Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| isoclinism | isoclinic class of S5 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×A5⋊C4 |
| unambiguous semidirect product | A5⋊(C4×C5) |
| derived | A5 ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2×C5 | S5 |
| commutator, hypocenter | A5 | C4×C5 |
| Frattini | C2 | C5×S5 |
| socle, prime radical | C2×C5×A5 | C2 |
| cosocle | C2×A5 | C2×C5 |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C2×C5 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 60 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C5×A5⋊C4, C2×C5×A5⋊C4, D5×A5⋊C4 |
| maximal normal subgroup of | C3×C5×A5⋊C4, C2×C5×A5⋊C4, D5×A5⋊C4 |
| maximal quotient of | C5×GL(2,𝔽5), C3×C5×A5⋊C4, C2×C5×A5⋊C4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 473);