<p>A graph is called an integral graph when all eigenvalues of its adjacency matrix are integers. We study which Cayley graphs over a non-abelian group <Equation ID="Equ13"> <EquationSource Format="TEX">\( T_{8n}=\left\langle a,b\mid a^{2n}=b^8=e,a^n=b^4,b^{-1}ab=a^{-1} \right\rangle \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>T</mi> <mrow> <mn>8</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mfenced close="〉" open="〈"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∣</mo> <msup> <mi>a</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mn>8</mn> </msup> <mo>=</mo> <mi>e</mi> <mo>,</mo> <msup> <mi>a</mi> <mi>n</mi> </msup> <mo>=</mo> <msup> <mi>b</mi> <mn>4</mn> </msup> <mo>,</mo> <msup> <mi>b</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>a</mi> <mi>b</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mfenced> </mrow> </math></EquationSource> </Equation>are integral graphs. Based on the group representation theory, we first give the irreducible matrix representations and characters of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(T_{8n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mrow> <mn>8</mn> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. Then, we give necessary and sufficient conditions for which Cayley graphs over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(T_{8n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mrow> <mn>8</mn> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> are integral graphs. As applications, we also characterize some families of connected integral Cayley graphs over <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(T_{8n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mrow> <mn>8</mn> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>.</p>