INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF PURE MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 384 Minimal number of generators: 65 Number of equivalence classes of cusps: 32 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/4 -1/2 -1/4 -3/13 -1/6 -1/7 -1/8 -1/9 0/1 1/6 3/16 1/5 1/4 3/11 2/7 5/16 1/3 3/8 2/5 7/16 1/2 9/16 5/8 31/48 2/3 11/16 11/15 3/4 13/16 7/8 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/0 -7/8 1/0 -6/7 -1/1 1/0 -5/6 0/1 -9/11 -1/4 0/1 -13/16 0/1 -4/5 0/1 1/2 -7/9 1/1 2/1 -10/13 2/1 1/0 -3/4 1/0 -5/7 -1/1 0/1 -7/10 0/1 -9/13 -1/2 0/1 -11/16 0/1 -2/3 0/1 1/0 -11/17 3/1 1/0 -9/14 -2/1 -7/11 0/1 1/0 -5/8 1/0 -3/5 -1/1 1/0 -4/7 -3/2 -1/1 -9/16 -1/1 -5/9 -1/1 -4/5 -6/11 -2/3 -1/2 -1/2 0/1 -4/9 -3/2 -1/1 -7/16 -1/1 -3/7 -1/1 -2/3 -5/12 -1/2 -12/29 -2/3 -1/2 -7/17 -1/2 -1/3 -2/5 -1/2 0/1 -7/18 -2/3 -19/49 -7/13 -1/2 -31/80 -1/2 -12/31 -1/2 -4/9 -5/13 -1/2 0/1 -3/8 0/1 -1/3 -1/1 0/1 -5/16 0/1 -4/13 0/1 1/0 -7/23 -1/1 0/1 -3/10 0/1 -5/17 -1/1 1/0 -7/24 1/0 -2/7 -1/1 1/0 -5/18 -2/3 -3/11 -1/2 0/1 -7/26 0/1 -11/41 0/1 1/1 -4/15 2/1 1/0 -1/4 -1/1 -4/17 -4/7 -1/2 -3/13 -1/2 0/1 -5/22 0/1 -2/9 -1/1 1/0 -1/5 -1/1 -1/2 -3/16 -1/2 -2/11 -1/2 0/1 -3/17 -1/2 -1/3 -1/6 0/1 -1/7 -1/1 0/1 -1/8 -1/1 -1/9 -1/1 -2/3 0/1 -1/2 0/1 1/6 0/1 2/11 -1/2 0/1 3/16 -1/2 1/5 -1/2 -1/3 1/4 -1/3 4/15 -1/4 -2/9 3/11 -1/2 0/1 2/7 -1/3 -1/4 7/24 -1/4 5/17 -1/3 -1/4 3/10 0/1 4/13 -1/4 0/1 5/16 0/1 1/3 -1/3 0/1 3/8 0/1 5/13 -1/2 0/1 12/31 -4/7 -1/2 7/18 -2/5 2/5 -1/2 0/1 3/7 -2/5 -1/3 7/16 -1/3 4/9 -1/3 -3/10 5/11 -2/7 -1/4 1/2 0/1 5/9 -4/11 -1/3 9/16 -1/3 4/7 -1/3 -3/10 11/19 -2/7 -1/4 18/31 -1/4 -2/9 7/12 -1/3 3/5 -1/3 -1/4 5/8 -1/4 7/11 -1/4 0/1 9/14 -2/7 20/31 -8/31 -1/4 31/48 -1/4 11/17 -1/4 -3/13 2/3 -1/4 0/1 11/16 0/1 9/13 -1/2 0/1 7/10 0/1 12/17 -1/2 0/1 29/41 -2/5 -1/3 17/24 -1/3 5/7 -1/3 0/1 8/11 -2/7 -1/4 19/26 -2/7 11/15 -3/11 -1/4 3/4 -1/4 13/17 -1/3 -1/4 10/13 -1/4 -2/9 17/22 -2/9 7/9 -2/9 -1/5 11/14 -2/11 4/5 -1/6 0/1 13/16 0/1 9/11 0/1 1/0 14/17 -1/2 -2/5 19/23 -1/3 0/1 5/6 0/1 11/13 -1/2 0/1 6/7 -1/3 -1/4 7/8 -1/4 1/1 -1/4 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(15,14,16,15) (-1/1,-7/8) -> (7/8,1/1) Hyperbolic Matrix(65,56,224,193) (-7/8,-6/7) -> (2/7,7/24) Hyperbolic Matrix(47,40,-208,-177) (-6/7,-5/6) -> (-5/22,-2/9) Hyperbolic Matrix(95,78,-352,-289) (-5/6,-9/11) -> (-3/11,-7/26) Hyperbolic