Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #90363⋊C3 |
| Hall | #90363 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #563 |
| Sylow | C7 ⋊ C3×#563 |
392 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 162 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:
2^4,3,2^2,7|a^2=g,c^2=g,d^2=g,c^a=cg,d^a=dfg,h^a=h^6,c^b=cf,d^b=dg,d^c=dg,h^e=h^271 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6696);