Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×#13052144 |
| Fitting | C2×C7×H(5,𝔽2) ⋊ C2×C3 |
| Hall | C2×#90016 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×B(4,𝔽2) |
| Sylow | C7 ⋊ C2×C3×B(4,𝔽2) |
2036 conjugacy classes of subgroups; only the 273 normal subgroups are shown.
This group has 2036 conjugacy classes of subgroups — too many to draw. Only its 273 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) = 112 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2,7)(3,4,6)(10,12)(11,14)(13,16)(15,17);(8,9);(2,3)(4,6)(5,7)(14,17);(10,11,13,15)(12,14,16,17)130 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,𝔽7)×B(4,𝔽2), D3×#13052144, C2×#72214654, … |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×B(4,𝔽2), #72276807, #72276818, … |
| maximal quotient of | D3×#13052144, #72276818, #72277880 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,7)(3,4,6)(10,12)(11,14)(13,16)(15,17), (8,9), (2,3)(4,6)(5,7)(14,17), (10,11,13,15)(12,14,16,17));