Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C2 |
| direct product | C7×D5⋊(C3⋊C8) |
| unambiguous semidirect product | D5⋊(C7×C3⋊C8) |
| derived, Hall | C3×C5 ⋊ C2×C7×C8 |
| Hall | C3×C5×C7 ⋊ C2×C8 |
| Hall | C5×C7 ⋊ C2×C3⋊C8 |
| Hall | C3×C7 ⋊ D5⋊C8 |
| Sylow | C5 ⋊ C2×C7×C3⋊C8 |
| Sylow | C3 ⋊ C7×D5⋊C8 |
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)(8,9,12,10,11)(13,14,15)(16,18,20,19)(17,21,23,22);(9,10,11,12)(14,15)(16,17,19,22,20,23,18,21)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 637);