Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C22×C22⋊C4×ΓL(1,𝔽8) |
| Fitting, Hall | C22×C7×C22⋊C4 ⋊ C3 |
| Sylow | C7 ⋊ C22×C3×C22⋊C4 |
1348 conjugacy classes of subgroups; only the 507 normal subgroups are shown.
This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 507 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 — 4 generators on 19 points:
(1,2)(14,15,16)(17,19,18);(3,4)(13,14,15,17,16,18,19);(5,6,8,10)(7,9,11,12);(6,9)(10,12)200 conjugacy classes — view the full character table.
| maximal subgroup of | C22×ΓL(1,𝔽8)×#120, C22×C22⋊C4×AGL(1,𝔽7), C22×#6445⋊C3, … |
| maximal normal subgroup of | C22×ΓL(1,𝔽8)×#120, C22×C22⋊C4×AGL(1,𝔽7), C22×#6445⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 9962);