Example PSL(2,7)
Group $G=\rm{PSL}(2,7)$ has 2 3-dimensional faithful representation $V_1,V_2$.
Two representations are given as follows:
\[\begin{align*} \mathbb{P}(V_1)= \langle \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0 \end{pmatrix}, \frac{1}{i\sqrt{7}} \begin{pmatrix} \zeta_7^3-\zeta_7^4 & \zeta_7^5-\zeta_7^2 & \zeta_7^6-\zeta_7\\ \zeta_7^5-\zeta_7^2 & \zeta_7^6-\zeta_7 & \zeta_7^3-\zeta_7^4\\ \zeta_7^6-\zeta_7 & \zeta_7^3-\zeta_7^4 & \zeta_7^5-\zeta_7^2\\ \end{pmatrix}, \begin{pmatrix} \zeta_7 & 0 & 0\\ 0 & \zeta_7^2 & 0\\ 0 & 0 & \zeta_7^4 \end{pmatrix} \rangle \end{align*}\]where $\zeta_7$ is a 7-th root of unity.
The Magma code is as follows:
FScale:=CyclotomicField(84); F:=FScale; Z7:=RootOfUnity(7); a:=Z7^4-Z7^3; b:=Z7^2-Z7^5; c:=Z7-Z7^6; isq7:= 2*Z7^4 + 2*Z7^2 + 2*Z7 + 1; G:=MatrixGroup<3,F| [0,1,0, 0,0,1, 1,0,0], [-a/isq7,-b/isq7,-c/isq7, -b/isq7,-c/isq7,-a/isq7, -c/isq7,-a/isq7,-b/isq7], [Z7,0,0, 0,Z7^2,0, 0,0,Z7^4]>;\[\begin{align*} \mathbb{P}(V_2)= \langle \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0 \end{pmatrix}, \frac{1}{i\sqrt{7}} \begin{pmatrix} \zeta_7^3-\zeta_7^4 & \zeta_7^5-\zeta_7^2 & \zeta_7^6-\zeta_7\\ \zeta_7^5-\zeta_7^2 & \zeta_7^6-\zeta_7 & \zeta_7^3-\zeta_7^4\\ \zeta_7^6-\zeta_7 & \zeta_7^3-\zeta_7^4 & \zeta_7^5-\zeta_7^2\\ \end{pmatrix}, \begin{pmatrix} \zeta_7^3 & 0 & 0\\ 0 & \zeta_7^6 & 0\\ 0 & 0 & \zeta_7^5 \end{pmatrix} \rangle \end{align*}\]
where $\zeta_7$ is a 7-th root of unity.
The Magma code is as follows:
FScale:=CyclotomicField(84); F:=FScale; Z7:=RootOfUnity(7); a:=Z7^4-Z7^3; b:=Z7^2-Z7^5; c:=Z7-Z7^6; isq7:= 2*Z7^4 + 2*Z7^2 + 2*Z7 + 1; G:=MatrixGroup<3,F| [0,1,0, 0,0,1, 1,0,0], [-a/isq7,-b/isq7,-c/isq7, -b/isq7,-c/isq7,-a/isq7, -c/isq7,-a/isq7,-b/isq7], [Z7^3,0,0, 0,Z7^6,0, 0,0,Z7^5]>;
The Burnside Symbols are
\[\begin{align*} [\mathbb{P}(V_1) \circlearrowleft G]=&(1,G \circlearrowright k(x,y),())\\ &+2(C_2, C_2^2\circlearrowright k(x),(1))+(C_3,1 \circlearrowright k,(1,1))\\ &+(C_7,1 \circlearrowright k,(3,6))+(C_7,1 \circlearrowright k,(1,2))\\ &+(1.C_2^2,1 \circlearrowright k,((0,1),(1,0)))+(2.C_2^2,1 \circlearrowright k,((0,1),(1,1)))\\ &+(C_4,1 \circlearrowright k,(1,1))+2(C_4,1 \circlearrowright k,(2,3)) \end{align*}\]and
