Finite Group DatabasebetaFinite Group Databasebeta| order | 3600 |
| isoclinism | isoclinic class of S5 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C3×C5×A5⋊C4 |
| derived | A5 ⋊ C3×C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2×C3×C5 | S5 |
| commutator, hypocenter | A5 | C3×C4×C5 |
| Frattini | C2 | C3×C5×S5 |
| socle, prime radical | C2×C3×C5×A5 | C2 |
| cosocle | C2×A5 | C2×C3×C5 |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C2×C3×C5 | |
154 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
210 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×C5×A5⋊C4 |
| maximal normal subgroup of | C2×C3×C5×A5⋊C4 |
| maximal quotient of | C3×C5×GL(2,𝔽5), C2×C3×C5×A5⋊C4 |
Past the SmallGroups library coverage — no SmallGroups id.