Finite Group DatabasebetaFinite Group Databasebeta| order | 22000 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C42×C5×C25 (order 8000) · Inn(G) = C1 · Out(G) = C22×C42×C5×C25 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C22000 |
| abelian | C11×C16×C125 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C8×C25 | C2×C5×C11 |
| socle, prime radical | C2×C5×C11 | C8×C25 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
40 conjugacy classes of subgroups.
22000 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | Dic11000, D22000 |
| maximal subgroup of | Dic11000, D22000 |
| maximal normal subgroup of | Dic11000, D22000 |
Past the SmallGroups library coverage — no SmallGroups id.