Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#1161 |
| Fitting | C3×C5×C7×Q8 ⋊ C2 |
| Hall | C3×C5×C7 ⋊ #41 |
| Hall | C5×C7 ⋊ C3×#41 |
| Hall | C3×C7 ⋊ C5×#41 |
| Sylow | C7 ⋊ C3×C5×#41 |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 840 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5,6,7,8)(9,10,12,14,15,13,11)(16,18,19,22)(17,20,21,23);(16,17,19,21)(18,23,22,20);(10,11)(12,13)(14,15)(18,22)(20,23)375 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 857);