Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#5834 |
| Fitting | C2×C3×C7×#41 ⋊ C2 |
| Hall | C3×C7 ⋊ #550 |
| Sylow | C3 ⋊ C7×#550 |
624 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3,4,5,6,7)(8,9,10)(12,14)(15,17,18,21)(16,20,19,22);(12,14)(16,19)(20,22);(12,14)(15,16)(17,20)(18,19)(21,22);(9,10)(11,12)(13,14)(17,21)(20,22)315 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10839);