\[\begin{align*} [\mathbb{P}(V_2) \circlearrowleft G]=&(1,G \circlearrowright k(x,y),())\\ &+2(C_2, C_2^2\circlearrowright k(x),(1))+(C_3,1 \circlearrowright k,(1,1))\\ &+(C_7,1 \circlearrowright k,(2,4))+(C_7,1 \circlearrowright k,(5,6))\\ &+(1.C_2^2,1 \circlearrowright k,((0,1),(1,0)))+(2.C_2^2,1 \circlearrowright k,((0,1),(1,0)))\\ &+(C_4,1 \circlearrowright k,(1,1))+2(C_4,1 \circlearrowright k,(2,3)) \end{align*}\]where
\[\begin{align*} C_2=\langle \frac{1}{7} \begin{pmatrix} -2\zeta_{84}^{22}+3\zeta_{84}^{16}-3\zeta_{84}^{12}+2\zeta_{84}^{8}+2\zeta_{84}^{6}-3\zeta_{84}^{2}-4 & 4\zeta_{84}^{22}-\zeta_{84}^{18}-\zeta_{84}^{16}+4\zeta_{84}^{12}-4\zeta_{84}^{8}-2\zeta_{84}^{6}+\zeta_{84}^{2}+2 & 2\zeta_{84}^{18}+4\zeta_{84}^{16}-3\zeta_{84}^{12}+3\zeta_{84}^{6}-4\zeta_{84}^2-2\\ \zeta_{84}^{22}+\zeta_{84}^{18}-3\zeta_{84}^{16}-\zeta_{84}^8-3\zeta_{84}^6+3\zeta_{84}^2+1 & 3\zeta_{84}^{22}-\zeta_{84}^{16}+\zeta_{84}^{12}-3\zeta_{84}^8-3\zeta_{84}^6+\zeta_{84}^2-1 & -3\zeta_{84}^{22}+3\zeta_{84}^{18}-2\zeta_{84}^{12}+3\zeta_{84}^8+4\zeta_{84}^6-2\\ -\zeta_{84}^{22}-2\zeta_{84}^{18}+\zeta_{84}^{16}-2\zeta_{84}^{12}+\zeta_{84}^8-2\zeta_{84}^6-\zeta_{84}^2 & -\zeta_{84}^{22}-3\zeta_{84}^{18}-\zeta_{84}^{16}+3\zeta_{84}^{12}+\zeta_{84}^8+\zeta_{84}^2+1 & -\zeta_{84}^{22}-2\zeta_{84}^{16}+2\zeta_{84}^{12}+\zeta_{84}^8+\zeta_{84}^6+2\zeta_{84}^2-2 \end{pmatrix} \rangle. \end{align*}\]The action on $\mathbb{P}^1$ is given by \(\begin{align*} C_2^2= \langle &\frac{1}{7} \begin{pmatrix} -3\zeta_{84}^{22}+2\zeta_{84}^{18}+5\zeta_{84}^{16}-4\zeta_{84}^{12}+3\zeta_{84}^8-\zeta_{84}^6-5\zeta_{84}^2-1 & -\zeta_{84}^{22}-4\zeta_{84}^{18}+4\zeta_{84}^{16}+\zeta_{84}^{12}+\zeta_{84}^8+2\zeta_{84}^6-4\zeta_{84}^2-5\\ -5\zeta_{84}^{22}+\zeta_{84}^18+6\zeta_{84}^{16}-9\zeta_{84}^{12}+5\zeta_{84}^8+3\zeta_{84}^6-6\zeta_{84}^2-4 & 3\zeta_{84}^{22}-2\zeta_{84}^{18}-5\zeta_{84}^{16}+4\zeta_{84}^{12}-3\zeta_{84}^8+\zeta_{84}^6+5\zeta_{84}^2+1 \end{pmatrix},\\ &\frac{1}{7} \begin{pmatrix} 2\zeta_{84}^{22}-6\zeta_{84}^{18}-\zeta_{84}^{16}+5\zeta_{84}^{12}-2\zeta_{84}^8-4\zeta_{84}^6+\zeta_{84}^2+3 & -4\zeta_{84}^{22}-2\zeta_{84}^{18}+2\zeta_{84}^{16}+4\zeta_{84}^{12}+4\zeta_{84}^8+\zeta_{84}^6-2\zeta_{84}^2+1\\ 8\zeta_{84}^{22}-3\zeta_{84}^{18}-4\zeta_{84}^{16}+6\zeta_{84}^{12}-8\zeta_{84}^8-2\zeta_{84}^6+4\zeta_{84}^2-2 & -2\zeta_{84}^{22}+6\zeta_{84}^{18}+\zeta_{84}^{16}-5\zeta_{84}^{12}+2\zeta_{84}^8+4\zeta_{84}^6-\zeta_{84}^2-3 \end{pmatrix} \rangle. \end{align*}\)
