Dimension1 Octahedral
Group $G=S_4$ does not have faithful 2-dimensional representation, its binary-extension $C_2.S_4$ has 2 2-dimensional faithful representation, one representation is as follows:
\[\begin{align*} G=\langle \begin{pmatrix} \zeta_8 & 0\\ 0 & \zeta_8^{-1} \end{pmatrix}, \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}, \begin{pmatrix} (1+\zeta_4)/2 & (-1+\zeta_4)/2\\ (1+\zeta_4)/2 & (1-\zeta_4)/2 \end{pmatrix} \rangle. \end{align*}\]where $\zeta_4$ is a 4-th root of unity, $\zeta_8$ is a 8-th root of unity.
The Magma code is as follows:
FScale:=CyclotomicField(24:Sparse:=true); F:=FScale; Z8:=F.1; G:=MatrixGroup<2,F| [Z8,0, 0,Z8^(-1)], [0,1, -1,0], [(Z8^2+1)/2,(Z8^2-1)/2, (Z8^2+1)/2,(1-Z8^2)/2]>;
The Burnside symbol is
\[\begin{align*} [\mathbb{P}^1 \circlearrowleft G]=&(1,G \circlearrowright k(x),())\\ &+(C_2,1 \circlearrowright k,(1))\\ &+(C_3,1 \circlearrowright k,(2))+(C_4,1 \circlearrowright k,(1)) \end{align*}\]where
\[\begin{align*} C_2=\langle \begin{pmatrix} (-\zeta_8-\zeta_8^3)/2 & (\zeta_8+\zeta_8^3)/2\\ (\zeta_8+\zeta_8^3)/2 & (\zeta_8+\zeta_8^3)/2 \end{pmatrix} \rangle, C_3=\langle \begin{pmatrix} (1-\zeta_4)/2 & (1-\zeta_4)/2\\ (-1-\zeta_4)/2 & (1+\zeta_4)/2 \end{pmatrix} \rangle, C_4=\langle \begin{pmatrix} (\zeta_8-\zeta_8^3)/2 & (\zeta_8-\zeta_8^3)/2\\ (-\zeta_8+\zeta_8^3)/2 & (\zeta_8-\zeta_8^3)/2 \end{pmatrix} \rangle \end{align*}\]There are conjugaction relations:
\[\begin{align*} (C_3,1 \circlearrowright k,(1))=(C_3,1 \circlearrowright k,(2))\\ (C_4,1 \circlearrowright k,(1))=(C_4,1 \circlearrowright k,(3)) \end{align*}\]The other projective linear representation of $S_4$ gives the same Burnside symbols.