Finite Group DatabasebetaFinite Group Databasebeta| order | 18000 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 115200 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23 |
| abelian | C22×C4×C9×C125 |
| multiplicative | (ℤ/23845)ˣ, (ℤ/27861)ˣ, (ℤ/33765)ˣ, (ℤ/33885)ˣ, (ℤ/35625)ˣ, (ℤ/36016)ˣ, (ℤ/37148)ˣ, (ℤ/45020)ˣ, (ℤ/47500)ˣ, (ℤ/47690)ˣ, (ℤ/55722)ˣ, (ℤ/67500)ˣ, (ℤ/67530)ˣ, (ℤ/67770)ˣ, (ℤ/71250)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C3×C25 | C23×C3×C5 |
| socle, prime radical | C23×C3×C5 | C2×C3×C25 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
324 conjugacy classes of subgroups.
18000 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id.