Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of C22⋊(C22⋊ΓL(1,𝔽8)) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5 · Schur multiplier (H2) = C24 |
| Fitting, socle, Hall | C24×C5×C7 ⋊ C3 |
| derived, Hall | C24×C7 ⋊ C3×C5 |
| Hall | C24×C5 ⋊ ΓL(1,𝔽8) |
| Hall | C5×C7 ⋊ C22⋊A4 |
| Sylow | C24 ⋊ C5×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C5×C22⋊A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | C22⋊(C22⋊ΓL(1,𝔽8)) |
| commutator, cosocle, hypocenter | C24×C7 | C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C24×C5×C7 | C3 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 ◁ C5 | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 420 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4,5)(7,8,9)(11,12,13)(15,16,17)(18,20,19);(10,13)(11,12)(14,15,16,18,17,19,20);(6,9)(7,8)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 997);