Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #90013⋊C3 |
| Fitting | C7×H(5,𝔽2) ⋊ C2×C3 |
| Hall | #90013 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #453 |
| Sylow | C7 ⋊ C3×#453 |
336 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 56 |
Polycyclic presentation:
2^3,3,2^3,7|f^2=g,b^a=bf,c^a=ce,f^a=fg,h^a=h^6,e^b=eg,f^b=fg,f^c=fg,h^d=h^265 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#453, #72214048, #72214911, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#453, C2×#90013⋊C3, #72229686, … |
| maximal quotient of | #72214911, C2×#90013⋊C3, C3×#90013⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1431);