Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | C8×Q8×ΓL(1,𝔽8) |
| Fitting, Hall | C7×C8×Q8 ⋊ C3 |
| Sylow | C7 ⋊ C3×C8×Q8 |
204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7,8)(18,19,20)(21,23,22);(9,10,11,12)(13,16,15,14)(17,18,19,21,20,22,23);(9,13,11,15)(10,14,12,16)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4265);