<p>In this paper, we provide a new equivalent characterization of the existence of the dual group inverse (DGI) of a dual matrix. We provide necessary and sufficient conditions for the existence of DGI for the product <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\widehat{A}}{\widehat{B}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>A</mi> <mo stretchy="true">^</mo> </mover> <mover accent="true"> <mi>B</mi> <mo stretchy="true">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation>, under the circumstances where either <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\widehat{A}}^\#\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mover accent="true"> <mi>A</mi> <mo stretchy="true">^</mo> </mover> </mrow> <mo>#</mo> </msup> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\widehat{B}}^\#\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mover accent="true"> <mi>B</mi> <mo stretchy="true">^</mo> </mover> </mrow> <mo>#</mo> </msup> </math></EquationSource> </InlineEquation> exists. Furthermore, we provide equivalent conditions of various mixed reverse order laws of DGI and weak dual group inverse (WDGI), and investigate the relation between <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\({\widehat{A}}\overset{D-\#}{\le }{\widehat{B}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>A</mi> <mo stretchy="true">^</mo> </mover> <mover> <mo>≤</mo> <mrow> <mi>D</mi> <mo>-</mo> <mo>#</mo> </mrow> </mover> <mover accent="true"> <mi>B</mi> <mo stretchy="true">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> and the reverse order law of DGI. Finally, we present an application of the mixed reverse order law of DGI and WDGI in solving dual equations.</p>