There are precisely three permutation equivalent $1$-parameter maximal affine Hadamard families stemming from symmetric matrix being a permuted "starting point" for a $1$-parameter orbit found by M. Petrescu [82]; $$P_7=\left[\begin{array}{lllllll} 1& 1& 1& 1& 1& 1& 1\\ 1& w& w^4& w^5& w^3& w^3& w\\ 1& w^4& w& w^3& w^5& w^3& w\\ 1& w^5& w^3& w& w^4& w& w^3\\ 1& w^3& w^5& w^4& w& w& w^3\\ 1& w^3& w^3& w& w& w^4& w^5\\ 1& w& w& w^3& w^3& w^5& w^4\end{array}\right]:\quad w=\exp(2\pi i/6).$$ They are $$P_7^{(1)}(a)=P_7\circ{\rm EXP}\left(i R_{P_7^{(1)}}(a)\right),$$ where $$R_{P_7^{(1)}}(a)= \left[\begin{array}{rrrrrrr} \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet\\ \bullet& a& a& \bullet& \bullet& \bullet& \bullet\\ \bullet& a& a& \bullet& \bullet& \bullet& \bullet\\ \bullet& \bullet& \bullet& -a& -a& \bullet& \bullet\\ \bullet& \bullet& \bullet& -a& -a& \bullet& \bullet\\ \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet\\ \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet\end{array}\right]$$ and $$\begin{aligned} P_1\cdot P_7^{(1)}(a)\cdot P_2^{\rm T} &= P_7\circ{\rm EXP}\left(i P_1\cdot R_{P_7^{(1)}(a)}\cdot P_2^{\rm T}\right)\quad:\quad P_1 = \left[e_1, e_7, e_3, e_4, e_6, e_5, e_2\right],\\ P_2\cdot P_7^{(1)}(a)\cdot P_1^{\rm T} &= P_7\circ{\rm EXP}\left(i P_2\cdot R_{P_7^{(1)}(a)}\cdot P_1^{\rm T}\right)\quad:\quad P_2 = \left[e_1, e_2, e_7, e_6, e_5, e_4, e_3\right], \end{aligned}$$ and $e_k$ denotes the $k^{\rm th}$ standard basis column vector.
Matrix $P_7\in BH(7,6)$ was first introduced by B. W. Brock in 1988 [177].