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 -6/1 -4/1 -2/1 -48/31 -4/3 0/1 1/1 8/7 16/13 4/3 16/11 3/2 8/5 16/9 2/1 16/7 5/2 8/3 3/1 16/5 7/2 11/3 4/1 13/3 5/1 16/3 17/3 6/1 7/1 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 0/1 -11/2 0/1 1/2 -16/3 1/2 -5/1 1/2 1/1 -4/1 1/1 -15/4 -2/1 1/0 -11/3 0/1 1/2 -7/2 1/1 1/0 -24/7 1/0 -17/5 1/1 1/0 -10/3 0/1 -13/4 0/1 1/0 -16/5 0/1 -3/1 0/1 1/1 -8/3 0/1 -13/5 0/1 1/2 -31/12 4/9 1/2 -18/7 2/3 -5/2 0/1 1/2 -7/3 2/3 1/1 -16/7 1/1 -9/4 1/1 3/2 -11/5 2/1 1/0 -2/1 0/1 -9/5 4/5 1/1 -16/9 1/1 -7/4 1/1 3/2 -19/11 2/1 1/0 -31/18 -2/1 1/0 -12/7 1/1 -5/3 1/1 1/0 -8/5 1/0 -11/7 0/1 1/0 -14/9 2/1 -31/20 8/1 1/0 -48/31 1/0 -17/11 -3/1 1/0 -3/2 0/1 1/0 -16/11 0/1 -13/9 0/1 1/2 -10/7 0/1 -17/12 0/1 1/2 -41/29 2/3 1/1 -24/17 1/1 -7/5 0/1 1/1 -11/8 2/1 1/0 -26/19 2/1 -15/11 3/1 1/0 -4/3 1/0 -17/13 1/1 1/0 -13/10 -2/1 1/0 -22/17 -2/1 -9/7 -2/1 -1/1 -14/11 -2/3 -5/4 -1/2 0/1 -16/13 0/1 -11/9 0/1 1/4 -17/14 1/2 2/3 -23/19 0/1 1/1 -6/5 0/1 -13/11 0/1 1/2 -7/6 1/1 1/0 -8/7 1/0 -1/1 0/1 1/0 0/1 0/1 1/1 0/1 1/4 8/7 1/4 7/6 1/4 1/3 6/5 0/1 11/9 0/1 1/0 16/13 0/1 5/4 0/1 1/6 9/7 1/5 2/9 13/10 2/9 1/4 4/3 1/4 7/5 0/1 1/3 10/7 0/1 13/9 0/1 1/2 16/11 0/1 3/2 0/1 1/4 17/11 3/13 1/4 14/9 2/7 11/7 0/1 1/4 8/5 1/4 5/3 1/4 1/3 7/4 3/10 1/3 16/9 1/3 9/5 1/3 4/11 11/6 2/5 1/2 2/1 0/1 9/4 3/10 1/3 16/7 1/3 7/3 1/3 2/5 12/5 1/2 29/12 2/5 1/2 17/7 1/2 1/1 5/2 0/1 1/2 18/7 2/5 49/19 7/15 1/2 80/31 1/2 31/12 1/2 4/7 13/5 0/1 1/2 8/3 0/1 3/1 0/1 1/3 16/5 0/1 13/4 0/1 1/4 23/7 0/1 1/3 10/3 0/1 17/5 1/4 1/3 24/7 1/4 7/2 1/4 1/3 18/5 2/5 11/3 0/1 1/2 26/7 0/1 41/11 0/1 1/5 15/4 2/9 1/4 4/1 1/3 17/4 4/9 1/2 13/3 0/1 1/2 22/5 0/1 9/2 1/4 1/3 5/1 1/3 1/2 16/3 1/2 11/2 0/1 1/2 17/3 1/2 1/1 6/1 0/1 7/1 0/1 1/3 8/1 1/3 9/1 1/3 2/5 1/0 0/1 1/2 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(17,112,-12,-79) (-6/1,1/0) -> (-10/7,-17/12) Hyperbolic Matrix(63,352,-46,-257) (-6/1,-11/2) -> (-11/8,-26/19) Hyperbolic Matrix(65,352,12,65) (-11/2,-16/3) -> (16/3,11/2) Hyperbolic Matrix(31,160,6,31) (-16/3,-5/1) -> (5/1,16/3) Hyperbolic Matrix(17,80,-10,-47) (-5/1,-4/1) -> (-12/7,-5/3) Hyperbolic Matrix(79,304,-46,-177) (-4/1,-15/4) -> (-31/18,-12/7) Hyperbolic Matrix(95,352,-78,-289) (-15/4,-11/3) -> (-11/9,-17/14) Hyperbolic