Finite Group DatabasebetaFinite Group Databasebeta| order | 7068 |
| automorphism | Aut(G) order 3816720 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic1767 |
| unambiguous semidirect product | C3⋊Dic589 |
| unambiguous semidirect product | C19⋊Dic93 |
| unambiguous semidirect product | C31⋊Dic57 |
| derived, Hall | C3×C19×C31 ⋊ C4 |
| Hall | C19×C31 ⋊ Dic3 |
| Hall | C3×C31 ⋊ Dic19 |
| Hall | C3×C19 ⋊ Dic31 |
| Sylow | C31 ⋊ Dic57 |
| Sylow | C19 ⋊ Dic93 |
| Sylow | C3 ⋊ Dic589 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D1767 |
| commutator, hypocenter | C3×C19×C31 | C4 |
| Fitting, socle | C2×C3×C19×C31 | C2 |
| LCS | G ▷ C3×C19×C31 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,19,31,2|a^2=e,b^a=b^2,c^a=c^18,d^a=d^301770 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Dic5301, C3⋊#20296620, C3⋊#20296619, … |
| maximal normal subgroup of | C3⋊#20296620, C3⋊#20296619, C3⋊((C19⋊C3)⋊Dic31), … |
| maximal quotient of | Dic5301, C3⋊Dic1767, C3×Dic1767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7068, 12);