Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×ΓL(1,𝔽8)×#104 |
| unambiguous semidirect product | (C2×C7×#104)⋊C3 |
| Fitting, Hall | C2×C7×#104 ⋊ C3 |
| Sylow | C7 ⋊ C2×C3×#104 |
340 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 123 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 — 3 generators on 17 points:
(1,2)(12,13,14)(15,17,16);(3,4,6,9,7,5,10,8)(11,12,13,15,14,16,17);(3,5)(4,7)(6,8)(9,10)140 conjugacy classes — view the full character table.
| maximal subgroup of | C2×ΓL(1,𝔽8)×#453, ΓL(1,𝔽8)×#2002, ΓL(1,𝔽8)×#1906, … |
| maximal normal subgroup of | C2×ΓL(1,𝔽8)×#453, ΓL(1,𝔽8)×#2002, ΓL(1,𝔽8)×#1906, … |
| maximal quotient of | D3×ΓL(1,𝔽8)×#104 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4240);