Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #120⋊ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #379⋊ΓL(1,𝔽8) |
| Fitting, Hall | C7×#379 ⋊ C3 |
| derived | C24×C7 ⋊ C3×C4 |
| Sylow | #379 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ #379⋊C3 |
212 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 84 |
Polycyclic presentation:
2,3,2^5,7|a^2=c,d^a=df,e^a=efg,d^b=de,e^b=d,f^b=g,g^b=fg,h^b=h^276 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#379⋊C3, #72277208, #72277823, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#379⋊C3, #72277208, #72277823, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4351);