Finite Group DatabasebetaFinite Group Databasebeta| order | 6834 |
| automorphism | Aut(G) order 7216704 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D3417 |
| unambiguous semidirect product | C3⋊D1139 |
| unambiguous semidirect product | C17⋊D201 |
| unambiguous semidirect product | C67⋊D51 |
| Fitting, socle, derived, Hall | C3×C17×C67 ⋊ C2 |
| Hall | C17×C67 ⋊ D3 |
| Hall | C3×C67 ⋊ D17 |
| Sylow | C67 ⋊ D51 |
| Hall | C3×C17 ⋊ D67 |
| Sylow | C17 ⋊ D201 |
| Sylow | C3 ⋊ D1139 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C17×C67 | C2 |
| LCS | G ▷ C3×C17×C67 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,17,67|b^a=b^2,c^a=c^16,d^a=d^661710 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D51×D67, D17×D201, D3×D1139, … |
| maximal normal subgroup of | D51×D67, D17×D201, D3×D1139, … |
| maximal quotient of | Dic3417, C2×D3417, D10251, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6834, 9);