Matrix(49,176,-22,-79) (-11/3,-7/2) -> (-9/4,-11/5) Hyperbolic Matrix(65,224,56,193) (-7/2,-24/7) -> (8/7,7/6) Hyperbolic Matrix(239,816,70,239) (-24/7,-17/5) -> (17/5,24/7) Hyperbolic Matrix(175,592,-128,-433) (-17/5,-10/3) -> (-26/19,-15/11) Hyperbolic Matrix(127,416,-98,-321) (-10/3,-13/4) -> (-13/10,-22/17) Hyperbolic Matrix(129,416,40,129) (-13/4,-16/5) -> (16/5,13/4) Hyperbolic Matrix(31,96,10,31) (-16/5,-3/1) -> (3/1,16/5) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(79,208,30,79) (-8/3,-13/5) -> (13/5,8/3) Hyperbolic Matrix(241,624,56,145) (-13/5,-31/12) -> (17/4,13/3) Hyperbolic Matrix(385,992,-248,-639) (-31/12,-18/7) -> (-14/9,-31/20) Hyperbolic Matrix(81,208,44,113) (-18/7,-5/2) -> (11/6,2/1) Hyperbolic Matrix(47,112,-34,-81) (-5/2,-7/3) -> (-7/5,-11/8) Hyperbolic Matrix(97,224,42,97) (-7/3,-16/7) -> (16/7,7/3) Hyperbolic Matrix(127,288,56,127) (-16/7,-9/4) -> (9/4,16/7) Hyperbolic Matrix(81,176,52,113) (-11/5,-2/1) -> (14/9,11/7) Hyperbolic Matrix(79,144,-62,-113) (-2/1,-9/5) -> (-9/7,-14/11) Hyperbolic Matrix(161,288,90,161) (-9/5,-16/9) -> (16/9,9/5) Hyperbolic Matrix(127,224,72,127) (-16/9,-7/4) -> (7/4,16/9) Hyperbolic Matrix(129,224,-110,-191) (-7/4,-19/11) -> (-13/11,-7/6) Hyperbolic Matrix(575,992,222,383) (-19/11,-31/18) -> (31/12,13/5) Hyperbolic Matrix(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(111,176,70,111) (-8/5,-11/7) -> (11/7,8/5) Hyperbolic Matrix(225,352,62,97) (-11/7,-14/9) -> (18/5,11/3) Hyperbolic Matrix(2561,3968,992,1537) (-31/20,-48/31) -> (80/31,31/12) Hyperbolic Matrix(2399,3712,930,1439) (-48/31,-17/11) -> (49/19,80/31) Hyperbolic Matrix(353,544,146,225) (-17/11,-3/2) -> (29/12,17/7) Hyperbolic Matrix(65,96,44,65) (-3/2,-16/11) -> (16/11,3/2) Hyperbolic Matrix(287,416,198,287) (-16/11,-13/9) -> (13/9,16/11) Hyperbolic Matrix(145,208,-122,-175) (-13/9,-10/7) -> (-6/5,-13/11) Hyperbolic Matrix(927,1312,248,351) (-17/12,-41/29) -> (41/11,15/4) Hyperbolic Matrix(385,544,46,65) (-41/29,-24/17) -> (8/1,9/1) Hyperbolic Matrix(159,224,22,31) (-24/17,-7/5) -> (7/1,8/1) Hyperbolic Matrix(95,128,-72,-97) (-15/11,-4/3) -> (-4/3,-17/13) Parabolic Matrix(209,272,136,177) (-17/13,-13/10) -> (3/2,17/11) Hyperbolic Matrix(433,560,-358,-463) (-22/17,-9/7) -> (-23/19,-6/5) Hyperbolic