Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #89919⋊C3 |
| Fitting | C2×C7×D8 ⋊ C2×C3 |
| Hall | #89919 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #509 |
| Sylow | C7 ⋊ C3×#509 |
180 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 58 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|b^2=g,e^2=fg,f^2=g,c^a=ce,e^a=ef,f^a=fg,h^a=h^6,c^b=cg,e^c=ef,f^c=fg,h^d=h^262 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#509, #72228671, #72228678, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#509, #72228671, #72228678, … |
| maximal quotient of | #72276619 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1343);