Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#4685 |
| Fitting | C3×C4×C7×C8 ⋊ C2 |
| Hall | C3×C7 ⋊ #431 |
| Sylow | C3 ⋊ C7×#431 |
236 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 672 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3,4,5,6,7)(8,9,10)(11,12,13,15,14,16,17,18)(20,24)(23,26);(19,21,22,25)(20,23,24,26);(9,10)(19,20)(21,23)(22,24)(25,26)504 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 5009);