Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | ΓL(1,𝔽8)×#385 |
| unambiguous semidirect product | (C7×#385)⋊C3 |
| Fitting, Hall | C7×#385 ⋊ C3 |
| Sylow | C7 ⋊ C3×#385 |
468 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 224 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,3,4)(5,7,6)(9,11)(13,15)(16,17,19,18);(1,2,3,5,4,6,7)(12,13)(14,15)(16,18,19,17);(8,9)(10,11)(12,13)(14,15)140 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#65699, C4×ΓL(1,𝔽8)×#120, #72218607, … |
| maximal normal subgroup of | C4×ΓL(1,𝔽8)×#120, #72218607, AGL(1,𝔽7)×#385 |
| maximal quotient of | ΓL(1,𝔽8)×#65699 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4205);