Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | ΓL(1,𝔽8)×#354 |
| unambiguous semidirect product | (C7×#354)⋊C3 |
| Fitting, Hall | C7×#354 ⋊ C3 |
| Sylow | C7 ⋊ C3×#354 |
140 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(2,3,4)(5,7,6)(8,9)(10,16,18,23)(11,19,21,14)(12,15)(13,17)(20,22);(1,2,3,5,4,6,7)(8,10,12,18)(9,13,15,22)(11,20,21,17)(14,23,19,16)65 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#354, #72276879, ΓL(1,𝔽8)×#2199 |
| maximal normal subgroup of | AGL(1,𝔽7)×#354, #72276879, ΓL(1,𝔽8)×#2199 |
| maximal quotient of | ΓL(1,𝔽8)×#1393 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 371);