Finite Group DatabasebetaFinite Group Databasebeta| order | 4320 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×AGL(1,𝔽5)×S4 |
| Fitting | C22×C5×C9 ⋊ C4×D3 |
| derived | C5×A4 ⋊ C2×C4×C9 |
| Sylow | C5 ⋊ C4×C9×S4 |
360 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 180 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,4,3)(6,7,8,9)(10,17,16,12,11,18,15,14,13);(2,4)(5,6,9,7,8)225 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×S4×3−1+2, D9×AGL(1,𝔽5)×S4 |
| maximal normal subgroup of | AGL(1,𝔽5)×S4×3−1+2, D9×AGL(1,𝔽5)×S4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3)(6,7,8,9)(10,17,16,12,11,18,15,14,13), (2,4)(5,6,9,7,8));