Finite Group DatabasebetaFinite Group Databasebeta| order | 18720 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 73728 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2×C3 |
| abelian | C2×C32×C5×C13×C16 |
| multiplicative | (ℤ/19039)ˣ, (ℤ/21847)ˣ, (ℤ/28089)ˣ, (ℤ/38078)ˣ, (ℤ/43694)ˣ, (ℤ/56178)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C8 | C22×C32×C5×C13 |
| socle, prime radical | C22×C32×C5×C13 | C8 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
336 conjugacy classes of subgroups.
18720 conjugacy classes. The character table is too large to display.
| automorphism group of | C79×C241, C7×C3121 |
| outer automorphism group of | C79×C241, C7×C3121 |
Past the SmallGroups library coverage — no SmallGroups id.