Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #89357⋊C3 |
| Hall | #89357 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #357 |
| Sylow | C7 ⋊ C3×#357 |
132 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 48 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:
2^2,3,2^4,7|a^2=e,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg,h^a=h^6,d^b=df,f^b=fg,h^c=h^280 conjugacy classes — view the full character table.
| maximal subgroup of | #72276619, AGL(1,𝔽7)×#357, #72276641 |
| maximal normal subgroup of | #72276619, AGL(1,𝔽7)×#357, #72276641 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 123);