First non-affine family of order six was found in May 2006 by K. Beauchamp and R. Nicoară [3] reads $$B_6^{(1)}(y)=\left[\begin{array}{rrrrrr} 1 & 1 & 1 & 1 & 1 & 1\\ 1 & -1 & -1/x & -y & y & 1/x\\ 1 & -x & 1 & y & 1/z & -1/t\\ 1 & -1/y & 1/y & -1 & -1/t & 1/t\\ 1 & 1/y & z & -t & 1 & -1/x\\ 1 & x & -t & t & -x & -1 \end{array}\right]$$ with $$\begin{aligned} x(y) &= \frac{1+2y+y^2\pm\sqrt{2}\sqrt{1+2y+2y^3+y^4}}{1+2y-y^2},\\ z(y) &= \frac{1}{y}\frac{1+2y-y^2}{y^2+2y-1},\\ t(y) &= xyz \end{aligned}$$ where $y$ is a free parameter $y=\exp(2\pi i s)$ and $s$ varies in the interval $$\frac{\arccos\frac{1}{2}\left(\sqrt{3}-1\right)}{2\pi}< s < 1-\frac{\arccos\frac{1}{2}\left(\sqrt{3}-1\right)}{2\pi}.$$
For $y =$ $d$$^2$ or its complex conjugate (defined here) one obtains a permutations of $C_6$. Also for $y = -1$ or $y =\pm i$ family $B_6^{(1)}$ is equivalent to $D_6^{(1)}$.
Numerical results show that the defect $d\left(B_6^{(1)}(y)\right) = 4$ for a generic value of the parameter $y$.