Coclass of \(\mathrm{Gal}(\Bbbk_2^{(2)}/\Bbbk)\) for some fields \(\Bbbk=\mathbb{Q}(\sqrt{p_1p_2q},\sqrt{-1})\) with 2-class groups of types (2,2,2) (Q2788566)

Let \(p_{1},p_{2}\) and \(q\) be primes with \(p_{1}\equiv1~(\text{mod}\) \(8)\), \(p_{2}\equiv5 (\text{mod}\) \( 8)\) and \(q\equiv3\) \((\text{mod}\) \(4)\). Then the biquadratic field \(k:=\) \(\mathbb{Q(} \sqrt{p_{1}p_{2}q},\sqrt{-1})\) has a \(2\)-class group of type \((2,2,2)\). Let \(k_{1}^{(1)}\) and \(k_{2}^{(2)}\) denote the first and second Hilbert \(2\)-class fields of \(k\), respectively. Continuing earlier work (see, for example, [\textit{A. Azizi} et al., Int. J. Number Theory 11, No. 4, 1177--1215 (2015; Zbl 1319.11079)]) the authors study the structure of \(G:=\mathrm{Gal}(k_{2}^{(2)}/k)\). Let \(n\) and \(m\) be defined by \(2^{n}=h(-p_{2}q)\) and \(2^{m+1}=h(-p_{1})\). Then the main result is the following (Theorem 2.2): the \(2\)-class field tower of \(k\) is \(2\); NEWLINE\[NEWLINEG=\left\langle \rho ,\tau,\sigma:\rho^{4}=\sigma^{2^{n+1}}=\tau^{2^{m+1}}=[\tau,\sigma]=1,\rho ^{2}=\sigma^{2^{n}},[\rho,\sigma]=\sigma^{2},[\rho,\tau]=\tau^{2}\right\rangle NEWLINE\]NEWLINE and has derived group \(G^{\prime}=\left\langle \sigma^{2},\tau ^{2}\right\rangle \); the order of \(G\) is \(2^{m+n+3}\) and its nilpotence class is \(\max(n,m)+1\).