Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | ΓL(1,𝔽8)×#539 |
| unambiguous semidirect product | (C7×#539)⋊C3 |
| Fitting, Hall | C7×#539 ⋊ C3 |
| Sylow | C7 ⋊ C3×#539 |
436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 23 points:
(2,3,4)(5,7,6)(8,9)(10,11)(12,13)(14,15)(18,22)(20,23);(1,2,3,5,4,6,7)(9,13)(10,14)(16,17,19,21)(18,20,22,23);(8,10,12,14)(9,11,13,15)(16,18)(17,20)(19,22)(21,23);(8,11)(9,14)(10,13)(12,15)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 9989);