Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | #89773⋊C3 |
| Fitting | C7×#117 ⋊ C2×C3 |
| Hall | #89773 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #418 |
| Sylow | C7 ⋊ C3×#418 |
380 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 104 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:
2^3,3,2^3,7|c^2=e,f^2=g,c^a=cf,f^a=fg,h^a=h^6,c^b=cg,f^c=fg,h^d=h^286 conjugacy classes — view the full character table.
| maximal subgroup of | #72218348, #72218555, C4×#6512⋊C3, … |
| maximal normal subgroup of | #72218555, C4×#6512⋊C3, AGL(1,𝔽7)×#418, … |
| maximal quotient of | #72218348 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1190);