Finite Group DatabasebetaFinite Group Databasebeta| order | 775 |
| isoclinism | isoclinic class of ΓL(1,𝔽32) |
| automorphism | ? |
| homology | Abelianization (H1) = C52 · Schur multiplier (H2) = C5 |
| direct product | C5×ΓL(1,𝔽32) |
| unambiguous semidirect product | C31⋊C52 |
| unambiguous semidirect product | (C5×C31)⋊C5 |
| Fitting, socle | C5×C31 ⋊ C5 |
| derived, Sylow | C31 ⋊ C52 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | ΓL(1,𝔽32) |
| commutator, cosocle, hypocenter | C31 | C52 |
| Frattini | C1 | G |
| Fitting, socle | C5×C31 | C5 |
| LCS | G ▷ C31 | |
| UCS | C1 ◁ C5 | |
16 conjugacy classes of subgroups.
| transitive degree | μt(G) = 155 |
55 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D31⋊C5, D5×ΓL(1,𝔽32), C5⋊(D31⋊C5), … |
| maximal normal subgroup of | C5×D31⋊C5, D5×ΓL(1,𝔽32), C5⋊(D31⋊C5), … |
| maximal quotient of | C2×C5×ΓL(1,𝔽32), C3×C5×ΓL(1,𝔽32), C25×ΓL(1,𝔽32), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(775, 3);