Finite Group DatabasebetaFinite Group Databasebeta| order | 8215 |
| isoclinism | isoclinic class of ΓL(1,𝔽32) |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C53 · Schur multiplier (H2) = C1 |
| direct product | C53×ΓL(1,𝔽32) |
| unambiguous semidirect product | C31⋊(C5×C53) |
| unambiguous semidirect product | (C31×C53)⋊C5 |
| Fitting, socle, Hall | C31×C53 ⋊ C5 |
| derived, Sylow | C31 ⋊ C5×C53 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C53 | ΓL(1,𝔽32) |
| commutator, hypocenter | C31 | C5×C53 |
| Frattini | C1 | G |
| Fitting, socle | C31×C53 | C5 |
| LCS | G ▷ C31 | |
| UCS | C1 ◁ C53 | |
8 conjugacy classes of subgroups.
583 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C53×ΓL(1,𝔽32), D53×ΓL(1,𝔽32), C53×D31⋊C5, … |
| maximal normal subgroup of | C2×C53×ΓL(1,𝔽32), D53×ΓL(1,𝔽32), C53×D31⋊C5, … |
| maximal quotient of | C2×C53×ΓL(1,𝔽32), C3×C53×ΓL(1,𝔽32) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8215, 1);