Finite Group DatabasebetaFinite Group Databasebeta| order | 22500 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 18000 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C22×C9×C625 |
| multiplicative | (ℤ/33753)ˣ, (ℤ/45004)ˣ, (ℤ/67506)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3×C125 | C22×C3×C5 |
| socle, prime radical | C22×C3×C5 | C3×C125 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
75 conjugacy classes of subgroups.
22500 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id.