The full symmetry group of Minkowski space as an affine pseudo-Euclidean space is the group ##\text{ISO}(1,3)^{\uparrow}##, i.e., the proper orthochronous Poincare group (I leave out the non-continuously connected parts, including time reversal and space reflections). They are the semidirect product of the proper orthochronous Lorentz group, represented by ##\mathrm{R}^{4 \times 4}## matrices, ##{\Lambda^{\mu}}_{\nu}## fulfilling
$$\eta_{\mu \nu} {\Lambda^{\mu}}_{\rho} {\Lambda^{\nu}}_{\sigma} =\eta_{\rho
\sigma}, \quad \mathrm{det} \hat{\Lambda}=+1, \quad {\Lambda^0}_{0} \geq 1.$$
with the pseudo-metric components ##(\eta_{\mu \nu})=\mathrm{diag}(1,-1,-1,-1)##, and translations ##x^{\mu} \rightarrow x^{\mu}+a^{\mu}## with ##a^{\mu}=\text{const}.##
The group element is ##(\hat{\Lambda},a)##, acting on the spacetime components as
$$(\hat{\Lambda}, a) \boldsymbol{x}=\hat{\Lambda} \boldsymbol{x} +a.$$
The group product thus is given by
$$(\hat{\Lambda}_2,a_2)(\hat{\Lambda}_1,a_1)=(\hat{\Lambda}_2 \hat{\Lambda}_1,a_2+\hat{\Lambda}_2 a_1).$$