\[\begin{align*} C_3= \langle \begin{pmatrix} 0 & 0 & \zeta_{84}^{22}-\zeta_{84}^8\\ -\zeta_{84}^{16}+\zeta_{84}^2 & 0 & 0\\ 0 & -\zeta_{84}^{18} & 0 \end{pmatrix} \rangle. \end{align*}\] \[\begin{align*} C_7= \langle \frac{1}{7} \begin{pmatrix} 2\zeta_{84}^{22} - 4\zeta_{84}^{18} - 4\zeta_{84}^{16} + 2\zeta_{84}^{12} - 2\zeta_{84}^8 - \zeta_{84}^6 + 4\zeta_{84}^2 + 1 & -3\zeta_{84}^{22} + 3\zeta_{84}^{18} - 2\zeta_{84}^{12} + 3\zeta_{84}^8 + 4\zeta_{84}^6 - 2 & -2\zeta_{84}^{22} + 3\zeta_{84}^{18} + 2\zeta_{84}^{16} - 4\zeta_{84}^{12} + 2\zeta_{84}^8 + 3\zeta_{84}^6 - 2\zeta_{84}^2\\ \zeta_{84}^{22} - 2\zeta_{84}^{18} - 2\zeta_{84}^{16} + \zeta_{84}^{12} - \zeta_{84}^8 - 4\zeta_{84}^6 + 2\zeta_{84}^2 + 4 & 2\zeta_{84}^{22} - 2\zeta_{84}^{18} - \zeta_{84}^{12} - 2\zeta_{84}^8 + 2\zeta_{84}^6 - 1 & -\zeta_{84}^{22} - 2\zeta_{84}^{18} + \zeta_{84}^{16} - 2\zeta_{84}^{12} + \zeta_{84}^8 - 2\zeta_{84}^6 - \zeta_{84}^2\\ 4\zeta_{84}^{22} - \zeta_{84}^{18} - \zeta_{84}^{16} + 4\zeta_{84}^{12} - 4\zeta_{84}^8 - 2\zeta_{84}^6 + \zeta_{84}^2 + 2 & \zeta_{84}^{22} - \zeta_{84}^{18} + 3\zeta_{84}^{12} - \zeta_{84}^8 + \zeta_{84}^6 + 3 & 3\zeta_{84}^{22} - \zeta_{84}^{18} - 3\zeta_{84}^{16} - \zeta_{84}^{12} - 3\zeta_{84}^8 - \zeta_{84}^6 + 3\zeta_{84}^2 \end{pmatrix} \rangle. \end{align*}\] \[\begin{align*} 1.C_2^2= \langle &\frac{1}{7} \begin{pmatrix} -2\zeta_{84}^{22} + 3\zeta_{84}^{16} - 3\zeta_{84}^{12} + 2\zeta_{84}^8 +2\zeta_{84}^6 - 3\zeta_{84}^2 - 4 & 4\zeta_{84}^{22} - \zeta_{84}^{18} -\zeta_{84}^{16} + 4\zeta_{84}^{12} - 4\zeta_{84}^8 - 2\zeta_{84}^6 + \zeta_{84}^2 +2 & 2\zeta_{84}^{18} + 4\zeta_{84}^{16} - 3\zeta_{84}^{12} + 3\zeta_{84}^6 - 4\zeta_{84}^2 - 2\\ \zeta_{84}^{22} + \zeta_{84}^{18} - 3\zeta_{84}^{16} - \zeta_{84}^8 - 3\zeta_{84}^6 + 3\zeta_{84}^2 + 1 & 3\zeta_{84}^{22} - \zeta_{84}^{16} + \zeta_{84}^{12} - 3\zeta_{84}^8 - 3\zeta_{84}^6 + \zeta_{84}^2 - 1 & -3\zeta_{84}^{22} + 3\zeta_{84}^{18} - 2\zeta_{84}^{12} + 3\zeta_{84}^8 + 4\zeta_{84}^6 - 2\\ -\zeta_{84}^{22} - 2\zeta_{84}^{18} + \zeta_{84}^{16} - 2\zeta_{84}^{12} + \zeta_{84}^8 - 2\zeta_{84}^6 - \zeta_{84}^2 & -\zeta_{84}^{22} - 3\zeta_{84}^{18} - \zeta_{84}^{16} + 3\zeta_{84}^{12} + \zeta_{84}^8 + \zeta_{84}^2 + 1 & -\zeta_{84}^{22} - 2\zeta_{84}^{16} + 2\zeta_{84}^{12} + \zeta_{84}^8 + \zeta_{84}^6 + 2\zeta_{84}^2 - 2 \end{pmatrix},\\ &\frac{1}{7} \begin{pmatrix} -\zeta_{84}^{22} - 2\zeta_{84}^{16} + 2\zeta_{84}^{12} + \zeta_{84}^8 + \zeta_{84}^6 + 2\zeta_{84}^2 - 2 & -\zeta_{84}^{22} - 2\zeta_{84}^{18} + \zeta_{84}^{16} - 2\zeta_{84}^{12} + \zeta_{84}^8 - 2\zeta_{84}^6 - \zeta_{84}^2 & -\zeta_{84}^{22} - 3\zeta_{84}^{18} - \zeta_{84}^{16} + 3\zeta_{84}^{12} + \zeta_{84}^8 + \zeta_{84}^2 + 1\\ 2\zeta_{84}^{18} + 4\zeta_{84}^{16} - 3\zeta_{84}^{12} + 3\zeta_{84}^6 - 4\zeta_{84}^2 - 2 & -2\zeta_{84}^{22} + 3\zeta_{84}^{16} - 3\zeta_{84}^{12} + 2\zeta_{84}^8 + 2\zeta_{84}^6 - 3\zeta_{84}^2 - 4 & 4\zeta_{84}^{22} - \zeta_{84}^{18} - \zeta_{84}^{16} + 4\zeta_{84}^{12} - 4\zeta_{84}^8 - 2\zeta_{84}^6 + \zeta_{84}^2 + 2\\ -3\zeta_{84}^{22} + 3\zeta_{84}^{18} - 2\zeta_{84}^{12} + 3\zeta_{84}^8 + 4\zeta_{84}^6 - 2 & \zeta_{84}^{22} + \zeta_{84}^{18} - 3\zeta_{84}^{16} - zeta_84^8 - 3\zeta_{84}^6 + 3\zeta_{84}^2 + 1 & 3\zeta_{84}^{22} - zeta_84^{16} + zeta_84^{12} - 3\zeta_{84}^8 - 3\zeta_{84}^6 + \zeta_{84}^2 - 1 \end{pmatrix} \rangle. \end{align*}\] \[\begin{align*} 2.C_2^2= \langle &\frac{1}{7} \begin{pmatrix} 3\zeta_{84}^{22} - \zeta_{84}^{16} + \zeta_{84}^{12} - 3\zeta_{84}^8 - 3\zeta_{84}^6 + \zeta_{84}^2 - 1 & 2\zeta_{84}^{22} + 2\zeta_{84}^{18} + \zeta_{84}^{16} - 2\zeta_{84}^8 + \zeta_{84}^6 - \zeta_{84}^2 + 2 & -2\zeta_{84}^{22} + 3\zeta_{84}^{18} + 2\zeta_{84}^{16} - 4\zeta_{84}^{12} + 2\zeta_{84}^8 + 3\zeta_{84}^6 - 2\zeta_{84}^2\\ \zeta_{84}^{22} - 2\zeta_{84}^{18} - 2\zeta_{84}^{16} + \zeta_{84}^{12} - \zeta_{84}^8 - 4\zeta_{84}^6 + 2\zeta_{84}^2 + 4 & -\zeta_{84}^{22} - 2\zeta_{84}^{16} + 2\zeta_{84}^{12} + \zeta_{84}^8 + \zeta_{84}^6 + 2\zeta_{84}^2 - 2 & -2\zeta_{84}^{22} + \zeta_{84}^{18} - 2\zeta_{84}^{16} - \zeta_{84}^{12} + 2\zeta_{84}^8 + 2\zeta_{84}^2 + 2\\ -3\zeta_{84}^{18} + \zeta_{84}^{16} + \zeta_{84}^{12} - \zeta_{84}^6 - \zeta_{84}^2 + 3 & \zeta_{84}^{22} - \zeta_{84}^{18} + 3\zeta_{84}^{12} - \zeta_{84}^8 + \zeta_{84}^6 + 3 & -2\zeta_{84}^{22} + 3\zeta_{84}^{16} - 3\zeta_{84}^{12} + 2\zeta_{84}^8 + 2\zeta_{84}^6 - 3\zeta_{84}^2 - 4 \end{pmatrix},\\ &\frac{1}{7} \begin{pmatrix} -\zeta_{84}^{22} - 2\zeta_{84}^{16} + 2\zeta_{84}^{12} + \zeta_{84}^8 + \zeta_{84}^6 + 2\zeta_{84}^2 - 2 & -\zeta_{84}^{22} - 2\zeta_{84}^{18} + \zeta_{84}^{16} - 2\zeta_{84}^{12} + \zeta_{84}^8 - 2\zeta_{84}^6 - \zeta_{84}^2 & -\zeta_{84}^{22} - 3\zeta_{84}^{18} - \zeta_{84}^{16} + 3\zeta_{84}^{12} + \zeta_{84}^8 + \zeta_{84}^2 + 1\\ 2\zeta_{84}^{18} + 4\zeta_{84}^{16} - 3\zeta_{84}^{12} + 3\zeta_{84}^6 - 4\zeta_{84}^2 - 2 & -2\zeta_{84}^{22} + 