Finite Group DatabasebetaFinite Group Databasebeta| order | 18000 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 18432000 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C3×C5 |
| abelian | C22×C32×C4×C5×C25 |
| multiplicative | (ℤ/21593)ˣ, (ℤ/21917)ˣ, (ℤ/23405)ˣ, (ℤ/26425)ˣ, (ℤ/27125)ˣ, (ℤ/27633)ˣ, (ℤ/28179)ˣ, (ℤ/33975)ˣ, (ℤ/34875)ˣ, (ℤ/36844)ˣ, (ℤ/43186)ˣ, (ℤ/43834)ˣ, (ℤ/46810)ˣ, (ℤ/52850)ˣ, (ℤ/54250)ˣ, (ℤ/55266)ˣ, (ℤ/56358)ˣ, (ℤ/67950)ˣ, (ℤ/69750)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C5 | C23×C32×C52 |
| socle, prime radical | C23×C32×C52 | C2×C5 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
2268 conjugacy classes of subgroups.
18000 conjugacy classes. The character table is too large to display.
| automorphism group of | C11×C13×C151, C7×C31×C101, C7×C31×C125 |
| outer automorphism group of | C11×C13×C151, C7×C31×C101, C7×C31×C125 |
Past the SmallGroups library coverage — no SmallGroups id.