Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | C4×#11724052 |
| Fitting | C24×C32×C4 ⋊ D3 |
| derived | C3×C22⋊A4 ⋊ C2×C3×C4 |
1959 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 1959 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 72 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,4,3,2)(5,6,8,7)(14,19,21,16,17,20)(15,18);(2,3,4)(6,8,7)(9,10,11,12)(13,14,16)(15,17,20)(18,21,19)184 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#72213301 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3,2)(5,6,8,7)(14,19,21,16,17,20)(15,18), (2,3,4)(6,8,7)(9,10,11,12)(13,14,16)(15,17,20)(18,21,19));