/*
2"M22 as permutations on 352a points.
On the cosets of [a conjugate of] M3' = A7, where
M3 = <x,yxy^2xy^-1xyxyxy^2> = 2 x A7.
*/

G<x,y>:=PermutationGroup<352|\[
2,1,5,6,3,4,9,11,7,14,8,17,19,10,18,22,12,15,13,26,28,16,31,33,35,
20,38,21,34,41,23,44,24,29,25,48,50,27,52,53,30,55,57,32,59,60,61,36,64,37,
66,39,40,70,42,73,43,75,45,46,47,80,82,49,85,51,88,90,92,54,95,97,56,100,58,
103,105,107,109,62,112,63,114,116,65,119,121,67,124,68,126,69,108,129,71,132,72,134,136,74,
139,141,76,144,77,146,78,93,79,130,150,81,153,83,152,84,157,159,86,162,87,164,166,89,168,
91,171,172,94,110,175,96,178,98,177,99,181,183,101,155,102,187,188,104,165,106,192,193,195,111,
196,115,113,179,140,189,117,202,118,204,206,120,208,122,145,123,180,125,190,213,127,128,216,218,131,
219,135,133,154,167,137,225,138,227,229,231,142,143,156,169,236,147,148,239,149,151,243,221,245,223,
224,158,246,160,248,161,230,163,238,253,255,256,170,237,232,173,261,174,176,264,198,266,200,201,182,
267,184,269,185,207,186,215,274,276,277,191,214,209,194,282,283,252,197,285,199,203,289,205,290,271,
292,242,210,294,211,212,286,288,298,296,217,302,273,220,304,222,226,308,228,309,250,311,263,233,313,
234,235,305,307,317,315,240,241,319,244,257,321,258,247,249,326,251,323,254,314,260,316,259,328,303,
330,262,300,265,278,325,279,268,270,334,272,332,275,295,281,297,280,336,284,338,287,337,293,340,306,
291,343,299,331,301,329,312,345,310,347,318,322,320,346,324,342,341,327,350,333,339,335,352,351,344,
349,348]
,\[
3,4,2,1,7,8,10,12,13,15,16,18,20,21,5,23,24,6,25,27,29,30,32,34,36,
37,9,39,40,42,43,11,45,46,47,49,31,51,33,14,54,56,26,58,28,17,62,63,19,65,
67,68,69,71,72,22,74,76,77,78,79,81,83,84,86,87,89,91,93,94,96,98,99,101,102,
104,106,108,110,111,35,113,115,117,118,120,122,123,38,125,127,128,129,130,131,41,133,135,137,138,
140,142,143,44,145,147,148,109,60,149,151,152,154,155,48,156,158,160,161,50,163,165,114,167,169,
170,52,173,53,174,176,177,179,162,55,180,182,184,185,57,186,168,134,189,190,191,59,194,61,159,
197,132,198,199,88,200,201,64,203,205,207,103,209,210,66,144,211,75,212,214,202,215,217,70,183,
220,112,221,222,223,224,73,226,228,230,232,233,124,234,235,237,225,238,240,241,242,80,244,82,227,
196,192,247,136,85,249,250,251,252,254,188,90,257,258,259,260,92,262,263,95,265,97,204,219,171,
268,116,100,270,271,272,273,275,166,105,278,279,280,281,107,284,157,195,286,287,288,150,206,291,119,
231,121,293,295,296,297,261,126,299,300,301,303,181,218,305,306,307,175,229,310,139,208,141,312,314,
315,316,282,146,318,319,320,253,243,276,153,322,323,324,325,248,245,327,304,164,294,328,246,172,329,
213,274,264,255,178,331,332,333,321,269,266,335,285,187,313,336,267,193,337,236,339,292,298,341,342,
338,283,344,216,326,311,317,346,330,302,348,239,334,349,343,350,351,345,256,347,352,340,277,309,289,
290,308]
>;
A:=x;B:=y;

M3:=sub<G|x,y*x*y^2*x*y^-1*x*y*x*y*x*y^2>;
DM3:=DerivedSubgroup(M3);
// A7 x 2 and A7.