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Problem #467

Permutation Order I
 Public ★(x6) 02/02/18 by Philippe_57721 7xp Programming 100.0%

Let $\mathfrak{S}(n)$ be the set of all permutations of [1..n]

The order of a permutation $\mathfrak{p}$ is the smallest integer e such as $\mathfrak{p}^e = Id$
We apply the permutation e times to itself and get the identity permutation {1,2,...,n}

For $\mathfrak{S}(4)$ :
 Order(1 2 3 4) = 1
Order(1 2 4 3) = 2
Order(1 3 2 4) = 2
Order(1 3 4 2) = 3
Order(1 4 2 3) = 3
Order(1 4 3 2) = 2
Order(2 1 3 4) = 2
Order(2 1 4 3) = 2
Order(2 3 1 4) = 3
Order(2 3 4 1) = 4
Order(2 4 1 3) = 4
Order(2 4 3 1) = 3
Order(3 1 2 4) = 3
Order(3 1 4 2) = 4
Order(3 2 1 4) = 2
Order(3 2 4 1) = 3
Order(3 4 1 2) = 2
Order(3 4 2 1) = 4
Order(4 1 2 3) = 4
Order(4 1 3 2) = 3
Order(4 2 1 3) = 3
Order(4 2 3 1) = 2
Order(4 3 1 2) = 4
Order(4 3 2 1) = 2

What is the largest order for the elements of $\mathfrak{S}(85)$?
For which permutation do we get this order? Give the index in lexicographic order (0-origin) of the 1st one

Answer format: Maximal order,Index modulo $10^9$

Example: 420,257453673 // For $\mathfrak{S}(20)$

[My timing: 40 sec]

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