Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D7⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×Dic7⋊A4 |
| unambiguous semidirect product | Dic7⋊(C5×A4) |
| unambiguous semidirect product | (C22⋊ΓL(1,𝔽8))⋊(C4×C5) |
| unambiguous semidirect product | (C5×Dic7)⋊A4 |
| derived | C22×C7 ⋊ C3×C4×C5 |
| Hall | C22×C5×Dic7 ⋊ C3 |
| Hall | C22×Dic7 ⋊ C3×C5 |
| Hall | C5×C7 ⋊ C4×A4 |
| Sylow | C7 ⋊ C4×C5×A4 |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 420 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4,5)(7,11,10,8,9,12)(13,14,15,16)(18,19,20);(6,7,10,11,9,12,8)(17,20)(18,19)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 610);