Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #89353⋊C3 |
| Hall | #89353 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #341 |
| Sylow | C7 ⋊ C3×#341 |
180 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 180 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.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^2,3,2^4,7|a^2=f,b^2=d,d^2=g,e^2=g,f^2=g,b^a=be,h^a=h^6,e^b=eg,f^b=fg,h^c=h^2104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 119);