Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | ΓL(1,𝔽8)×#518 |
| unambiguous semidirect product | (C7×#518)⋊C3 |
| Fitting, Hall | C7×#518 ⋊ C3 |
| Sylow | C7 ⋊ C3×#518 |
676 conjugacy classes of subgroups; only the 363 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 363 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 19 points:
(2,3,4)(5,7,6)(8,9,10,11)(12,13)(14,18)(15,16)(17,19);(1,2,3,5,4,6,7)(13,16)(17,18);(12,13)(14,17)(15,16)(18,19);(12,14)(13,17)(15,19)(16,18)170 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#3484, AGL(1,𝔽7)×#518, #72229295, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#3484, AGL(1,𝔽7)×#518, #72229295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 9968);