Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #89322⋊C3 |
| Hall | #89322 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #342 |
| Sylow | C7 ⋊ C3×#342 |
284 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 80 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:
2^2,3,2^4,7|a^2=f,b^2=d,b^a=be,d^a=dg,h^a=h^6,e^b=eg,h^c=h^286 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#342, #72276505, #72276608, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#342, #72276505, #72276608, … |
| maximal quotient of | #72277053, #72277603 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 87);