Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | ΓL(1,𝔽8)×#389 |
| unambiguous semidirect product | (C7×#389)⋊C3 |
| Fitting, Hall | C7×#389 ⋊ C3 |
| Sylow | C7 ⋊ C3×#389 |
340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,3,4)(5,7,6)(9,11)(13,15)(16,17,19,18);(1,2,3,5,4,6,7)(12,13)(14,15)(16,18,19,17);(8,9,10,11)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4209);