Matrix(287,234,352,287) (-9/11,-13/16) -> (13/16,9/11) Hyperbolic Matrix(129,104,160,129) (-13/16,-4/5) -> (4/5,13/16) Hyperbolic Matrix(79,62,-144,-113) (-4/5,-7/9) -> (-5/9,-6/11) Hyperbolic Matrix(127,98,-416,-321) (-7/9,-10/13) -> (-4/13,-7/23) Hyperbolic Matrix(113,86,-272,-207) (-10/13,-3/4) -> (-5/12,-12/29) Hyperbolic Matrix(47,34,-112,-81) (-3/4,-5/7) -> (-3/7,-5/12) Hyperbolic Matrix(17,12,-112,-79) (-5/7,-7/10) -> (-1/6,-1/7) Hyperbolic Matrix(95,66,-416,-289) (-7/10,-9/13) -> (-3/13,-5/22) Hyperbolic Matrix(287,198,416,287) (-9/13,-11/16) -> (11/16,9/13) Hyperbolic Matrix(65,44,96,65) (-11/16,-2/3) -> (2/3,11/16) Hyperbolic Matrix(209,136,272,177) (-2/3,-11/17) -> (13/17,10/13) Hyperbolic Matrix(385,248,-992,-639) (-11/17,-9/14) -> (-7/18,-19/49) Hyperbolic Matrix(81,52,176,113) (-9/14,-7/11) -> (5/11,1/2) Hyperbolic Matrix(111,70,176,111) (-7/11,-5/8) -> (5/8,7/11) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(17,10,-80,-47) (-3/5,-4/7) -> (-2/9,-1/5) Hyperbolic Matrix(127,72,224,127) (-4/7,-9/16) -> (9/16,4/7) Hyperbolic Matrix(161,90,288,161) (-9/16,-5/9) -> (5/9,9/16) Hyperbolic Matrix(81,44,208,113) (-6/11,-1/2) -> (7/18,2/5) Hyperbolic Matrix(49,22,-176,-79) (-1/2,-4/9) -> (-2/7,-5/18) Hyperbolic Matrix(127,56,288,127) (-4/9,-7/16) -> (7/16,4/9) Hyperbolic Matrix(97,42,224,97) (-7/16,-3/7) -> (3/7,7/16) Hyperbolic Matrix(353,146,544,225) (-12/29,-7/17) -> (11/17,2/3) Hyperbolic Matrix(49,20,-272,-111) (-7/17,-2/5) -> (-2/11,-3/17) Hyperbolic Matrix(127,50,160,63) (-2/5,-7/18) -> (11/14,4/5) Hyperbolic Matrix(2399,930,3712,1439) (-19/49,-31/80) -> (31/48,11/17) Hyperbolic Matrix(2561,992,3968,1537) (-31/80,-12/31) -> (20/31,31/48) Hyperbolic Matrix(575,222,992,383) (-12/31,-5/13) -> (11/19,18/31) Hyperbolic Matrix(79,30,208,79) (-5/13,-3/8) -> (3/8,5/13) Hyperbolic Matrix(17,6,48,17) (-3/8,-1/3) -> (1/3,3/8) Hyperbolic Matrix(31,10,96,31) (-1/3,-5/16) -> (5/16,1/3) Hyperbolic Matrix(129,40,416,129) (-5/16,-4/13) -> (4/13,5/16) Hyperbolic Matrix(271,82,-1008,-305) (-7/23,-3/10) -> (-7/26,-11/41) Hyperbolic Matrix(47,14,-272,-81) (-3/10,-5/17) -> (-3/17,-1/6) Hyperbolic Matrix(239,70,816,239) (-5/17,-7/24) -> (7/24,5/17) Hyperbolic Matrix(193,56,224,65) (-7/24,-2/7) -> (6/7,7/8) Hyperbolic Matrix(225,62,352,97) (-5/18,-3/11) -> (7/11,9/14) Hyperbolic Matrix(927,248,1312,351) (-11/41,-4/15) -> (12/17,29/41) Hyperbolic