Matrix(127,160,50,63) (-14/11,-5/4) -> (5/2,18/7) Hyperbolic Matrix(129,160,104,129) (-5/4,-16/13) -> (16/13,5/4) Hyperbolic Matrix(287,352,234,287) (-16/13,-11/9) -> (11/9,16/13) Hyperbolic Matrix(145,176,14,17) (-17/14,-23/19) -> (9/1,1/0) Hyperbolic Matrix(193,224,56,65) (-7/6,-8/7) -> (24/7,7/2) Hyperbolic Matrix(15,16,14,15) (-8/7,-1/1) -> (1/1,8/7) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(177,-208,40,-47) (7/6,6/5) -> (22/5,9/2) Hyperbolic Matrix(289,-352,78,-95) (6/5,11/9) -> (11/3,26/7) Hyperbolic Matrix(113,-144,62,-79) (5/4,9/7) -> (9/5,11/6) Hyperbolic Matrix(321,-416,98,-127) (9/7,13/10) -> (13/4,23/7) Hyperbolic Matrix(207,-272,86,-113) (13/10,4/3) -> (12/5,29/12) Hyperbolic Matrix(81,-112,34,-47) (4/3,7/5) -> (7/3,12/5) Hyperbolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(289,-416,66,-95) (10/7,13/9) -> (13/3,22/5) Hyperbolic Matrix(639,-992,248,-385) (17/11,14/9) -> (18/7,49/19) Hyperbolic Matrix(47,-80,10,-17) (5/3,7/4) -> (9/2,5/1) Hyperbolic Matrix(79,-176,22,-49) (2/1,9/4) -> (7/2,18/5) Hyperbolic Matrix(111,-272,20,-49) (17/7,5/2) -> (11/2,17/3) Hyperbolic Matrix(305,-1008,82,-271) (23/7,10/3) -> (26/7,41/11) Hyperbolic Matrix(81,-272,14,-47) (10/3,17/5) -> (17/3,6/1) Hyperbolic Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(17,112,-12,-79) -> Matrix(1,0,0,1) Matrix(63,352,-46,-257) -> Matrix(3,-2,2,-1) Matrix(65,352,12,65) -> Matrix(1,0,0,1) Matrix(31,160,6,31) -> Matrix(3,-2,8,-5) Matrix(17,80,-10,-47) -> Matrix(3,-2,2,-1) Matrix(79,304,-46,-177) -> Matrix(1,0,0,1) Matrix(95,352,-78,-289) -> Matrix(1,0,2,1) Matrix(49,176,-22,-79) -> Matrix(3,-2,2,-1) Matrix(65,224,56,193) -> Matrix(1,-2,4,-7) Matrix(239,816,70,239) -> Matrix(1,-2,4,-7) Matrix(175,592,-128,-433) -> Matrix(1,2,0,1) Matrix(127,416,-98,-321) -> Matrix(1,-2,0,1) Matrix(129,416,40,129) -> Matrix(1,0,4,1) Matrix(31,96,10,31) -> Matrix(1,0,2,1) Matrix(17,48,6,17) -> Matrix(1,0,2,1) Matrix(79,208,30,79) -> Matrix(1,0,0,1) Matrix(241,624,56,145) -> Matrix(1,0,0,1) Matrix(385,992,-248,-639) -> Matrix(7,-4,2,-1) Matrix(81,208,44,113) -> Matrix(3,-2,8,-5) Matrix(47,112,-34,-81) -> Matrix(3,-2,2,-1) Matrix(97,224,42,97) -> Matrix(5,-4,14,-11) Matrix(127,288,56,127) -> Matrix(5,-6,16,-19) Matrix(81,176,52,113) -> Matrix(1,-2,4,-7) Matrix(79,144,-62,-113) -> Matrix