Finite Group DatabasebetaFinite Group Databasebeta| order | 27216 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 1119744 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C3 |
| abelian | C22×C3×C4×C7×C81 |
| multiplicative | (ℤ/33089)ˣ, (ℤ/35045)ˣ, (ℤ/39935)ˣ, (ℤ/42543)ˣ, (ℤ/47649)ˣ, (ℤ/49329)ˣ, (ℤ/52245)ˣ, (ℤ/59535)ˣ, (ℤ/63532)ˣ, (ℤ/66178)ˣ, (ℤ/70090)ˣ, (ℤ/79870)ˣ, (ℤ/81684)ˣ, (ℤ/85086)ˣ, (ℤ/95298)ˣ, (ℤ/98658)ˣ, (ℤ/104490)ˣ, (ℤ/119070)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C27 | C23×C32×C7 |
| socle, prime radical | C23×C32×C7 | C2×C27 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
972 conjugacy classes of subgroups.
27216 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id.