Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | Aut(G) = C23×D3×AGL(1,𝔽5) (order 960) · Inn(G) = C2×D3×AGL(1,𝔽5) · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | AGL(1,𝔽5)×#67 |
| Fitting | C22×C3×C5 ⋊ C2×C4 |
| Hall | C3×C5 ⋊ C4×D4 |
| Sylow | C5 ⋊ C4×#67 |
| Sylow | C3 ⋊ D4×AGL(1,𝔽5) |
188 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 188 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) = 12 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3)(5,7)(8,9,10,12,11);(2,3)(4,5)(6,7);(9,10,11,12)45 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#235, #11756836, D3×D4×AGL(1,𝔽5), … |
| maximal normal subgroup of | AGL(1,𝔽5)×#235, #11756836, D3×D4×AGL(1,𝔽5), … |
| maximal quotient of | #11751745, #11751763, AGL(1,𝔽5)×Dic3⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 1010);