Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | ΓL(1,𝔽8)×#440 |
| unambiguous semidirect product | (C7×#440)⋊C3 |
| Fitting, Hall | C7×#440 ⋊ C3 |
| Sylow | C7 ⋊ C3×#440 |
240 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 105 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)(8,9,11,10)(12,13)(14,16)(15,17)(18,19);(1,2,3,5,4,6,7)(13,16)(14,19)(17,18);(8,10,11,9)(13,17)(16,18)110 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#65735, C4×ΓL(1,𝔽8)×AGL(1,ℤ/8), #72218565, … |
| maximal normal subgroup of | C4×ΓL(1,𝔽8)×AGL(1,ℤ/8), #72218565, AGL(1,𝔽7)×#440 |
| maximal quotient of | ΓL(1,𝔽8)×#65735 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4260);