Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| isoclinism | isoclinic class of A4×AGL(1,𝔽5) |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×S4 (order 960) · Inn(G) = A4×AGL(1,𝔽5) · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×A4×AGL(1,𝔽5) |
| unambiguous semidirect product | D5⋊(C4×A4) |
| unambiguous semidirect product | (C23×AGL(1,𝔽5))⋊C3 |
| Fitting, socle | C23×C5 ⋊ C3×C4 |
| derived | C22×C5 ⋊ C2×C3×C4 |
| Hall | C23×AGL(1,𝔽5) ⋊ C3 |
| Sylow | C5 ⋊ C2×C4×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×AGL(1,𝔽5) |
| commutator, hypocenter | C22×C5 | C2×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C23×C5 | C3×C4 |
| prime radical | C2×D5×A4 | C2 |
| cosocle | C22×D5 | C22×C3 |
| LCS | G ▷ C22×C5 | |
| UCS | C1 ◁ C2 | |
132 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 11 points:
(4,5,6)(8,9,10,11);(1,2)(3,6)(4,5)(7,8,9,11,10)40 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 1192);