gap> f1:=(22,33,44,55,66)(77,88,99,110,121) > (21,32,43,54,65)(17,28,39,50,61) > (13,24,35,46,57)(74,83,94,105,116) > (72,81,92,103,114)(68,87,98,109,120) > (12,23,34,45,56)(16,27,38,49,60) > (14,25,36,47,58)(75,84,95,106,117) > (73,82,93,104,115)(67,86,97,108,119); ( 12, 23, 34, 45, 56)( 13, 24, 35, 46, 57)( 14, 25, 36, 47, 58) ( 16, 27, 38, 49, 60)( 17, 28, 39, 50, 61)( 21, 32, 43, 54, 65) ( 22, 33, 44, 55, 66)( 67, 86, 97,108,119)( 68, 87, 98,109,120) ( 72, 81, 92,103,114)( 73, 82, 93,104,115)( 74, 83, 94,105,116) ( 75, 84, 95,106,117)( 77, 88, 99,110,121) gap> gap> f2:=(11,66,77,88,33)(44,55,121,132,99) > (10,59,72,83,30)(4,63,76,87,24) > (6,65,68,79,26)(41,50,114,125,96) > (8,57,70,81,28)(52,116,127,98,43) > (5,64,67,78,25)(29,9,58,71,82)(42,51,115,126,97) > (7,56,69,80,27)(40,49,113,124,95)(53,117,128,89,34) > (39,48,112,123,94)(35,54,118,129,90); ( 4, 63, 76, 87, 24)( 5, 64, 67, 78, 25)( 6, 65, 68, 79, 26) ( 7, 56, 69, 80, 27)( 8, 57, 70, 81, 28)( 9, 58, 71, 82, 29) ( 10, 59, 72, 83, 30)( 11, 66, 77, 88, 33)( 34, 53,117,128, 89) ( 35, 54,118,129, 90)( 39, 48,112,123, 94)( 40, 49,113,124, 95) ( 41, 50,114,125, 96)( 42, 51,115,126, 97)( 43, 52,116,127, 98) ( 44, 55,121,132, 99) gap> gap> r2:=(11,66,77,88,33)(44,55,121,132,99) > (10,59,72,83,30)(4,63,76,87,24) > (6,65,68,79,26)(41,50,114,125,96) > (8,57,70,81,28)(52,116,127,98,43) > (5,64,67,78,25)(29,9,58,71,82)(42,51,115,126,97) > (7,56,69,80,27)(40,49,113,124,95)(53,117,128,89,34) > (39,48,112,123,94)(35,54,118,129,90) > (13,15,17,19,21)(109,107,105,103,101) > (12,14,16,18,20)(108,106,104,102,100) > (1,60,73,84,31)(3,62,75,86,23) > (2,61,74,85,32)(120,131,92,37,46) > (38,47,111,122,93)(36,45,119,130,91); ( 1, 60, 73, 84, 31)( 2, 61, 74, 85, 32)( 3, 62, 75, 86, 23) ( 4, 63, 76, 87, 24)( 5, 64, 67, 78, 25)( 6, 65, 68, 79, 26) ( 7, 56, 69, 80, 27)( 8, 57, 70, 81, 28)( 9, 58, 71, 82, 29) ( 10, 59, 72, 83, 30)( 11, 66, 77, 88, 33)( 12, 14, 16, 18, 20) ( 13, 15, 17, 19, 21)( 34, 53,117,128, 89)( 35, 54,118,129, 90) ( 36, 45,119,130, 91)( 37, 46,120,131, 92)( 38, 47,111,122, 93) ( 39, 48,112,123, 94)( 40, 49,113,124, 95)( 41, 50,114,125, 96) ( 42, 51,115,126, 97)( 43, 52,116,127, 98)( 44, 55,121,132, 99) (100,108,106,104,102)(101,109,107,105,103) gap> gap> f3:=r2*f1*r2^-1; ( 1, 14, 80, 93, 40)( 2, 15, 81, 94, 41)( 6, 19, 85, 98, 35) ( 7, 20, 86, 89, 36)( 8, 21, 87, 90, 37)( 9, 12, 78, 91, 38) ( 11, 22, 88, 99, 44)( 45, 64, 75,126,100)( 46, 65, 76,127,101) ( 50, 59, 70,131,105)( 51, 60, 71,122,106)( 52, 61, 72,123,107) ( 53, 62, 73,124,108)( 55, 66, 77,132,110) gap> f8:=r2^2*f1*r2^-2; ( 1, 58,111,108, 42)( 2, 59,112,109, 43)( 3, 60,113,100, 34) ( 5, 62,115,102, 36)( 6, 63,116,103, 37)( 10, 57,120,107, 41) ( 11, 66,121,110, 44)( 12, 69,122, 95, 31)( 13, 70,123, 96, 32) ( 17, 74,127, 90, 26)( 18, 75,128, 91, 27)( 19, 76,129, 92, 28) ( 20, 67,130, 93, 29)( 22, 77,132, 99, 33) gap> f7:=r2^3*f1*r2^-3; ( 1, 49,102, 89, 23)( 3, 51,104, 91, 25)( 