Finite Group DatabasebetaFinite Group Databasebeta| order | 8268 |
| automorphism | Aut(G) order 5159232 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic2067 |
| unambiguous semidirect product | C3⋊Dic689 |
| derived, Hall | C3×C13×C53 ⋊ C4 |
| Hall | C13×C53 ⋊ Dic3 |
| Hall | C3×C53 ⋊ Dic13 |
| Sylow | C53 ⋊ Dic39 |
| Hall | C3×C13 ⋊ Dic53 |
| Sylow | C13 ⋊ Dic159 |
| Sylow | C3 ⋊ Dic689 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D2067 |
| commutator, hypocenter | C3×C13×C53 | C4 |
| Fitting, socle | C2×C3×C13×C53 | C2 |
| LCS | G ▷ C3×C13×C53 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,13,53,2|a^2=e,b^a=b^2,c^a=c^12,d^a=d^522070 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Dic6201, C3⋊(Dic689⋊C3), C3⋊Dic2067, … |
| maximal normal subgroup of | C3⋊(Dic689⋊C3), C3×Dic2067, Dic4134 |
| maximal quotient of | Dic6201, C3⋊Dic2067, C3×Dic2067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8268, 10);