Matrix(31,8,-128,-33) (-4/15,-1/4) -> (-1/4,-4/17) Parabolic Matrix(241,56,624,145) (-4/17,-3/13) -> (5/13,12/31) Hyperbolic Matrix(31,6,160,31) (-1/5,-3/16) -> (3/16,1/5) Hyperbolic Matrix(65,12,352,65) (-3/16,-2/11) -> (2/11,3/16) Hyperbolic Matrix(159,22,224,31) (-1/7,-1/8) -> (17/24,5/7) Hyperbolic Matrix(385,46,544,65) (-1/8,-1/9) -> (29/41,17/24) Hyperbolic Matrix(145,14,176,17) (-1/9,0/1) -> (14/17,19/23) Hyperbolic Matrix(79,-12,112,-17) (0/1,1/6) -> (7/10,12/17) Hyperbolic Matrix(257,-46,352,-63) (1/6,2/11) -> (8/11,19/26) Hyperbolic Matrix(47,-10,80,-17) (1/5,1/4) -> (7/12,3/5) Hyperbolic Matrix(177,-46,304,-79) (1/4,4/15) -> (18/31,7/12) Hyperbolic Matrix(289,-78,352,-95) (4/15,3/11) -> (9/11,14/17) Hyperbolic Matrix(79,-22,176,-49) (3/11,2/7) -> (4/9,5/11) Hyperbolic Matrix(433,-128,592,-175) (5/17,3/10) -> (19/26,11/15) Hyperbolic Matrix(321,-98,416,-127) (3/10,4/13) -> (10/13,17/22) Hyperbolic Matrix(639,-248,992,-385) (12/31,7/18) -> (9/14,20/31) Hyperbolic Matrix(81,-34,112,-47) (2/5,3/7) -> (5/7,8/11) Hyperbolic Matrix(113,-62,144,-79) (1/2,5/9) -> (7/9,11/14) Hyperbolic Matrix(191,-110,224,-129) (4/7,11/19) -> (11/13,6/7) Hyperbolic Matrix(175,-122,208,-145) (9/13,7/10) -> (5/6,11/13) Hyperbolic Matrix(97,-72,128,-95) (11/15,3/4) -> (3/4,13/17) Parabolic Matrix(463,-358,560,-433) (17/22,7/9) -> (19/23,5/6) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(15,14,16,15) -> Matrix(1,0,-4,1) Matrix(65,56,224,193) -> Matrix(1,2,-4,-7) Matrix(47,40,-208,-177) -> Matrix(1,0,0,1) Matrix(95,78,-352,-289) -> Matrix(1,0,2,1) Matrix(287,234,352,287) -> Matrix(1,0,4,1) Matrix(129,104,160,129) -> Matrix(1,0,-8,1) Matrix(79,62,-144,-113) -> Matrix(3,-2,-4,3) Matrix(127,98,-416,-321) -> Matrix(1,-2,0,1) Matrix(113,86,-272,-207) -> Matrix(1,0,-2,1) Matrix(47,34,-112,-81) -> Matrix(1,2,-2,-3) Matrix(17,12,-112,-79) -> Matrix(1,0,0,1) Matrix(95,66,-416,-289) -> Matrix(1,0,0,1) Matrix(287,198,416,287) -> Matrix(1,0,0,1) Matrix(65,44,96,65) -> Matrix(1,0,-4,1) Matrix(209,136,272,177) -> Matrix(1,-2,-4,9) Matrix(385,248,-992,-639) -> Matrix(1,4,-2,-7) Matrix(81,52,176,113) -> Matrix(1,2,-4,-7) Matrix(111,70,176,111) -> Matrix(1,0,-4,1) Matrix(49,30,80,49) -> Matrix(1,2,-4,-7) Matrix(17,10,-80,-47) -> Matrix(1,2,-2,-3) Matrix(127,72,224,127) -> Matrix(5,6,-16,-19) Matrix(161,90,288,161) -> Matrix(9,8,-26,-23) Matrix(81,44,208,113) -> Matrix(3,2,-8,-5) Matrix(49,22,-176,-79) -> Matrix(1,2,-2,-3) Matrix(127,56,288,127) -> Matrix(5,6,-16,-19) Matrix(97,42,224,97) -> Matrix(5,4,-14,-11) Matrix(353,146,544,225) -> Matrix(3,2,-14,-9) Matrix(49,20,-272,-111) -> Matrix(1,0,0,1) Matrix(127,50,160,63) -> Matrix(1,0,-4,1) Matrix(2399,930,3712,1439) -> Matrix(19,10,-78,-41) Matrix(2561,992,3968,1537) -> Matrix(25,12,-98,-47) Matrix(575,222,992,383) -> Matrix(5,2,-18,-7) Matrix(79,30,208,79) -> Matrix(1,0,0,1) Matrix(17,6,48,17) -> Matrix(1,0,-2,1) Matrix(31,10,96,31) -> Matrix(1,0,-2,1) Matrix(129,40,416,129) -> Matrix(1,0,-4,1) Matrix(271,82,-1008,-305) -> Matrix(1,0,2,1) Matrix(47,14,-272,-81) -> Matrix(1,0,-2,1) Matrix(239,70,816,239) -> Matrix(1,2,-4,-7) Matrix(193,56,224,65) -> Matrix(1,2,-4,-7) Matrix(225,62,352,97) -> Matrix(1,0,-2,1) Matrix(927,248,1312,351) -> Matrix(1,-2,-2,5) Matrix(31,8,-128,-33) -> Matrix(1,2,-2,-3) Matrix(241,56,624,145) -> Matrix(1,0,0,1) Matrix(31,6,160,31) -> Matrix(3,2,-8,-5) Matrix(65,12,352,65) -> Matrix(1,0,0,1) Matrix(159,22,224,31) -> Matrix(1,0,-2,1) Matrix(385,46,544,65) -> Matrix(5,4,-14,-11) Matrix(145,14,176,17) -> Matrix(3,2,-8,-5) Matrix(79,-12,112,-17) -> Matrix(1,0,0,1) Matrix(257,-46,352,-63) -> Matrix(5,2,-18,-7) Matrix(47,-10,80,-17) -> Matrix(5,2,-18,-7) Matrix(177,-46,304,-79) -> Matrix(1,0,0,1) Matrix(289,-78,352,-95) -> Matrix(1,0,2,1) Matrix(79,-22,176,-49) -> Matrix(5,2,-18,-7) Matrix(433,-128,592,-175) -> Matrix(9,2,-32,-7) Matrix(321,-98,416,-127) -> Matrix(7,2,-32,-9) Matrix(639,-248,992,-385) -> Matrix(9,4,-34,-15) Matrix(81,-34,112,-47) -> Matrix(5,2,-18,-7) Matrix(113,-62,144,-79) -> Matrix(5,2,-28,-11) Matrix(191,-110,224,-129) -> Matrix(7,2,-18,-5) Matrix(175,-122,208,-145) -> Matrix(1,0,0,1) Matrix(97,-72,128,-95) -> Matrix(7,2,-32,-9) Matrix(463,-358,560,-433) -> Matrix(9,2,-32,-7) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 16 Degree of the the map X: 16 Degree of the the map Y: 64 Permutation triple for Y: ((1,4,16,34,62,51,50,63,64,55,44,61,46,17,5,2)(3,10,35,58,26,8,7,25,57,32,42,54,53,56,36,11)(6,21,31,9,30,59,28,27,41,14,13,33,45,48,47,22)(12,37,29,52,20,19,23,40,39,15,43,60,49,18,24,38); (1,2,8,28,60,29,9,3)(4,14,6,5,20,54,42,15)(7,18,17,48,45,16,12,11)(10,13,40,23,22,26,46,34)(19,51,59,30,44,39,25,36)(21,41,50,49,58,35,37,55)(24,27,53,62,61,32,31,38)(33,57,64,63,56,47,52,43); (2,6,23,36,63,41,24,7)(3,12,31,55,57,39,13,4)(5,18,50,19)(8,22,56,27)(9,32,33,10)(11,25)(14,21)(15,44,37,16)(17,26,49,28,51,53,20,47)(29,35,34,45,43,42,61,30)(46,62)(52,60)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 (-1/2,0/1) 0 16 1/6 0/1 1 8 2/11 (-1/2,0/1) 0 16 3/16 -1/2 1 1 1/5 (-1/2,-1/3) 0 16 1/4 -1/3 1 4 4/15 (-1/4,-2/9) 0 16 3/11 (-1/2,0/1) 0 16 2/7 (-1/3,-1/4) 0 16 7/24 -1/4 1 2 5/17 (-1/3,-1/4) 0 16 3/10 0/1 1 8 4/13 (-1/4,0/1) 0 16 5/16 0/1 1 1 1/3 (-1/3,0/1) 0 16 3/8 0/1 1 2 5/13 (-1/2,0/1) 0 16 12/31 (-4/7,-1/2) 0 16 7/18 -2/5 1 8 2/5 (-1/2,0/1) 0 16 3/7 (-2/5,-1/3) 0 16 7/16 -1/3 5 1 4/9 (-1/3,-3/10) 0 16 5/11 (-2/7,-1/4) 0 16 1/2 0/1 1 8 5/9 (-4/11,-1/3) 0 16 9/16 -1/3 7 1 4/7 (-1/3,-3/10) 0 16 11/19 (-2/7,-1/4) 0 16 18/31 (-1/4,-2/9) 0 16 7/12 -1/3 1 4 3/5 (-1/3,-1/4) 0 16 5/8 -1/4 1 2 7/11 (-1/4,0/1) 0 16 9/14 -2/7 1 8 20/31 (-8/31,-1/4) 0 16 31/48 -1/4 11 1 11/17 (-1/4,-3/13) 0 16 2/3 (-1/4,0/1) 0 16 11/16 0/1 2 1 9/13 (-1/2,0/1) 0 16 7/10 0/1 1 8 12/17 (-1/2,0/1) 0 16 29/41 (-2/5,-1/3) 0 16 17/24 -1/3 2 2 5/7 (-1/3,0/1) 0 16 8/11 (-2/7,-1/4) 0 16 19/26 -2/7 1 8 11/15 (-3/11,-1/4) 0 16 3/4 -1/4 1 4 13/17 (-1/3,-1/4) 0 16 10/13 (-1/4,-2/9) 0 16 17/22 -2/9 1 8 7/9 (-2/9,-1/5) 0 16 11/14 -2/11 1 8 4/5 (-1/6,0/1) 0 16 13/16 0/1 6 1 9/11 (0/1,1/0) 0 16 14/17 (-1/2,-2/5) 0 16 19/23 (-1/3,0/1) 0 16 5/6 0/1 1 8 11/13 (-1/2,0/1) 0 16 6/7 (-1/3,-1/4) 0 16 7/8 -1/4 1 2 1/1 (-1/4,0/1) 0 16 1/0 0/1 2 1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(79,-12,112,-17) (0/1,1/6) -> (7/10,12/17) Hyperbolic Matrix(257,-46,352,-63) (1/6,2/11) -> (8/11,19/26) Hyperbolic Matrix(65,-12,352,-65) (2/11,3/16) -> (2/11,3/16) Reflection Matrix(31,-6,160,-31) (3/16,1/5) -> (3/16,1/5) Reflection Matrix(47,-10,80,-17) (1/5,1/4) -> (7/12,3/5) Hyperbolic Matrix(177,-46,304,-79) (1/4,4/15) -> (18/31,7/12) Hyperbolic Matrix(289,-78,352,-95) (4/15,3/11) -> (9/11,14/17) Hyperbolic Matrix(79,-22,176,-49) (3/11,2/7) -> (4/9,5/11) Hyperbolic Matrix(193,-56,224,-65) (2/7,7/24) -> (6/7,7/8) Glide Reflection Matrix(239,-70,816,-239) (7/24,5/17) -> (7/24,5/17) Reflection Matrix(433,-128,592,-175) (5/17,3/10) -> (19/26,11/15) Hyperbolic Matrix(321,-98,416,-127) (3/10,4/13) -> (10/13,17/22) Hyperbolic Matrix(129,-40,416,-129) (4/13,5/16) -> (4/13,5/16) Reflection Matrix(31,-10,96,-31) (5/16,1/3) -> (5/16,1/3) Reflection