Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #89352⋊C3 |
| Hall | #89352 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ H(3,ℤ/4) |
| Sylow | C7 ⋊ C3×H(3,ℤ/4) |
188 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 74 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,d^2=g,e^2=g,b^a=be,d^a=dg,h^a=h^6,f^b=fg,h^c=h^286 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×H(3,ℤ/4), #72277216, #72277255, … |
| maximal normal subgroup of | AGL(1,𝔽7)×H(3,ℤ/4), #72277216, #72277255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 117);