Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #578⋊ΓL(1,𝔽8) |
| Fitting, Hall | C7×#578 ⋊ C3 |
| Sylow | #578 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ #578⋊C3 |
146 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 32 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:
2^2,3,2^3,7,2|d^2=h,e^2=h,f^2=h,b^a=bd,d^a=dh,d^b=dh,e^c=ef,f^c=e,g^c=g^4,f^e=fh70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10085);