Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊(C7⋊GL(2,𝔽3)) |
| Fitting | C4×C7×Q8 ⋊ D3 |
| Sylow | C7 ⋊ C4⋊GL(2,𝔽3) |
182 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 31 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:
2^3,3,7,2^3|b^2=c,f^2=h,g^2=h,b^a=bc,d^a=d^2,e^a=e^6,f^a=fg,g^a=gh,f^d=g,g^d=fgh,g^f=gh107 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6451);