The block cipher introduced in our Module $3$ lecture notes encrypts

five bits at a time. Now consider the analogous cipher that encrypts

seven bits at a time. Here a key $($w$,$ P$,$ k$)$ consists of: $(1)$ a bit string w

of length $7,(2)$ a permutation on the digits d$6$d$5$d$4$d$3$d$2$d$1$d$0,$ and $(3)$ an

odd integer k modulo $27=128.$ How many keys of the form $($w$,$ P$,$ k$)$

are there?