A family of complex Hadamard martrices found by M. Matolcsi et al. [34] in June 2006; $$S_{12}^{(5)}(a,b,c,d,e)=S_{12}\circ{\rm EXP}\left(i R_{S_{12}^{(5)}}(a,b,c,d,e)\right),$$ where $$R_{S_{12}^{(5)}}(a,b,c,d,e)=\left[\begin{array}{rrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & a & a & a & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & b & b & b & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & d & e & \bullet & \bullet & \bullet & \bullet & d & e & \bullet & \bullet & \bullet\\ \bullet & d & e & a & a & a & \bullet & d & e & \bullet & \bullet & \bullet\\ \bullet & d & e & b & b & b & \bullet & d & e & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & c & \bullet & \bullet & c & \bullet & \bullet\\ \bullet & \bullet & \bullet & a & a & a & c & \bullet & \bullet & c & \bullet & \bullet\\ \bullet & \bullet & \bullet & b & b & b & c & \bullet & \bullet & c & \bullet & \bullet\\ \bullet & d & e & \bullet & \bullet & \bullet & c & d & e & c & \bullet & \bullet\\ \bullet & d & e & a & a & a & c & d & e & c & \bullet & \bullet\\ \bullet & d & e & b & b & b & c & d & e & c & \bullet & \bullet \end{array}\right].$$ It is generated by the spectral matrix $S_{12}$ containing $36^{\rm th}$-roots of unity which log-Hadamard form reads: $${\rm LOG}\left(S_{12}\right)=\frac{2}{36}\pi \left[\begin{array}{rrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & 12 & 24 & 28 & 4 & 16 & \bullet & 12 & 24 & \bullet & 12 & 24\\ \bullet & 24 & 12 & 20 & 8 & 32 & \bullet & 24 & 12 & \bullet & 24 & 12\\ \bullet & 27 & \bullet & \bullet & \bullet & \bullet & 18 & 9 & 18 & 18 & 18 & 18\\ \bullet & 3 & 24 & 28 & 4 & 16 & 18 & 21 & 6 & 18 & 30 & 6\\ \bullet & 15 & 12 & 20 & 8 & 32 & 18 & 33 & 30 & 18 & 6 & 30\\ \bullet & \bullet & \bullet & 18 & 18 & 18 & 9 & \bullet & \bullet & 27 & 18 & 18\\ \bullet & 12 & 24 & 10 & 22 & 34 & 9 & 12 & 24 & 27 & 30 & 6\\ \bullet & 24 & 12 & 2 & 26 & 14 & 9 & 24 & 12 & 27 & 6 & 30\\ \bullet & 27 & \bullet & 18 & 18 & 18 & 27 & 9 & 18 & 9 & \bullet & \bullet\\ \bullet & 3 & 24 & 10 & 22 & 34 & 27 & 21 & 6 & 9 & 12 & 24\\ \bullet & 15 & 12 & 2 & 26 & 14 & 27 & 33 & 30 & 9 & 24 & 12 \end{array}\right].$$