Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of A5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×C5×A5 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2×C3×C5 | A5 |
| commutator, hypocenter | A5 | C2×C3×C5 |
| Frattini | C1 | G |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C2×C3×C5 | |
108 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 150 |
150 conjugacy classes — view the full character table.
| maximal subgroup of | C22×C3×C5×A5, C3×C5×A5⋊C4, C2×C3×C5×S5, … |
| maximal normal subgroup of | C22×C3×C5×A5, C3×C5×A5⋊C4, C2×C3×C5×S5, … |
| maximal quotient of | C3×C5×#6732, C2×C3×C5×SL(2,𝔽5), C22×C3×C5×A5 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 563);