Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D5×AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#63304⋊C3 |
| unambiguous semidirect product | (C2×#63304)⋊C3 |
| derived, Hall | C5×C7 ⋊ C22×C3×C4 |
| Hall | C2×#63304 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C22×C4 |
| Hall | ΓL(1,𝔽8) ⋊ C2×C4×D5 |
| Sylow | C7 ⋊ C2×C3×C4×D5 |
| Sylow | C5 ⋊ C2×C4×AGL(1,𝔽7) |
216 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 280 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×D5×AGL(1,𝔽7), C2×#10587963⋊C3, #72218149, … |
| maximal normal subgroup of | C2×C4×D5×AGL(1,𝔽7), C2×#10587963⋊C3, #72218149, … |
| maximal quotient of | #72217890, #72218149 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 450);