Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | ΓL(1,𝔽8)×#353 |
| unambiguous semidirect product | (C7×#353)⋊C3 |
| Fitting, Hall | C7×#353 ⋊ C3 |
| Sylow | C7 ⋊ C3×#353 |
172 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 172 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) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 15 points:
(2,3,4)(5,7,6)(8,9)(10,13,15,12)(11,14);(1,2,3,5,4,6,7)(9,12)(10,15)(13,14)65 conjugacy classes — view the full character table.
| maximal subgroup of | #72276454, AGL(1,𝔽7)×#353, ΓL(1,𝔽8)×#2196 |
| maximal normal subgroup of | #72276454, AGL(1,𝔽7)×#353, ΓL(1,𝔽8)×#2196 |
| maximal quotient of | ΓL(1,𝔽8)×#1393 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 370);