Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #90000⋊C3 |
| Hall | #90000 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #451 |
| Sylow | C7 ⋊ C3×#451 |
296 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 82 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=f,b^2=g,c^2=g,e^2=g,b^a=bf,c^a=ce,e^a=eg,h^a=h^6,c^b=cg,e^c=eg,h^d=h^277 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1415);