Finite Group DatabasebetaFinite Group Databasebeta| order | 12456 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 173376 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23 |
| abelian | C23×C9×C173 |
| multiplicative | (ℤ/19779)ˣ, (ℤ/26372)ˣ, (ℤ/37476)ˣ, (ℤ/39558)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C23×C3×C173 |
| socle, prime radical | C23×C3×C173 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
96 conjugacy classes of subgroups.
12456 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×C4×C9×C173, C24×C9×C173 |
| maximal normal subgroup of | C22×C4×C9×C173, C24×C9×C173 |
| maximal quotient of | C22×C4×C9×C173, C24×C9×C173 |
Past the SmallGroups library coverage — no SmallGroups id.