Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | ΓL(1,𝔽8)×#407 |
| unambiguous semidirect product | (C7×#407)⋊C3 |
| Fitting, Hall | C7×#407 ⋊ C3 |
| Sylow | C7 ⋊ C3×#407 |
380 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 135 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 — 3 generators on 19 points:
(2,3,4)(5,7,6)(8,9)(10,11)(12,13,14,17,15,16,18,19);(1,2,3,5,4,6,7)(9,11);(13,16)(17,19)140 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#1783, ΓL(1,𝔽8)×#3003, AGL(1,𝔽7)×#407, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#1783, ΓL(1,𝔽8)×#3003, AGL(1,𝔽7)×#407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4227);