Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #89979⋊C3 |
| Hall | #89979 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #407 |
| Sylow | C7 ⋊ C3×#407 |
380 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 124 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|a^2=f,f^2=g,b^a=bg,c^a=ce,h^a=h^6,h^d=h^2116 conjugacy classes — view the full character table.
| maximal subgroup of | #72230218, AGL(1,𝔽7)×#407, #72276592, … |
| maximal normal subgroup of | #72230218, AGL(1,𝔽7)×#407, #72276592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1396);