Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | A4×#1094⋊C4 |
| derived | C22×C22⋊C9 ⋊ C32×C4 |
874 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 874 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4)(6,8,7)(9,10,12,11)(13,14)(15,21)(17,18)(19,20);(5,8)(6,7)(10,11,12)(13,17,18,16,15,20,14,19,21)152 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#63694⋊C3 |
| maximal normal subgroup of | A4×#63694⋊C3 |
| maximal quotient of | #72245336 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4)(6,8,7)(9,10,12,11)(13,14)(15,21)(17,18)(19,20), (5,8)(6,7)(10,11,12)(13,17,18,16,15,20,14,19,21));