Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | Aut(G) = C22×S5 (order 480) · Inn(G) = C2×S5 · Out(G) = C2 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊S5 |
| Fitting | C22 ⋊ S5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×S5 |
| commutator, socle, cosocle | C2×A5 | C22 |
| Fitting, hypercenter | C22 | S5 |
| hypocenter | A5 | D4 |
| LCS | G ▷ C2×A5 ▷ A5 | |
| UCS | C1 ◁ C2 ◁ C22 | |
106 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 9 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)(5,6,7)(8,9);(1,3)(2,4)(5,6,7,8,9)19 conjugacy classes — view the full character table.
| maximal subgroup of | #11756669, D4×S5, #11756679, … |
| maximal normal subgroup of | #11756669, D4×S5, #11756679, … |
| maximal quotient of | (C2×S5)⋊C4, (A5⋊C4)⋊C4, #11751513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 951);