There are precisely five permutation equivalent $1$-parameter maximal affine Hadamard families stemming from the symmetric matrix $$D_6=\left[\begin{array}{rrrrrr} 1& 1& 1& 1& 1& 1\\ 1& -1& i& -i& -i& i\\ 1& i& -1& i& -i& -i\\ 1& -i& i& -1& i& -i\\ 1& -i& -i& i& -1& i\\ 1& i& -i& -i& i& -1\\ \end{array}\right]$$ namely $D_6^{(1)}(c)=D_6\circ{\rm EXP}\left(i R_{D_6^{(1)}}(c)\right)$ with $$R_{D_6^{(1)}}(c)=\left[\begin{array}{rrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & c & c & \bullet\\ \bullet & \bullet & -c & \bullet & \bullet & -c\\ \bullet & \bullet & -c & \bullet & \bullet & -c\\ \bullet & \bullet & \bullet & c & c & \bullet \end{array}\right]$$ and $$\begin{aligned} P_1\cdot D_6^{(1)}(c)\cdot P_1^{\rm T} &= D_6\circ{\rm EXP}\left(i P_1\cdot R_{D_6^{(1)}}(c)\cdot P_1^{\rm T}\right)\quad:\quad P_1=\left[e_1,e_3,e_2,e_6,e_5,e_4\right],\\ P_2\cdot D_6^{(1)}(c)\cdot P_2^{\rm T} &= D_6\circ{\rm EXP}\left(i P_2\cdot R_{D_6^{(1)}}(c)\cdot P_2^{\rm T}\right)\quad:\quad P_2=\left[e_1,e_4,e_3,e_2,e_6,e_5\right],\\ P_3\cdot D_6^{(1)}(c)\cdot P_3^{\rm T} &= D_6\circ{\rm EXP}\left(i P_3\cdot R_{D_6^{(1)}}(c)\cdot P_3^{\rm T}\right)\quad:\quad P_3=\left[e_1,e_6,e_2,e_3,e_4,e_5\right],\\ P_4\cdot D_6^{(1)}(c)\cdot P_4^{\rm T} &= D_6\circ{\rm EXP}\left(i P_4\cdot R_{D_6^{(1)}}(c)\cdot P_4^{\rm T}\right)\quad:\quad P_4=\left[e_1,e_5,e_4,e_3,e_2,e_6\right], \end{aligned}$$ where $e_i$ denotes the $i^{\rm th}$ standard basis column vector.
Numerical results show that defect $d\left(D_6^{(1)}(c)\right)=4$ for a generic value of the parameter $c$.
There are equivalence relations within the family: $D(c) \simeq D(c + 1/2) \simeq D(1/4 - c)$ which allow to take the paremeter $c \in [-1/8, 1/8]$. Moreover $\Big(D(c)\Big)^{\rm T} \simeq D(-c)$.