Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of #6715 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#6715 |
| Fitting | C3×C5×C7×D4 ⋊ C2 |
| Hall | C3×C5×C7 ⋊ D8 |
| Hall | C5×C7 ⋊ #213 |
| Hall | C3×C7 ⋊ #686 |
| Hall | C3×C5 ⋊ C7×D8 |
| Sylow | C5 ⋊ C7×#213 |
| Sylow | C3 ⋊ C7×#686 |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 840 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,4,5,6,7)(8,9,11,12,10)(13,14,15)(17,20)(18,23)(21,22);(9,10)(11,12)(14,15)(16,17)(18,20)(19,22)(21,23)294 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 368);