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Abstract Let Rr0,Rr1:S1⟶S1 $R_{r_{0}}, R_{r_{1}}: \mathbb{S}^{1}\longrightarrow \mathbb{S} ^{1}$ be rotations on the unit circle S1 $\mathbb{S}^{1}$ and define f:Σ2×S1⟶Σ2×S1 $f: \varSigma _{2}\times \mathbb{S}^{1}\longrightarrow \varSigma _{2}\times \mathbb{S}^{1}$ as f(x,t)=(σ(x),Rrx1(t)), $$ f(x, t)=\bigl(\sigma (x), R_{r_{x_{1}}}(t)\bigr), $$ for x=x1x2⋯∈Σ2:={0,1}N $x=x_{1}x_{2}\cdots \in \varSigma _{2}:=\{0, 1\}^{\mathbb{N}}$, t∈S1 $t\in \mathbb{S}^{1}$, where σ:Σ2⟶Σ2 $\sigma: \varSigma _{2}\longrightarrow \varSigma _{2}$ is the shift, and r0 $r_{0}$ and r1 $r_{1}$ are rotational angles. It is first proved that the system (Σ2×S1,f) $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ exhibits maximal distributional chaos for any r0,r1∈R $r_{0}, r_{1}\in \mathbb{R}$ (no assumption of r0,r1∈R∖Q $r_{0}, r_{1}\in \mathbb{R}\setminus \mathbb{Q}$), generalizing Theorem 1 in Wu and Chen (Topol. Appl. 162:91–99, 2014). It is also obtained that (Σ2×S1,f) $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is cofinitely sensitive and (Mˆ1,Mˆ1) $(\hat{\mathscr{M}} ^{1}, \hat{\mathscr{M}}^{1})$-sensitive and that (Σ2×S1,f) $(\varSigma _{2}\times \mathbb{S}^{1}, f)$ is densely chaotic if and only if r1−r0∈R∖Q $r_{1}-r_{0} \in \mathbb{R}\setminus \mathbb{Q}$.