Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #89924⋊C3 |
| Hall | #89924 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #442 |
| Sylow | C7 ⋊ C3×#442 |
264 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 92 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,e^2=g,c^a=cg,h^a=h^6,c^b=ce,e^b=eg,e^c=eg,h^d=h^280 conjugacy classes — view the full character table.
| maximal subgroup of | #72218030, AGL(1,ℤ/8)×ΓL(1,𝔽8)⋊C4, C4×#6474⋊C3, … |
| maximal normal subgroup of | AGL(1,ℤ/8)×ΓL(1,𝔽8)⋊C4, C4×#6474⋊C3, AGL(1,𝔽7)×#442, … |
| maximal quotient of | #72218030, #72228667, #72260015 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1338);