What is the order of the group?

Question

I am trying to calculate the order of the subgroup of the group of unitary matrices generated by the matrices $$ \frac{1}{\sqrt{2}}\left[ {\begin{array}{cc} 1 & 1 \\ 1 & -1 \ \end{array} } \right] , \left[ {\begin{array}{cc} 1 & 0 \\ 0 & i \ \end{array} } \right] .$$ I know so far that should be a multiple of 6.

Answer 1

The group has order 192 (which was obtained using GAP:

gap> M1:=[[1,1],[1,-1]]*1/ER(2);;
gap> M2:=[[1,0],[0,E(4)]];;
gap> g:=Group(M1,M2);;
gap> Size(g);
192

I'm not sure how to prove this by hand without going into substantial boring arithmetic: The representation is irreducible (otherwise the group would be abelian). Most "obvious" vectors have an orbit length of 48.

Somewhat cheating (as I got this with the computer as well), you could verify that the group satisfies the following relators $$F_1^2,\qquad F_2^4,\qquad F_2^{-1}(F_1F_2)^3(F_1F_2^{-1})^2F_1,\qquad (F_2^{-1}F_1F_2F_1)^6 $$ and use these to show its order is at most 192.