(3,-2,-4,3) Matrix(161,288,90,161) -> Matrix(9,-8,26,-23) Matrix(127,224,72,127) -> Matrix(5,-6,16,-19) Matrix(129,224,-110,-191) -> Matrix(1,-2,2,-3) Matrix(575,992,222,383) -> Matrix(1,-2,2,-3) Matrix(49,80,30,49) -> Matrix(1,-2,4,-7) Matrix(111,176,70,111) -> Matrix(1,0,4,1) Matrix(225,352,62,97) -> Matrix(1,0,2,1) Matrix(2561,3968,992,1537) -> Matrix(1,-12,2,-23) Matrix(2399,3712,930,1439) -> Matrix(1,10,2,21) Matrix(353,544,146,225) -> Matrix(1,2,2,5) Matrix(65,96,44,65) -> Matrix(1,0,4,1) Matrix(287,416,198,287) -> Matrix(1,0,0,1) Matrix(145,208,-122,-175) -> Matrix(1,0,0,1) Matrix(927,1312,248,351) -> Matrix(3,-2,14,-9) Matrix(385,544,46,65) -> Matrix(5,-4,14,-11) Matrix(159,224,22,31) -> Matrix(1,0,2,1) Matrix(95,128,-72,-97) -> Matrix(1,-2,0,1) Matrix(209,272,136,177) -> Matrix(1,2,4,9) Matrix(433,560,-358,-463) -> Matrix(1,2,0,1) Matrix(127,160,50,63) -> Matrix(1,0,4,1) Matrix(129,160,104,129) -> Matrix(1,0,8,1) Matrix(287,352,234,287) -> Matrix(1,0,-4,1) Matrix(145,176,14,17) -> Matrix(3,-2,8,-5) Matrix(193,224,56,65) -> Matrix(1,-2,4,-7) Matrix(15,16,14,15) -> Matrix(1,0,4,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(177,-208,40,-47) -> Matrix(1,0,0,1) Matrix(289,-352,78,-95) -> Matrix(1,0,2,1) Matrix(113,-144,62,-79) -> Matrix(11,-2,28,-5) Matrix(321,-416,98,-127) -> Matrix(9,-2,32,-7) Matrix(207,-272,86,-113) -> Matrix(1,0,-2,1) Matrix(81,-112,34,-47) -> Matrix(7,-2,18,-5) Matrix(79,-112,12,-17) -> Matrix(1,0,0,1) Matrix(289,-416,66,-95) -> Matrix(1,0,0,1) Matrix(639,-992,248,-385) -> Matrix(15,-4,34,-9) Matrix(47,-80,10,-17) -> Matrix(7,-2,18,-5) Matrix(79,-176,22,-49) -> Matrix(7,-2,18,-5) Matrix(111,-272,20,-49) -> Matrix(1,0,0,1) Matrix(305,-1008,82,-271) -> Matrix(1,0,2,1) Matrix(81,-272,14,-47) -> Matrix(1,0,-2,1) Matrix(33,-128,8,-31) -> Matrix(7,-2,18,-5) 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: ((2,6,23,36,63,41,24,7)(3,12,33,55,57,39,13,4)(5,18,51,19)(8,27,56,22)(9,31,32,10)(11,25)(14,21)(15,37,43,16)(17,47,20,28,52,26,50,48)(29,42,45,44,62,35,34,30)(46,59)(53,61); (1,4,16,45,46,17,5,2)(3,10,35,58,26,8,7,11)(6,14,13,9,30,60,28,22)(12,15,29,53,20,19,24,38)(18,23,40,39,43,62,61,50)(21,33,32,44,49,48,27,41)(25,57,31,42,54,47,56,36)(34,59,52,51,63,64,55,37); (1,2,8,28,53,62,32,57,64,63,56,48,61,29,9,3)(4,14,41,51,50,58,35,43,55,21,6,5,20,54,42,15)(7,19,52,60,30,37,12,11,36,18,17,49,44,16,39,25)(10,33,38,24,27,26,59,45,31,13,40,23,22,47,46,34)) ----------------------------------------------------------------------- 