3\zeta_{84}^{16} - 3\zeta_{84}^{12} + 2\zeta_{84}^8 + 2\zeta_{84}^6 - 3\zeta_{84}^2 - 4 & 4\zeta_{84}^{22} - \zeta_{84}^{18} - \zeta_{84}^{16} + 4\zeta_{84}^{12} - 4\zeta_{84}^8 - 2\zeta_{84}^6 + \zeta_{84}^2 + 2\\ -3\zeta_{84}^{22} + 3\zeta_{84}^{18} - 2\zeta_{84}^{12} + 3\zeta_{84}^8 + 4\zeta_{84}^6 - 2 & \zeta_{84}^{22} + \zeta_{84}^{18} - 3\zeta_{84}^{16} - \zeta_{84}^8 - 3\zeta_{84}^6 + 3\zeta_{84}^2 + 1 & 3\zeta_{84}^{22} - \zeta_{84}^{16} + \zeta_{84}^{12} - 3\zeta_{84}^8 - 3\zeta_{84}^6 + \zeta_{84}^2 - 1 \end{pmatrix} \rangle. \end{align*}\] \[\begin{align*} C_4= \langle \frac{1}{7} \begin{pmatrix} -2\zeta_{84}^{22} + \zeta_{84}^{18} - 2\zeta_{84}^{16} - \zeta_{84}^{12} + 2\zeta_{84}^8 + 2\zeta_{84}^2 + 2 & \zeta_{84}^{22} - 2\zeta_{84}^{18} - 2\zeta_{84}^{16} + \zeta_{84}^{12} - \zeta_{84}^8 - 4\zeta_{84}^6 + 2\zeta_{84}^2 + 4 & -\zeta_{84}^{22} - 2\zeta_{84}^{16} + 2\zeta_{84}^{12} + \zeta_{84}^8 + \zeta_{84}^6 + 2\zeta_{84}^2 - 2\\ -2\zeta_{84}^{22} + 3\zeta_{84}^{16} - 3\zeta_{84}^{12} + 2\zeta_{84}^8 + 2\zeta_{84}^6 - 3\zeta_{84}^2 - 4 & -3\zeta_{84}^{18} + \zeta_{84}^{16} + \zeta_{84}^{12} - \zeta_{84}^6 - \zeta_{84}^2 + 3 & \zeta_{84}^{22} - \zeta_{84}^{18} + 3\zeta_{84}^{12} - \zeta_{84}^8 + \zeta_{84}^6 + 3\\ -2\zeta_{84}^{22} + 3\zeta_{84}^{18} + 2\zeta_{84}^{16} - 4\zeta_{84}^{12} + 2\zeta_{84}^8 + 3\zeta_{84}^6 - 2\zeta_{84}^2 & 3\zeta_{84}^{22} - \zeta_{84}^{16} + \zeta_{84}^{12} - 3\zeta_{84}^8 - 3\zeta_{84}^6 + \zeta_{84}^2 - 1 & 2\zeta_{84}^{22} + 2\zeta_{84}^{18} + \zeta_{84}^{16} - 2\zeta_{84}^8 + \zeta_{84}^6 - \zeta_{84}^2 + 2 \end{pmatrix} \rangle. \end{align*}\]There are conjugation relations: \(\begin{align*} &(C_3,1 \circlearrowright k,(1,1))=(C_3,1 \circlearrowright k,(2,2)),\\ &(C_7,1 \circlearrowright k,(3,6))=(C_7,1 \circlearrowright k,(3,5))=(C_7,1 \circlearrowright k,(5,6)),\\ &(C_7,1 \circlearrowright k,(1,2))=(C_7,1 \circlearrowright k,(1,4))=(C_7,1 \circlearrowright k,(2,4)),\\ &(1.C_2^2,1 \circlearrowright k,((0,1),(1,0)))=(1.C_2^2,1 \circlearrowright k,((0,1),(1,1)))=(1.C_2^2,1 \circlearrowright k,((1,0),(1,1))),\\ &(2.C_2^2,1 \circlearrowright k,((0,1),(1,1)))=(2.C_2^2,1 \circlearrowright k,((0,1),(1,0)))=(2.C_2^2,1 \circlearrowright k,((1,0),(1,1))),\\ &(C_4,1 \circlearrowright k,(1,1))=(C_4,1 \circlearrowright k,(3,3)),\\ &(C_4,1 \circlearrowright k,(2,3))=(C_4,1 \circlearrowright k,(1,2)). \end{align*}\)