Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C3×ΓL(1,𝔽8)×#120 |
| Fitting | C3×C7×#120 ⋊ C3 |
| Hall | C7×#120 ⋊ C32 |
| Sylow | C7 ⋊ C32×#120 |
780 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,4,6,3,5,7)(13,15)(16,17,18);(2,3,4)(5,7,6)(9,11)(12,13)(14,15);(8,9)(10,11)210 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,6,3,5,7)(13,15)(16,17,18), (2,3,4)(5,7,6)(9,11)(12,13)(14,15), (8,9)(10,11));