Finite Group DatabasebetaFinite Group Databasebeta| order | 6780 |
| automorphism | Aut(G) order 3037440 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic1695 |
| unambiguous semidirect product | C3⋊Dic565 |
| derived, Hall | C3×C5×C113 ⋊ C4 |
| Hall | C5×C113 ⋊ Dic3 |
| Hall | C3×C113 ⋊ Dic5 |
| Sylow | C113 ⋊ Dic15 |
| Hall | C3×C5 ⋊ Dic113 |
| Sylow | C5 ⋊ Dic339 |
| Sylow | C3 ⋊ Dic565 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D1695 |
| commutator, hypocenter | C3×C5×C113 | C4 |
| Fitting, socle | C2×C3×C5×C113 | C2 |
| LCS | G ▷ C3×C5×C113 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,5,113,2|a^2=e,b^a=b^2,c^a=c^4,d^a=d^1121698 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Dic5085, C3⋊Dic1695, C3×Dic1695, … |
| maximal normal subgroup of | C3×Dic1695, C5×Dic1695, Dic3390 |
| maximal quotient of | Dic5085, C3⋊Dic1695, C3×Dic1695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6780, 8);