Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×A4×AΓL(1,𝔽9) |
| Fitting, socle | C23×C32 ⋊ C3×SD16 |
998 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 998 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,9,6,2)(3,7,5,8)(13,14,15);(2,4)(3,8)(6,7)(12,15)(13,14);(10,11)72 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4×AGL(2,𝔽3), C2×S4×AΓL(1,𝔽9) |
| maximal normal subgroup of | C2×S4×AΓL(1,𝔽9) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,9,6,2)(3,7,5,8)(13,14,15), (2,4)(3,8)(6,7)(12,15)(13,14), (10,11));