Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#89701 |
| Hall | C3×C7 ⋊ #391 |
| Sylow | C7 ⋊ C3×#391 |
372 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1344 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(4,5,7,9,10,8,6)(11,12,13,14)(16,18);(19,21,22,26)(20,23,24,25);(5,6)(7,8)(9,10)(15,16)(17,18)(19,20,22,24)(21,25,26,23)372 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4649);