Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| isoclinism | isoclinic class of AGL(1,𝔽5)×S4 |
| automorphism | Aut(G) = AGL(1,𝔽5)×S4 (order 480) · Inn(G) = G · Out(G) = C1 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | AGL(1,𝔽5)×S4 |
| unambiguous semidirect product | (A4⋊AGL(1,𝔽5))⋊C2 |
| unambiguous semidirect product | (A4×AGL(1,𝔽5))⋊C2 |
| Fitting, socle | C22×C5 ⋊ C4×D3 |
| derived | C5×A4 ⋊ C2×C4 |
| Sylow | C5 ⋊ C4×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C5×A4 | C2×C4 |
| Fitting, socle | C22×C5 | C4×D3 |
| prime radical | D5×S4 | C2 |
| cosocle | D5×A4 | C22 |
| LCS | G ▷ C5×A4 | |
| UCS | C1 | |
112 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 2 generators on 9 points:
(1,2,4,3)(5,6,7,9,8);(2,3,4)(6,7,8,9)25 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 1189);