This paper deals with a special class of 6th order surfaces with a quadruple straight line in a three-dimensional Euclidean space. These surfaces, denoted by $mathcal P_4^6$, are the pedal surfaces of one special 1st order 4th class congruence $mathcal C_4^1$. The parametric and implicit equations of $mathcal P_4^6$ are derived, some of their properties are proved and their visualizations are given. The singularities of $mathcal P_4^6$ are classified according to the shapes of their tangent cones. The methods applied are analytic, synthetic and algebraic, supported by the program Mathematica 6.