Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | ΓL(1,𝔽8)×#417 |
| unambiguous semidirect product | (C7×#417)⋊C3 |
| Fitting, Hall | C7×#417 ⋊ C3 |
| Sylow | C7 ⋊ C3×#417 |
324 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 123 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)(8,9,10,11)(13,16)(14,18)(17,19);(1,2,3,5,4,6,7)(12,13)(14,19)(15,17)(16,18);(13,17)(16,19)110 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#65793, C4×ΓL(1,𝔽8)×AGL(1,ℤ/8), #72218610, … |
| maximal normal subgroup of | C4×ΓL(1,𝔽8)×AGL(1,ℤ/8), #72218610, AGL(1,𝔽7)×#417, … |
| maximal quotient of | ΓL(1,𝔽8)×#65793 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4237);