Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #90212⋊C3 |
| Fitting | C7×#117 ⋊ C2×C3 |
| Hall | #90212 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #518 |
| Sylow | C7 ⋊ C3×#518 |
676 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 272 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^4,3,2^2,7|c^2=g,d^2=f,c^a=cg,h^a=h^6,d^b=dg,h^e=h^2122 conjugacy classes — view the full character table.
| maximal subgroup of | #72229268, AGL(1,𝔽7)×#518, #72229315, … |
| maximal normal subgroup of | #72229268, AGL(1,𝔽7)×#518, #72229315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6547);