Normal extensions and induced characters of \(2\)-groups \(M_n\) (Q5937622)

Let \(D_n\), \(Q_n\) and \(SD_n\) be the dihedral group, the generalized quaternion group and semidihedral group of order \(2^{n+1}\), respectively. Let \(M_n=\langle a,b\mid a^{2^n}=b^2=1,\;a^b=a^{1+2^{n-1}},\;n\geq 3\rangle\) be the \(2\)-group of class \(2\) and order \(2^{n+1}\) with cyclic subgroup of index \(2\). We define \(2\)-groups \(G_t(D_n)\), \(G_t(Q_n)\) (\(0\leq t\leq n-2\)), \(G_t(M_n)^+\) and \(G_t(M_n)^-\) (\(0\leq t\leq n-3\)) as follows NEWLINE\[NEWLINE\begin{aligned} G_t(D_n)&=\langle a,b,x\mid a^{2^n}=b^2=x^{2^t}=1,\;a^b=a^{-1},\;a^x=a^{1+2^{n-t}},\;b^x=b\rangle,\\ G_t(Q_n)&=\langle a,b,x\mid a^{2^n}=x^{2^t}=1,\;b^2=a^{2^{n-1}},\;a^b=a^{-1},\;a^x=a^{1+2^{n-t}},\;b^x=b\rangle,\\ G_t(M_n)^+&=\langle a,b\mid a^{2^n}=b^{2^{t+1}}=1,\;a^b=a^{1+2^{n-t-1}}\rangle,\\ G_t(M_n)^-&=\langle a,b\mid a^{2^n}=b^{2^{t+1}}=1,\;a^b=a^{-1+2^{n-t-1}}\rangle,\end{aligned}NEWLINE\]NEWLINE Obviously, \(G_0(D_n)=D_n\), \(G_0(Q_n)=Q_n\), \(G_0(M_n)^+=M_n\) and \(G_0(M_n)^-=SD_n\). We formulate only one result.NEWLINENEWLINENEWLINETheorem 5. Let \(M_n=\langle a,b\rangle\) be a normal subgroup of index \(2^t\) in a \(2\)-group \(G\) (\(t\geq 1\)) such that \(\langle a\rangle\) is normal in \(G\). Let \(\phi\) be a faithful irreducible character of \(M_n\) such that \(\phi^G\in\text{Irr}(G)\). Then \(G\in\{G_t(D_n),D_t(Q_n),G_t(M_n)^+,G_t(M_n)^-\}\).