Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×ΓL(1,𝔽8)×SL(2,ℤ/4) |
| derived | C7×A4 ⋊ C2×C3×C4 |
| Sylow | C7 ⋊ C2×C3×SL(2,ℤ/4) |
272 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 168 |
A minimal faithful permutation representation — 2 generators on 17 points:
(3,4,5,6)(7,8,10,9)(12,13,14)(15,17,16);(1,2)(8,10,9)(11,12,14,16,13,15,17)100 conjugacy classes — view the full character table.
| maximal subgroup of | C2×ΓL(1,𝔽8)×GL(2,ℤ/4), C2×AGL(1,𝔽7)×SL(2,ℤ/4), C2×#72222255 |
| maximal normal subgroup of | C2×ΓL(1,𝔽8)×GL(2,ℤ/4), C2×AGL(1,𝔽7)×SL(2,ℤ/4), C2×#72222255 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,5,6)(7,8,10,9)(12,13,14)(15,17,16), (1,2)(8,10,9)(11,12,14,16,13,15,17));