Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C34 |
| direct product | C3×D3×#6900 |
| Fitting, Sylow | C32×#6900 ⋊ C2 |
4984 conjugacy classes of subgroups; only the 930 normal subgroups are shown.
This group has 4984 conjugacy classes of subgroups — too many to draw. Only its 930 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 486 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3)(5,6);(4,5,6)(16,22,23,21,19,20,18,17,24);(7,13,14,12,10,11,9,8,15);(8,10,13)(11,15,14)(17,19,22)(20,24,23)459 conjugacy classes. The character table is too large to display.
| maximal subgroup of | AGL(1,ℤ/9)×#6900, C3×D3×#92314, C3×D3×#92265, … |
| maximal normal subgroup of | C3×D3×#92314, C3×D3×#92265, D32×#6900, … |
| maximal quotient of | AGL(1,ℤ/9)×#6900 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(5,6), (4,5,6)(16,22,23,21,19,20,18,17,24), (7,13,14,12,10,11,9,8,15), (8,10,13)(11,15,14)(17,19,22)(20,24,23));