Matrix(17,-6,48,-17) (1/3,3/8) -> (1/3,3/8) Reflection Matrix(79,-30,208,-79) (3/8,5/13) -> (3/8,5/13) Reflection Matrix(575,-222,992,-383) (5/13,12/31) -> (11/19,18/31) Glide Reflection Matrix(639,-248,992,-385) (12/31,7/18) -> (9/14,20/31) Hyperbolic Matrix(127,-50,160,-63) (7/18,2/5) -> (11/14,4/5) Glide Reflection Matrix(81,-34,112,-47) (2/5,3/7) -> (5/7,8/11) Hyperbolic Matrix(97,-42,224,-97) (3/7,7/16) -> (3/7,7/16) Reflection Matrix(127,-56,288,-127) (7/16,4/9) -> (7/16,4/9) Reflection Matrix(113,-52,176,-81) (5/11,1/2) -> (7/11,9/14) Glide Reflection Matrix(113,-62,144,-79) (1/2,5/9) -> (7/9,11/14) Hyperbolic Matrix(161,-90,288,-161) (5/9,9/16) -> (5/9,9/16) Reflection Matrix(127,-72,224,-127) (9/16,4/7) -> (9/16,4/7) Reflection Matrix(191,-110,224,-129) (4/7,11/19) -> (11/13,6/7) Hyperbolic Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(111,-70,176,-111) (5/8,7/11) -> (5/8,7/11) Reflection Matrix(1921,-1240,2976,-1921) (20/31,31/48) -> (20/31,31/48) Reflection Matrix(1055,-682,1632,-1055) (31/48,11/17) -> (31/48,11/17) Reflection Matrix(209,-136,272,-177) (11/17,2/3) -> (13/17,10/13) Glide Reflection Matrix(65,-44,96,-65) (2/3,11/16) -> (2/3,11/16) Reflection Matrix(287,-198,416,-287) (11/16,9/13) -> (11/16,9/13) Reflection Matrix(175,-122,208,-145) (9/13,7/10) -> (5/6,11/13) Hyperbolic Matrix(897,-634,1088,-769) (12/17,29/41) -> (14/17,19/23) Glide Reflection Matrix(1393,-986,1968,-1393) (29/41,17/24) -> (29/41,17/24) Reflection Matrix(239,-170,336,-239) (17/24,5/7) -> (17/24,5/7) Reflection Matrix(97,-72,128,-95) (11/15,3/4) -> (3/4,13/17) Parabolic Matrix(463,-358,560,-433) (17/22,7/9) -> (19/23,5/6) Hyperbolic Matrix(129,-104,160,-129) (4/5,13/16) -> (4/5,13/16) Reflection Matrix(287,-234,352,-287) (13/16,9/11) -> (13/16,9/11) Reflection Matrix(15,-14,16,-15) (7/8,1/1) -> (7/8,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,4,1) (0/1,1/0) -> (-1/2,0/1) Matrix(79,-12,112,-17) -> Matrix(1,0,0,1) Matrix(257,-46,352,-63) -> Matrix(5,2,-18,-7) -1/3 Matrix(65,-12,352,-65) -> Matrix(-1,0,4,1) (2/11,3/16) -> (-1/2,0/1) Matrix(31,-6,160,-31) -> Matrix(5,2,-12,-5) (3/16,1/5) -> (-1/2,-1/3) Matrix(47,-10,80,-17) -> Matrix(5,2,-18,-7) -1/3 Matrix(177,-46,304,-79) -> Matrix(1,0,0,1) Matrix(289,-78,352,-95) -> Matrix(1,0,2,1) 0/1 Matrix(79,-22,176,-49) -> Matrix(5,2,-18,-7) -1/3 