Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C3×#10620789 |
| Fitting | C24×C3×C7 ⋊ C2×C3 |
| Hall | #6516 ⋊ C32 |
| Hall | C3×ΓL(1,𝔽8) ⋊ #120 |
| Sylow | C7 ⋊ C32×#120 |
780 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 186 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:
(1,2,5,6,4,7,3)(13,15)(16,17,18);(2,6,5,3,4,7)(9,11)(12,13)(14,15);(8,9)(10,11)174 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#120×AGL(1,𝔽7), D3×#10620789, #72244206, … |
| maximal normal subgroup of | C3×#120×AGL(1,𝔽7), D3×#10620789, #72244206, … |
| maximal quotient of | C3×#90013⋊C3, #72246101, C3×#13052144, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,5,6,4,7,3)(13,15)(16,17,18), (2,6,5,3,4,7)(9,11)(12,13)(14,15), (8,9)(10,11));