Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×ΓL(1,𝔽8)×#120 |
| Fitting, Hall | C2×C7×#120 ⋊ C3 |
| Sylow | C7 ⋊ C2×C3×#120 |
1324 conjugacy classes of subgroups; only the 315 normal subgroups are shown.
This group has 1324 conjugacy classes of subgroups — too many to draw. Only its 315 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(12,13,14)(15,17,16);(3,4)(5,7)(6,8)(9,10)(11,12,13,15,14,16,17);(4,7)(8,10);(4,8)(7,10)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 9971);