Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #89519⋊C3 |
| Fitting | C7×#100 ⋊ C2×C3 |
| Hall | #89519 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #453 |
| Sylow | C7 ⋊ C3×#453 |
336 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 66 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|c^2=fg,f^2=g,c^a=cf,e^a=eg,f^a=fg,h^a=h^6,c^b=ce,f^b=fg,e^c=eg,h^d=h^259 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#453, #72214911, #72229499, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#453, #72229499, C2×#89519⋊C3, … |
| maximal quotient of | #72214911, #72229493, C2×#89519⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 936);