Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | ΓL(1,𝔽8)×#344 |
| unambiguous semidirect product | (C7×#344)⋊C3 |
| Fitting, Hall | C7×#344 ⋊ C3 |
| Sylow | C7 ⋊ C3×#344 |
180 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(2,3,4)(5,7,6)(8,9)(10,16,17,13)(11,15)(12,14)(18,22,23,20)(19,21);(1,2,3,5,4,6,7)(8,10,11,18,12,17,19,23)(9,13,15,20,14,16,21,22)110 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#1914, AGL(1,𝔽7)×#344, #72229510, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#1914, AGL(1,𝔽7)×#344, #72229510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 361);