Since it is not possible for me to reply directly in that new Maple Primes:
I branched. Feel free to re-join for a reasonable structure. What a mess.
I am rusty on such (may be it is 'obvious' via Lie theory). Your group is just the
group of invertible matrices over the integers (this follows from algebra). And as
det = -1 the possible 'root' K must be one for n=odd.
Looking at Eigenvalues of K they must be conjugated or purely real and different (dim=2)
conjugate(a)^n+a^n = 29, conjugate(a)^n*a^n = -1 should hold for the Eigenvalues of K.
which should follow from looking at the characteristic polynomials of products.
The last is abs(a)^(2*n) = - 1. So I would say: no, not possible.