Galilean transformation has part(s) rotation matrix
G(R′, v′, a′, s′) ⋅ G(R, v, a, s) = G(R′ R, R′ v + v′, R′ a + a′ + v′ s, s′ + s).The set of all Galilean transformations Gal(3) forms a group with composition as the group operation.
The group is sometimes represented as a matrix group with spacetime events (x, t, 1) as vectors where t is real and x ∈ R3 is a position in space. 
The action is given by(
            
              
                
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    {\displaystyle {\begin{pmatrix}R&v&a\\0&1&s\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\t\\1\end{pmatrix}}={\begin{pmatrix}Rx+vt+a\\t+s\\1\end{pmatrix}},}
  where s is real and v, x, a ∈ R3 and R is a rotation matrix. 
The composition of transformations is then accomplished through matrix multiplication. Care must be taken in the discussion whether one restricts oneself to the connected component group of the orthogonal transformations.
Gal(3) has named subgroups. The identity component is denoted SGal(3).
Let m represent the transformation matrix with parameters v, R, s, a:
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