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 0/1 2 1 1/1 (0/1,1/4) 0 16 8/7 1/4 1 2 7/6 (1/4,1/3) 0 16 6/5 0/1 1 8 11/9 (0/1,1/0) 0 16 16/13 0/1 6 1 5/4 (0/1,1/6) 0 16 9/7 (1/5,2/9) 0 16 13/10 (2/9,1/4) 0 16 4/3 1/4 1 4 7/5 (0/1,1/3) 0 16 10/7 0/1 1 8 13/9 (0/1,1/2) 0 16 16/11 0/1 2 1 3/2 (0/1,1/4) 0 16 17/11 (3/13,1/4) 0 16 14/9 2/7 1 8 11/7 (0/1,1/4) 0 16 8/5 1/4 1 2 5/3 (1/4,1/3) 0 16 7/4 (3/10,1/3) 0 16 16/9 1/3 7 1 9/5 (1/3,4/11) 0 16 11/6 (2/5,1/2) 0 16 2/1 0/1 1 8 9/4 (3/10,1/3) 0 16 16/7 1/3 5 1 7/3 (1/3,2/5) 0 16 12/5 1/2 1 4 29/12 (2/5,1/2) 0 16 17/7 (1/2,1/1) 0 16 5/2 (0/1,1/2) 0 16 18/7 2/5 1 8 49/19 (7/15,1/2) 0 16 80/31 1/2 11 1 31/12 (1/2,4/7) 0 16 13/5 (0/1,1/2) 0 16 8/3 0/1 1 2 3/1 (0/1,1/3) 0 16 16/5 0/1 1 1 13/4 (0/1,1/4) 0 16 23/7 (0/1,1/3) 0 16 10/3 0/1 1 8 17/5 (1/4,1/3) 0 16 24/7 1/4 1 2 7/2 (1/4,1/3) 0 16 18/5 2/5 1 8 11/3 (0/1,1/2) 0 16 26/7 0/1 1 8 41/11 (0/1,1/5) 0 16 15/4 (2/9,1/4) 0 16 4/1 1/3 1 4 17/4 (4/9,1/2) 0 16 13/3 (0/1,1/2) 0 16 22/5 0/1 1 8 9/2 (1/4,1/3) 0 16 5/1 (1/3,1/2) 0 16 16/3 1/2 1 1 11/2 (0/1,1/2) 0 16 17/3 (1/2,1/1) 0 16 6/1 0/1 1 8 7/1 (0/1,1/3) 0 16 8/1 1/3 2 2 9/1 (1/3,2/5) 0 16 1/0 (0/1,1/2) 0 16 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(15,-16,14,-15) (1/1,8/7) -> (1/1,8/7) Reflection Matrix(193,-224,56,-65) (8/7,7/6) -> (24/7,7/2) Glide Reflection Matrix(177,-208,40,-47) (7/6,6/5) -> (22/5,9/2) Hyperbolic Matrix(289,-352,78,-95) (6/5,11/9) -> (11/3,26/7) Hyperbolic Matrix(287,-352,234,-287) (11/9,16/13) -> (11/9,16/13) Reflection Matrix(129,-160,104,-129) (16/13,5/4) -> (16/13,5/4) Reflection Matrix(113,-144,62,-79) (5/4,9/7) -> (9/5,11/6) Hyperbolic Matrix(321,-416,98,-127) (9/7,13/10) -> (13/4,23/7) Hyperbolic Matrix(207,-272,86,-113) (13/10,4/3) -> (12/5,29/12) Hyperbolic Matrix(81,-112,34,-47) (4/3,7/5) -> (7/3,12/5) Hyperbolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(289,-416,66,-95) (10/7,13/9) -> (13/3,22/5) Hyperbolic Matrix(287,-416,198,-287) (13/9,16/11) -> (13/9,16/11) Reflection Matrix(65,-96,44,-65) (16/11,3/2) -> (16/11,3/2) Reflection