Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of SL(2,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×SL(2,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, socle, hypercenter | C2×C3×C5 | A5 |
| commutator, hypocenter | SL(2,𝔽5) | C3×C5 |
| Frattini, cosocle | C2 | C3×C5×A5 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2×C3×C5 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C5×#6732, C2×C3×C5×SL(2,𝔽5), C3×D5×SL(2,𝔽5), … |
| maximal normal subgroup of | C3×C5×#6732, C2×C3×C5×SL(2,𝔽5), C3×D5×SL(2,𝔽5), … |
| maximal quotient of | C2×C3×C5×SL(2,𝔽5) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 328);