Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C22×#41×ΓL(1,𝔽8) |
| Fitting, Hall | C22×C7×#41 ⋊ C3 |
| Sylow | C7 ⋊ C22×C3×#41 |
1780 conjugacy classes of subgroups; only the 1155 normal subgroups are shown.
This group has 1780 conjugacy classes of subgroups — too many to draw. Only its 1155 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 — 5 generators on 19 points:
(1,2)(14,15,16)(17,19,18);(3,4)(13,14,15,17,16,18,19);(5,6)(7,10)(8,9)(11,12);(6,9)(10,12);(5,7,8,11)(6,10,9,12)200 conjugacy classes — view the full character table.
| maximal subgroup of | C22×ΓL(1,𝔽8)×H(5,𝔽2), C22×#41×AGL(1,𝔽7), C22×#10621518 |
| maximal normal subgroup of | C22×ΓL(1,𝔽8)×H(5,𝔽2), C22×#41×AGL(1,𝔽7), C22×#10621518 |
| maximal quotient of | ΓL(1,𝔽8)×#3490 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11585);