Matrix(353,-544,146,-225) (3/2,17/11) -> (29/12,17/7) Glide Reflection Matrix(639,-992,248,-385) (17/11,14/9) -> (18/7,49/19) Hyperbolic Matrix(225,-352,62,-97) (14/9,11/7) -> (18/5,11/3) Glide Reflection Matrix(111,-176,70,-111) (11/7,8/5) -> (11/7,8/5) Reflection Matrix(49,-80,30,-49) (8/5,5/3) -> (8/5,5/3) Reflection Matrix(47,-80,10,-17) (5/3,7/4) -> (9/2,5/1) Hyperbolic Matrix(127,-224,72,-127) (7/4,16/9) -> (7/4,16/9) Reflection Matrix(161,-288,90,-161) (16/9,9/5) -> (16/9,9/5) Reflection Matrix(113,-208,44,-81) (11/6,2/1) -> (5/2,18/7) Glide Reflection Matrix(79,-176,22,-49) (2/1,9/4) -> (7/2,18/5) Hyperbolic Matrix(127,-288,56,-127) (9/4,16/7) -> (9/4,16/7) Reflection Matrix(97,-224,42,-97) (16/7,7/3) -> (16/7,7/3) Reflection Matrix(111,-272,20,-49) (17/7,5/2) -> (11/2,17/3) Hyperbolic Matrix(3039,-7840,1178,-3039) (49/19,80/31) -> (49/19,80/31) Reflection Matrix(1921,-4960,744,-1921) (80/31,31/12) -> (80/31,31/12) Reflection Matrix(241,-624,56,-145) (31/12,13/5) -> (17/4,13/3) Glide Reflection Matrix(79,-208,30,-79) (13/5,8/3) -> (13/5,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(31,-96,10,-31) (3/1,16/5) -> (3/1,16/5) Reflection Matrix(129,-416,40,-129) (16/5,13/4) -> (16/5,13/4) Reflection Matrix(305,-1008,82,-271) (23/7,10/3) -> (26/7,41/11) Hyperbolic Matrix(81,-272,14,-47) (10/3,17/5) -> (17/3,6/1) Hyperbolic Matrix(239,-816,70,-239) (17/5,24/7) -> (17/5,24/7) Reflection Matrix(47,-176,4,-15) (41/11,15/4) -> (9/1,1/0) Glide Reflection Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic Matrix(31,-160,6,-31) (5/1,16/3) -> (5/1,16/3) Reflection Matrix(65,-352,12,-65) (16/3,11/2) -> (16/3,11/2) Reflection Matrix(15,-112,2,-15) (7/1,8/1) -> (7/1,8/1) Reflection Matrix(17,-144,2,-17) (8/1,9/1) -> (8/1,9/1) 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) -> (0/1,1/2) Matrix(1,0,2,-1) -> Matrix(1,0,8,-1) (0/1,1/1) -> (0/1,1/4) Matrix(15,-16,14,-15) -> Matrix(1,0,8,-1) (1/1,8/7) -> (0/1,1/4) Matrix(193,-224,56,-65) -> Matrix(7,-2,24,-7) *** -> (1/4,1/3) Matrix(177,-208,40,-47) -> Matrix(1,0,0,1) Matrix(289,-352,78,-95) -> Matrix(1,0,2,1) 0/1 Matrix(287,-352,234,-287) -> Matrix(1,0,0,-1) (11/9,16/13) -> (0/1,1/0) Matrix(129,-160,104,-129) -> Matrix(1,0,12,-1) (16/13,5/4) -> (0/1,1/6) Matrix(113,-144,62,-79) -> Matrix(11,-2,28,-5) Matrix(321,-416,98,-127) -> Matrix(9,-2,32,-7) 