Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of Dic42 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×#66622 |
| Fitting | C3×C5×C7×C8 ⋊ C2 |
| Hall | C3×C5×C7 ⋊ M16 |
| Hall | C5×C7 ⋊ ΓL(1,𝔽25) |
| Hall | C3×C7 ⋊ C5×M16 |
| Hall | C3×C5 ⋊ #1131 |
| Sylow | C7 ⋊ C5×ΓL(1,𝔽25) |
| Sylow | C3 ⋊ C5×#1131 |
80 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)(6,7,9,11,12,10,8)(13,14,15)(16,17,18,21,19,20,22,23);(7,8)(9,10)(11,12)(14,15)(17,20)(21,23)450 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 342);