Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C23 |
| direct product | ΓL(1,𝔽8)×#342 |
| unambiguous semidirect product | (C7×#342)⋊C3 |
| Fitting, Hall | C7×#342 ⋊ C3 |
| Sylow | C7 ⋊ C3×#342 |
284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 112 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,3,4)(5,7,6)(12,13)(14,18,19,16)(15,17);(1,2,3,5,4,6,7)(8,9,10,11)(12,14)(13,16)(15,19)(17,18)110 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#342, #72276565, ΓL(1,𝔽8)×#2021, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#342, #72276565, ΓL(1,𝔽8)×#2021, … |
| maximal quotient of | ΓL(1,𝔽8)×#1393, ΓL(1,𝔽8)×#1304 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 359);