1/4 Matrix(207,-272,86,-113) -> Matrix(1,0,-2,1) 0/1 Matrix(81,-112,34,-47) -> Matrix(7,-2,18,-5) 1/3 Matrix(79,-112,12,-17) -> Matrix(1,0,0,1) Matrix(289,-416,66,-95) -> Matrix(1,0,0,1) Matrix(287,-416,198,-287) -> Matrix(1,0,4,-1) (13/9,16/11) -> (0/1,1/2) Matrix(65,-96,44,-65) -> Matrix(1,0,8,-1) (16/11,3/2) -> (0/1,1/4) Matrix(353,-544,146,-225) -> Matrix(9,-2,22,-5) Matrix(639,-992,248,-385) -> Matrix(15,-4,34,-9) Matrix(225,-352,62,-97) -> Matrix(1,0,6,-1) *** -> (0/1,1/3) Matrix(111,-176,70,-111) -> Matrix(1,0,8,-1) (11/7,8/5) -> (0/1,1/4) Matrix(49,-80,30,-49) -> Matrix(7,-2,24,-7) (8/5,5/3) -> (1/4,1/3) Matrix(47,-80,10,-17) -> Matrix(7,-2,18,-5) 1/3 Matrix(127,-224,72,-127) -> Matrix(19,-6,60,-19) (7/4,16/9) -> (3/10,1/3) Matrix(161,-288,90,-161) -> Matrix(23,-8,66,-23) (16/9,9/5) -> (1/3,4/11) Matrix(113,-208,44,-81) -> Matrix(5,-2,12,-5) *** -> (1/3,1/2) Matrix(79,-176,22,-49) -> Matrix(7,-2,18,-5) 1/3 Matrix(127,-288,56,-127) -> Matrix(19,-6,60,-19) (9/4,16/7) -> (3/10,1/3) Matrix(97,-224,42,-97) -> Matrix(11,-4,30,-11) (16/7,7/3) -> (1/3,2/5) Matrix(111,-272,20,-49) -> Matrix(1,0,0,1) Matrix(3039,-7840,1178,-3039) -> Matrix(29,-14,60,-29) (49/19,80/31) -> (7/15,1/2) Matrix(1921,-4960,744,-1921) -> Matrix(15,-8,28,-15) (80/31,31/12) -> (1/2,4/7) Matrix(241,-624,56,-145) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(79,-208,30,-79) -> Matrix(1,0,4,-1) (13/5,8/3) -> (0/1,1/2) Matrix(17,-48,6,-17) -> Matrix(1,0,6,-1) (8/3,3/1) -> (0/1,1/3) Matrix(31,-96,10,-31) -> Matrix(1,0,6,-1) (3/1,16/5) -> (0/1,1/3) Matrix(129,-416,40,-129) -> Matrix(1,0,8,-1) (16/5,13/4) -> (0/1,1/4) Matrix(305,-1008,82,-271) -> Matrix(1,0,2,1) 0/1 Matrix(81,-272,14,-47) -> Matrix(1,0,-2,1) 0/1 Matrix(239,-816,70,-239) -> Matrix(7,-2,24,-7) (17/5,24/7) -> (1/4,1/3) Matrix(47,-176,4,-15) -> Matrix(9,-2,22,-5) Matrix(33,-128,8,-31) -> Matrix(7,-2,18,-5) 1/3 Matrix(31,-160,6,-31) -> Matrix(5,-2,12,-5) (5/1,16/3) -> (1/3,1/2) Matrix(65,-352,12,-65) -> Matrix(1,0,4,-1) (16/3,11/2) -> (0/1,1/2) Matrix(15,-112,2,-15) -> Matrix(1,0,6,-1) (7/1,8/1) -> (0/1,1/3) Matrix(17,-144,2,-17) -> Matrix(11,-4,30,-11) (8/1,9/1) -> (1/3,2/5) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.