4, 52,105, 92, 26) ( 8, 46,109, 96, 30)( 9, 47,100, 97, 31)( 10, 48,101, 98, 32) ( 11, 55,110, 99, 33)( 15, 61,116,129, 79)( 16, 62,117,130, 80) ( 17, 63,118,131, 81)( 18, 64,119,122, 82)( 20, 56,111,124, 84) ( 21, 57,112,125, 85)( 22, 66,121,132, 88) gap> f6:=r2^4*f1*r2^-4; ( 2, 52,118, 68, 13)( 3, 53,119, 69, 14)( 4, 54,120, 70, 15) ( 5, 45,111, 71, 16)( 7, 47,113, 73, 18)( 8, 48,114, 74, 19) ( 11, 55,121, 77, 22)( 23, 42,106,130, 78)( 24, 43,107,131, 79) ( 28, 37,101,125, 83)( 29, 38,102,126, 84)( 30, 39,103,127, 85) ( 31, 40,104,128, 86)( 33, 44,110,132, 88) gap> gap> slice:=Group(f1,f2,f3,f6,f7,f8); Group( ( 12, 23, 34, 45, 56)( 13, 24, 35, 46, 57)( 14, 25, 36, 47, 58) ( 16, 27, 38, 49, 60)( 17, 28, 39, 50, 61)( 21, 32, 43, 54, 65) ( 22, 33, 44, 55, 66)( 67, 86, 97,108,119)( 68, 87, 98,109,120) ( 72, 81, 92,103,114)( 73, 82, 93,104,115)( 74, 83, 94,105,116) ( 75, 84, 95,106,117)( 77, 88, 99,110,121), ( 4, 63, 76, 87, 24) ( 5, 64, 67, 78, 25)( 6, 65, 68, 79, 26)( 7, 56, 69, 80, 27) ( 8, 57, 70, 81, 28)( 9, 58, 71, 82, 29)( 10, 59, 72, 83, 30) ( 11, 66, 77, 88, 33)( 34, 53,117,128, 89)( 35, 54,118,129, 90) ( 39, 48,112,123, 94)( 40, 49,113,124, 95)( 41, 50,114,125, 96) ( 42, 51,115,126, 97)( 43, 52,116,127, 98)( 44, 55,121,132, 99), ( 1, 14, 80, 93, 40)( 2, 15, 81, 94, 41)( 6, 19, 85, 98, 35) ( 7, 20, 86, 89, 36)( 8, 21, 87, 90, 37)( 9, 12, 78, 91, 38) ( 11, 22, 88, 99, 44)( 45, 64, 75,126,100)( 46, 65, 76,127,101) ( 50, 59, 70,131,105)( 51, 60, 71,122,106)( 52, 61, 72,123,107) ( 53, 62, 73,124,108)( 55, 66, 77,132,110), ( 2, 52,118, 68, 13) ( 3, 53,119, 69, 14)( 4, 54,120, 70, 15)( 5, 45,111, 71, 16) ( 7, 47,113, 73, 18)( 8, 48,114, 74, 19)( 11, 55,121, 77, 22) ( 23, 42,106,130, 78)( 24, 43,107,131, 79)( 28, 37,101,125, 83) ( 29, 38,102,126, 84)( 30, 39,103,127, 85)( 31, 40,104,128, 86) ( 33, 44,110,132, 88), ( 1, 49,102, 89, 23)( 3, 51,104, 91, 25) ( 4, 52,105, 92, 26)( 8, 46,109, 96, 30)( 9, 47,100, 97, 31) ( 10, 48,101, 98, 32)( 11, 55,110, 99, 33)( 15, 61,116,129, 79) ( 16, 62,117,130, 80)( 17, 63,118,131, 81)( 18, 64,119,122, 82) ( 20, 56,111,124, 84)( 21, 57,112,125, 85)( 22, 66,121,132, 88), ( 1, 58,111,108, 42)( 2, 59,112,109, 43)( 3, 60,113,100, 34) ( 5, 62,115,102, 36)( 6, 63,116,103, 37)( 10, 57,120,107, 41) ( 11, 66,121,110, 44)( 12, 69,122, 95, 31)( 13, 70,123, 96, 32) ( 17, 74,127, 90, 26)( 18, 75,128, 91, 27)( 19, 76,129, 92, 28) ( 20, 67,130, 93, 29)( 22, 77,132, 99, 33) ) gap> n:=Size(slice); 234350966676931601388773460576233202938711900160000000000 gap> Factors(n); [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 11, 11, 13, 13, 17, 19, 23, 29 ] gap> f1; ( 12, 23, 34, 45, 56)( 13, 24, 35, 46, 57)( 14, 25, 36, 47, 58) ( 16, 27, 38, 49, 60)( 17, 28, 39, 50, 61)( 21, 32, 43, 54, 65) ( 22, 33, 44, 55, 66)( 67, 86, 97,108,119)( 68, 87, 98,109,120) ( 72, 81, 92,103,114)( 73, 82, 93,104,115)( 74, 83, 94,105,116) ( 75, 84, 95,106,117)( 77, 88, 99,110,121) gap> quit;