Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C4 |
| unambiguous semidirect product | #89993⋊C3 |
| Hall | #89993 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #390 |
| Sylow | C7 ⋊ C3×#390 |
516 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 144 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|a^2=g,b^a=bf,c^a=cf,h^a=h^6,c^b=ce,h^d=h^2104 conjugacy classes — view the full character table.
| maximal subgroup of | #72259738, AGL(1,𝔽7)×#390, #72276674, … |
| maximal normal subgroup of | #72259738, AGL(1,𝔽7)×#390, #72276674, … |
| maximal quotient of | #72230204 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1409);