Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #89917⋊C3 |
| Fitting | C7×#131 ⋊ C2×C3 |
| Hall | #89917 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #444 |
| Sylow | C7 ⋊ C3×#444 |
208 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 208 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) = 23 |
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^3,3,2^3,7|c^2=e,e^2=g,f^2=g,b^a=bf,f^a=fg,h^a=h^6,c^b=cg,f^b=fg,h^d=h^286 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#444, #72229627, #72229686 |
| maximal normal subgroup of | AGL(1,𝔽7)×#444, #72229627, #72229686 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1334);