Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| isoclinism | isoclinic class of D5×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×D5×S4 |
| unambiguous semidirect product | (C5×S4)⋊C22 |
| unambiguous semidirect product | (C5⋊S4)⋊C22 |
| unambiguous semidirect product | (D5×A4)⋊C22 |
| Fitting, socle | C23×C5 ⋊ C2×D3 |
| derived | C5×A4 ⋊ C23 |
| Sylow | C5 ⋊ C22×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D5×S4 |
| commutator, cosocle, hypocenter | C5×A4 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C23×C5 | C2×D3 |
| LCS | G ▷ C5×A4 | |
| UCS | C1 ◁ C2 | |
262 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 35 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 — 3 generators on 11 points:
(3,4,6,5)(7,8,10,11,9);(1,2)(4,5,6)(8,9)(10,11);(1,2)40 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×AGL(1,𝔽5))⋊S4, C4×D5×S4, D5×C4⋊S4, … |
| maximal normal subgroup of | (C2×AGL(1,𝔽5))⋊S4, C4×D5×S4, D5×C4⋊S4, … |
| maximal quotient of | S4×Dic10, #11756746, #11756747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 1193);