are perfectly isometric in the sense of Broué [@Broue].

By [@sasaki], if $p$ is odd and $P$ is metacyclic, then $\tilde{P}$ is cyclic. If $p = 3$ and $G = SL(2, 2^{3^n}) \rtimes C_3$, then $P$ is isomorphic to the metacylic $3$-group $M_{(n + 1 )+ 1}(3)$ (cf. [@HKK], (4.3)).

Now the above theorem is a generalization of [@JWata], Proposition 6. If $\tilde{P} = 1$, then $G$ is $p$-nilpotent, hence the above is clear. We will determine the ordinary irreducible characters in $b(G)$ (Propositions 5 in $\S 2$ below), and the generalized decomposition numbers associated with $b(G)$ (Propositions 6 and 7 in $\S 3$ below), and then we obtain Theorem 1 from Theorem 2 below.

We refer [@NT], [@T] and [@Gorenstein] for the notation and terminology. Let $H$ be a subgroup of $G$. For $x \in G$, we denote by $x^H$ the $H$-conjugacy class containing $x$. For a character $\chi$ of $G$, the restriction of $\chi$ to $H$ is denoted by $\chi \!\downarrow_{H}$. For a character $\zeta$ of $H$, the induced character $\zeta$ to $G$ is denoted by $\zeta\!\uparrow_{H}^{G}$. For characters $\chi$, $\chi'$ of $G$, the innner product of $\chi$ and $\chi'$ is denoted by $(\chi, \chi')$. For a complex 