For $$E({\mathbb {T}})$$ E(T) being the endomorphism group of the circle group $${\mathbb {T}}$$ T , the Furstenberg–Ellis–Namioka Structure Theorem of the CHART group $$G=E({\mathbb {T}})\times {\mathbb {T}}$$ G=E(T)×T with the product $$(f,u)(g,v)=(fg,uvf\circ g(\mathrm{e}^{i}))$$ (f,u)(g,v)=(fg,uvf∘g(ei)) is known to be equal to $$\{G,1_{\mathbb {T}}\times {\mathbb {T}},\{(1_{\mathbb {T}},1)\}\}$$ {G,1T×T,{(1T,1)}} . A somewhat similar group structure is known to exist on $$E({\mathbb {T}})\times E({\mathbb {T}})\times {\mathbb {T}}$$ E(T)×E(T)×T , studied by Milnes. We give an explicit characterization of the Furstenberg–Ellis–Namioka Structure Theorem for an admissible subgroup $$\Sigma $$ Σ of $$E({\mathbb {T}})\times E({\mathbb {T}})\times {\mathbb {T}}$$ E(T)×E(T)×T , where $$\Sigma $$ Σ is the Ellis group of the Hahn-type skew product dynamical system on the 3-torus $${\mathbb {T}}^3$$ T3 .