |
| Fitting, socle, derived, Hall | C3×C13×C137 ⋊ C2 |
| Hall | C13×C137 ⋊ D3 |
| Hall | C3×C137 ⋊ D13 |
| Sylow | C137 ⋊ D39 |
| Hall | C3×C13 ⋊ D137 |
| Sylow | C13 ⋊ D411 |
| Sylow | C3 ⋊ D1781 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C13×C137 | C2 |
| LCS | G ▷ C3×C13×C137 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,13,137|b^a=b^2,c^a=c^12,d^a=d^1362673 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D39×D137, D13×D411, D3×D1781, … |
| maximal normal subgroup of | D39×D137, D13×D411, D3×D1781, … |
| maximal quotient of | Dic5343, C2×D5343, D16029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10686, 9);