Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| isoclinism | isoclinic class of D3×AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| Fitting, socle | C22×C3×C5 ⋊ C2×C4 |
| derived, Hall | C3×C5 ⋊ C23×C4 |
| Sylow | C5 ⋊ C22×C4×D3 |
| Sylow | C3 ⋊ C23×AGL(1,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×AGL(1,𝔽5) |
| commutator, hypocenter | C3×C5 | C23×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C5 | C2×C4 |
| prime radical | C22×D3×D5 | C2 |
| cosocle | C3×D5 | C24 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C22 | |
472 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 166 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 — 4 generators on 12 points:
(5,6,7)(9,10,11,12);(1,2)(3,4)(6,7)(8,9,10,12,11);(3,4);(1,2)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 1197);