Finite Group DatabasebetaFinite Group Databasebeta| order | 8722 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×D623 |
| unambiguous semidirect product | C7⋊(C7×D89) |
| unambiguous semidirect product | C89⋊(C7×D7) |
| Fitting, socle, Hall | C72×C89 ⋊ C2 |
| derived | C7×C89 ⋊ C2×C7 |
| Sylow | C89 ⋊ C7×D7 |
| Sylow | C72 ⋊ D89 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | D623 |
| commutator, hypocenter | C7×C89 | C2×C7 |
| Frattini | C1 | G |
| Fitting, socle | C72×C89 | C2 |
| LCS | G ▷ C7×C89 | |
| UCS | C1 ◁ C7 | |
22 conjugacy classes of subgroups.
2191 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C7×D7×D89, D7×D623, (C7⋊D7)⋊D89, … |
| maximal normal subgroup of | C7×D7×D89, D7×D623, (C7⋊D7)⋊D89, … |
| maximal quotient of | C7×Dic623, C2×C7×D623, C3×C7×D623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8722, 8);