Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×ΓL(1,𝔽8)×C22⋊Q8 |
| unambiguous semidirect product | (C2×C7×C22⋊Q8)⋊C3 |
| Fitting, Hall | C2×C7×C22⋊Q8 ⋊ C3 |
| Sylow | C7 ⋊ C2×C3×C22⋊Q8 |
644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(16,17,18)(19,21,20);(4,6)(7,8,10,12)(9,13,14,11)(15,16,17,19,18,20,21);(7,9,10,14)(8,11,12,13);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 9973);