Finite Group DatabasebetaFinite Group Databasebeta| order | 8722 |
| automorphism | Aut(G) order 16118256 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D4361 |
| unambiguous semidirect product | C49⋊D89 |
| unambiguous semidirect product | C89⋊D49 |
| Fitting, derived, Hall | C49×C89 ⋊ C2 |
| Sylow | C89 ⋊ D49 |
| Sylow | C49 ⋊ D89 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C49×C89 | C2 |
| Frattini | C7 | D623 |
| socle | C7×C89 | D7 |
| LCS | G ▷ C49×C89 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
Polycyclic presentation:
2,7,89,7|b^7=d^6,b^a=b^6d,c^a=c^88,d^a=d^62182 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D49×D89, C2×D4361, C3×D4361, … |
| maximal normal subgroup of | D49×D89, C2×D4361, C3×D4361, … |
| maximal quotient of | Dic4361, C2×D4361, C3×D4361, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8722, 4);