Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C4 |
| unambiguous semidirect product | #89711⋊C3 |
| Fitting | C2×C42×C7 ⋊ C2×C3 |
| Hall | #89711 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #390 |
| Sylow | C7 ⋊ C3×#390 |
516 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^3,3,2^3,7|b^2=g,c^2=e,b^a=bg,c^a=cf,h^a=h^6,h^d=h^2116 conjugacy classes — view the full character table.
| maximal subgroup of | #72259738, AGL(1,𝔽7)×#390, #72259933, … |
| maximal normal subgroup of | #72259738, AGL(1,𝔽7)×#390, #72259933, … |
| maximal quotient of | #72228933, #72276814 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1128);