Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of D5×A5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D5×A5 |
| unambiguous semidirect product | C5⋊(C2×C3×A5) |
| Fitting | C3×C5 ⋊ C2×A5 |
| socle | C3×C5×A5 ⋊ C2 |
| derived | C5×A5 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D5×A5 |
| commutator, hypocenter | C5×A5 | C2×C3 |
| Frattini | C1 | G |
| Fitting | C3×C5 | C2×A5 |
| socle | C3×C5×A5 | C2 |
| cosocle | C5 | C2×C3×A5 |
| LCS | G ▷ C5×A5 | |
| UCS | C1 ◁ C3 | |
108 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 75 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×D5×A5, D5×#11761914, A5×#11761914, … |
| maximal normal subgroup of | C2×C3×D5×A5, C3×A5⋊AGL(1,𝔽5), D3×D5×A5, … |
| maximal quotient of | C3×D5×SL(2,𝔽5), C2×C3×D5×A5, C3×C5⋊#6732 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 561);