K4<i> := CyclotomicField(4); // provides i with i^2 = -1
GL2 := GL(2,K4); // the general linear group
I := elt< GL2 | i,0,0,-i >;
J := elt< GL2 | 0,1,-1,0 >;
K := I*J;
H8 := sub< GL2 | I,J >;
Order(H8); [I^2 eq -Id(GL2), J^2 eq -Id(GL2),
  K^2 eq -Id(GL2), I*J*K eq -Id(GL2)];
A := elt< GL2 | (1+i)/2,(1+i)/2,(-1+i)/2,(1-i)/2 >;
B := elt< GL2 | (1+i)/2,(-1+i)/2,(1+i)/2,(1-i)/2 >;
A*I eq J*A; A*J eq K*A; A*K eq I*A;
BT24 := sub<GL2 | [A,I] >; Order(BT24); // Checks that BT24 is correct
B in BT24;
A*B*A eq B^2; A*B*A eq B^2;
T12 := quo< BT24 | -Id(GL2) >; T12 eq Alt(4);
// Although BT24 is a matrix group, Magma makes the quotient
// BT24/(centre) as a permutation group, that is identical to A4.