Finite Group DatabasebetaFinite Group Databasebeta| order | 8292 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊(C691⋊C3) |
| unambiguous semidirect product | C691⋊A4 |
| Fitting, socle, derived, Hall | C22×C691 ⋊ C3 |
| Sylow | C691 ⋊ A4 |
| Sylow | C22 ⋊ C691⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C22×C691 | C3 |
| LCS | G ▷ C22×C691 | |
| UCS | C1 | |
10 conjugacy classes of subgroups.
Polycyclic presentation:
3,2^2,691|b^a=c,c^a=bc,d^a=d^437924 conjugacy classes — view the full character table.
| maximal subgroup of | A4×C691⋊C3, C3×C22⋊(C691⋊C3) |
| maximal normal subgroup of | A4×C691⋊C3, C3×C22⋊(C691⋊C3) |
| maximal quotient of | C22⋊(C691⋊C9), C3×C22⋊(C691⋊C3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8292, 10);