Finite Group DatabasebetaFinite Group Databasebeta| order | 7860 |
| isoclinism | isoclinic class of ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C131 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽16)×C131 |
| unambiguous semidirect product | C3⋊(AGL(1,𝔽5)×C131) |
| Fitting, socle, Hall | C3×C5×C131 ⋊ C4 |
| derived, Hall | C3×C5 ⋊ C4×C131 |
| Hall | C5×C131 ⋊ Dic3 |
| Hall | C3×C131 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ Dic3×C131 |
| Sylow | C3 ⋊ AGL(1,𝔽5)×C131 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C131 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C4×C131 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C131 | C4 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C131 | |
24 conjugacy classes of subgroups.
1179 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C131×C9⋊AGL(1,𝔽5), C3×ΓL(1,𝔽16)×C131, C131×C3⋊ΓL(1,𝔽16) |
| maximal normal subgroup of | C3×ΓL(1,𝔽16)×C131 |
| maximal quotient of | C131×C9⋊AGL(1,𝔽5), C3×ΓL(1,𝔽16)×C131, C131×C3⋊ΓL(1,𝔽16) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7860, 24);