Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of S5 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×A5⋊C4 |
| unambiguous semidirect product | A5⋊(C4×C7) |
| derived | A5 ⋊ C4×C7 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2×C7 | S5 |
| commutator, hypocenter | A5 | C4×C7 |
| Frattini | C2 | C7×S5 |
| socle, prime radical | C2×C7×A5 | C2 |
| cosocle | C2×A5 | C2×C7 |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C2×C7 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,4,5,6,7)(12,13,14,15,16);(8,9,10,11)(13,15)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 398);