Matrix(193,-56,224,-65) -> Matrix(7,2,-24,-7) *** -> (-1/3,-1/4) Matrix(239,-70,816,-239) -> Matrix(7,2,-24,-7) (7/24,5/17) -> (-1/3,-1/4) Matrix(433,-128,592,-175) -> Matrix(9,2,-32,-7) -1/4 Matrix(321,-98,416,-127) -> Matrix(7,2,-32,-9) -1/4 Matrix(129,-40,416,-129) -> Matrix(-1,0,8,1) (4/13,5/16) -> (-1/4,0/1) Matrix(31,-10,96,-31) -> Matrix(-1,0,6,1) (5/16,1/3) -> (-1/3,0/1) Matrix(17,-6,48,-17) -> Matrix(-1,0,6,1) (1/3,3/8) -> (-1/3,0/1) Matrix(79,-30,208,-79) -> Matrix(-1,0,4,1) (3/8,5/13) -> (-1/2,0/1) Matrix(575,-222,992,-383) -> Matrix(3,2,-10,-7) Matrix(639,-248,992,-385) -> Matrix(9,4,-34,-15) Matrix(127,-50,160,-63) -> Matrix(-1,0,8,1) *** -> (-1/4,0/1) Matrix(81,-34,112,-47) -> Matrix(5,2,-18,-7) -1/3 Matrix(97,-42,224,-97) -> Matrix(11,4,-30,-11) (3/7,7/16) -> (-2/5,-1/3) Matrix(127,-56,288,-127) -> Matrix(19,6,-60,-19) (7/16,4/9) -> (-1/3,-3/10) Matrix(113,-52,176,-81) -> Matrix(7,2,-24,-7) *** -> (-1/3,-1/4) Matrix(113,-62,144,-79) -> Matrix(5,2,-28,-11) Matrix(161,-90,288,-161) -> Matrix(23,8,-66,-23) (5/9,9/16) -> (-4/11,-1/3) Matrix(127,-72,224,-127) -> Matrix(19,6,-60,-19) (9/16,4/7) -> (-1/3,-3/10) Matrix(191,-110,224,-129) -> Matrix(7,2,-18,-5) -1/3 Matrix(49,-30,80,-49) -> Matrix(7,2,-24,-7) (3/5,5/8) -> (-1/3,-1/4) Matrix(111,-70,176,-111) -> Matrix(-1,0,8,1) (5/8,7/11) -> (-1/4,0/1) Matrix(1921,-1240,2976,-1921) -> Matrix(63,16,-248,-63) (20/31,31/48) -> (-8/31,-1/4) Matrix(1055,-682,1632,-1055) -> Matrix(25,6,-104,-25) (31/48,11/17) -> (-1/4,-3/13) Matrix(209,-136,272,-177) -> Matrix(9,2,-40,-9) *** -> (-1/4,-1/5) Matrix(65,-44,96,-65) -> Matrix(-1,0,8,1) (2/3,11/16) -> (-1/4,0/1) Matrix(287,-198,416,-287) -> Matrix(-1,0,4,1) (11/16,9/13) -> (-1/2,0/1) Matrix(175,-122,208,-145) -> Matrix(1,0,0,1) Matrix(897,-634,1088,-769) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(1393,-986,1968,-1393) -> Matrix(11,4,-30,-11) (29/41,17/24) -> (-2/5,-1/3) Matrix(239,-170,336,-239) -> Matrix(-1,0,6,1) (17/24,5/7) -> (-1/3,0/1) Matrix(97,-72,128,-95) -> Matrix(7,2,-32,-9) -1/4 Matrix(463,-358,560,-433) -> Matrix(9,2,-32,-7) -1/4 Matrix(129,-104,160,-129) -> Matrix(-1,0,12,1) (4/5,13/16) -> (-1/6,0/1) Matrix(287,-234,352,-287) -> Matrix(1,0,0,-1) (13/16,9/11) -> (0/1,1/0) Matrix(15,-14,16,-15) -> Matrix(-1,0,8,1) (7/8,1/1) -> (-1/4,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,8,1) (1/1,1/0) -> (-1/4,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.