Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊Q8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#1157⋊C3 |
| unambiguous semidirect product | #1157⋊(C3×C5) |
| unambiguous semidirect product | (C5×#1157)⋊C3 |
| Fitting | C2×C4×C5×C7 ⋊ C2×C3 |
| Hall | C5×#1157 ⋊ C3 |
| Hall | #1157 ⋊ C3×C5 |
| Hall | C5×ΓL(1,𝔽8) ⋊ #41 |
| Hall | C5×C7 ⋊ C3×#41 |
| Hall | ΓL(1,𝔽8) ⋊ C5×#41 |
| Sylow | C7 ⋊ C3×C5×#41 |
160 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4,5)(6,7,12)(8,9,11)(14,18)(17,20);(13,15,16,19)(14,17,18,20);(7,8)(9,11)(10,12)(13,14)(15,17)(16,18)(19,20)190 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1157⋊C3, C5×#6471⋊C3, #72218047, … |
| maximal normal subgroup of | D5×#1157⋊C3, C5×#6471⋊C3, #72218047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 578);