Finite Group DatabasebetaFinite Group Databasebeta| order | 15360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
| direct product | AGL(1,𝔽5)×#11715753 |
| Fitting | C5×#3532 ⋊ C4×D3 |
| derived | C5×#5464 ⋊ C22×C4 |
| Sylow | C5 ⋊ C4×#11715753 |
At least 8660 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has at least 8660 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(2,12,16,7,3,9)(4,13,5)(6,11,8,15,10,14)(17,18,21,19,20);(1,15,5,6)(2,14,10,3)(4,16,13,9)(7,12,11,8)(18,19,20,21);(2,6)(4,5)(7,15)(8,14)(9,10)(11,16)120 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×AGL(2,ℤ/4) |
| maximal normal subgroup of | AGL(1,𝔽5)×AGL(2,ℤ/4) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,12,16,7,3,9)(4,13,5)(6,11,8,15,10,14)(17,18,21,19,20), (1,15,5,6)(2,14,10,3)(4,16,13,9)(7,12,11,8)(18,19,20,21), (2,6)(4,5)(7,15)(8,14)(9,10)(11,16));