/*
L2(27) presented on its (2,3,14;14)-generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^3,
(x*y)^14,
(x*y*x*y*x*y^-1*x*y*x*y^-1)^3,
(x*y)^3*(x*y^-1)^2*(x*y)^2*(x*y^-1)^3*(x*y)^2*(x*y^-1)^2
>;
// The last relation is equivalent to [y,xyxyxy^-1xy^-1xyxyx] = 1.

M1:=sub<G|y,x*y*x*y*x*y*x*y*x*y*x*y*x*y^-1*x>;
M2:=sub<G|x,x^y>;
M3:=sub<G|x,x^(y*x*y)>;
M4:=sub<G|x,y^(x*y*x*y^-1)>;

H1:=sub<G|x,x^(y*x*y^-1)>;
H2:=sub<G|y,x^(y*x*y^-1*x)>;

M1a:=sub<G|y*x*y*x*y*x*y*x*y*x*y*x*y*x*y^-1*x,x*y*x*y*x*y*x*y*x*y*x*y*x*y^-1*x*y>;
M1b:=sub<G|y,x*y*x*y*x*y*x*y*x*y*x*y*x*y^-1*x*(x*y*x*y*x*y^-1*x*y*x*y^-1)^6>;
M1ab:=sub<G|y*x*y*x*y*x*y*x*y*x*y*x*y*x*y^-1*x*(x*y*x*y*x*y^-1*x*y*x*y^-1)^6,x*y*x*y*x*y*x*y*x*y*x*y*x*y^-1*x*(x*y*x*y*x*y^-1*x*y*x*y^-1)^6*y>;

/*
M1a and M1ab give faithful perm actions of degree 84 when (x*y)^14 is absent
and we have G = 3 x L2(27).
M1b and M1ab give faithful perm actions of degree 112 when (xyxyxy^-1xyxy^-1)^3
is absent and we have G = 4 o L2(27).
In the latter case, the action of G on the cosets of M1 has degree 56 and
gives 2 x L2(27).

We have M1ab < M1a < M1 and M1ab < M1b < M1.
*/