Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #10587987⋊C3 |
| Fitting | C2×C4×C5×C7 ⋊ C2×C3 |
| Hall | #10587987 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ #41 |
| Hall | C5×C7 ⋊ C3×#41 |
| Hall | ΓL(1,𝔽8) ⋊ #709 |
| Sylow | C7 ⋊ C3×#709 |
| Sylow | C5 ⋊ #1157⋊C3 |
160 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 280 |
Polycyclic presentation:
2^3,3,2,5,7|b^2=e,c^a=ce,f^a=f^4,g^a=g^6,g^d=g^2118 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1157⋊C3, #72218044, #72218046, … |
| maximal normal subgroup of | D5×#1157⋊C3, #72218044, #72218046, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 598);