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: 40 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -8/1 -6/1 -4/1 -10/3 -8/3 -12/5 -2/1 -12/7 -8/5 -4/3 -8/7 0/1 1/1 8/7 4/3 3/2 8/5 12/7 2/1 16/7 12/5 5/2 8/3 14/5 48/17 3/1 16/5 10/3 64/19 7/2 4/1 32/7 14/3 5/1 16/3 6/1 32/5 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 1/0 -7/1 -2/1 -1/1 -6/1 1/0 -5/1 -1/1 -2/3 -4/1 -1/2 -11/3 -1/3 0/1 -18/5 -1/2 -7/2 -2/5 -1/3 -24/7 -1/2 -17/5 -2/5 -1/3 -10/3 -3/8 -3/1 -1/3 -2/7 -8/3 -1/4 -13/5 -5/21 -4/17 -31/12 -4/17 -3/13 -18/7 -1/4 -5/2 -3/13 -2/9 -22/9 -7/32 -17/7 -2/9 -5/23 -12/5 -3/14 -31/13 -4/19 -1/5 -19/8 -7/33 -4/19 -26/11 -5/24 -7/3 -6/29 -1/5 -2/1 -1/6 -9/5 -8/55 -1/7 -16/9 -1/7 -7/4 -1/7 -6/43 -19/11 -4/29 -7/51 -12/7 -3/22 -29/17 -1/7 -2/15 -17/10 -5/37 -2/15 -5/3 -2/15 -3/23 -8/5 -1/8 -11/7 -4/33 -3/25 -14/9 -1/8 -31/20 -4/33 -3/25 -48/31 -3/25 -17/11 -3/25 -2/17 -3/2 -2/17 -1/9 -16/11 -1/9 -13/9 -1/9 -8/73 -10/7 -3/28 -37/26 -13/123 -2/19 -64/45 -2/19 -27/19 -2/19 -7/67 -17/12 -1/9 -2/19 -24/17 -1/10 -7/5 -1/9 -2/19 -4/3 -1/10 -9/7 -1/11 0/1 -32/25 0/1 -23/18 -1/9 0/1 -14/11 -1/10 -5/4 -2/21 -1/11 -16/13 -1/11 -11/9 -1/11 -4/45 -6/5 -1/12 -19/16 -1/15 0/1 -32/27 0/1 -13/11 -1/9 0/1 -7/6 -1/11 -2/23 -8/7 -1/12 -1/1 -1/13 0/1 0/1 0/1 1/1 0/1 1/15 8/7 1/14 7/6 2/27 1/13 6/5 1/14 5/4 1/13 2/25 4/3 1/12 11/8 0/1 1/11 18/13 1/12 7/5 2/23 1/11 24/17 1/12 17/12 2/23 1/11 10/7 3/34 3/2 1/11 2/21 8/5 1/10 13/8 5/49 4/39 31/19 4/39 3/29 18/11 1/10 5/3 3/29 2/19 22/13 7/66 17/10 2/19 5/47 12/7 3/28 31/18 4/37 1/9 19/11 7/65 4/37 26/15 5/46 7/4 6/55 1/9 2/1 1/8 9/4 8/57 1/7 16/7 1/7 7/3 1/7 6/41 19/8 4/27 7/47 12/5 3/20 29/12 1/7 2/13 17/7 5/33 2/13 5/2 2/13 3/19 8/3 1/6 11/4 4/23 3/17 14/5 1/6 31/11 4/23 3/17 48/17 3/17 17/6 3/17 2/11 3/1 2/11 1/5 16/5 1/5 13/4 1/5 8/39 10/3 3/14 37/11 13/59 2/9 64/19 2/9 27/8 2/9 7/31 17/5 1/5 2/9 24/7 1/4 7/2 1/5 2/9 4/1 1/4 9/2 0/1 1/3 32/7 0/1 23/5 0/1 1/5 14/3 1/4 5/1 2/7 1/3 16/3 1/3 11/2 1/3 4/11 6/1 1/2 19/3 -1/1 0/1 32/5 0/1 13/2 0/1 1/5 7/1 1/3 2/5 8/1 1/2 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(17,160,12,113) (-8/1,1/0) -> (24/17,17/12) Hyperbolic Matrix(31,224,22,159) (-8/1,-7/1) -> (7/5,24/17) Hyperbolic Matrix(33,224,-14,-95) (-7/1,-6/1) -> (-26/11,-7/3) Hyperbolic Matrix(17,96,-14,-79) (-6/1,-5/1) -> (-11/9,-6/5) Hyperbolic Matrix(15,64,-4,-17) (-5/1,-4/1) -> (-4/1,-11/3) Parabolic Matrix(79,288,48,175) (-11/3,-18/5) -> (18/11,5/3) Hyperbolic Matrix(143,512,-112,-401) (-18/5,-7/2) -> (-23/18,-14/11) Hyperbolic Matrix(65,224,56,193) (-7/2,-24/7) -> (8/7,7/6) Hyperbolic Matrix(47,160,42,143) (-24/7,-17/5) -> (1/1,8/7) Hyperbolic Matrix(161,544,-66,-223) (-17/5,-10/3) -> (-22/9,-17/7) Hyperbolic Matrix(49,160,-34,-111) (-10/3,-3/1) -> (-13/9,-10/7) Hyperbolic Matrix(47,128,-18,-49) (-3/1,-8/3) -> (-8/3,-13/5) Parabolic Matrix(383,992,222,575) (-13/5,-31/12) -> (31/18,19/11) Hyperbolic Matrix(385,992,-248,-639) (-31/12,-18/7) -> (-14/9,-31/20) Hyperbolic Matrix(113,288,82,209) (-18/7,-5/2) -> (11/8,18/13) Hyperbolic Matrix(353,864,-248,-607) (-5/2,-22/9) -> (-10/7,-37/26) Hyperbolic Matrix(239,576,-100,-241) (-17/7,-12/5) -> (-12/5,-31/13) Parabolic Matrix(417,992,256,609) (-31/13,-19/8) -> (13/8,31/19) Hyperbolic Matrix(257,608,-216,-511) (-19/8,-26/11) -> (-6/5,-19/16) Hyperbolic Matrix(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic Matrix(143,256,62,111) (-9/5,-16/9) -> (16/7,7/3) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(129,224,-110,-191) (-7/4,-19/11) -> (-13/11,-7/6) Hyperbolic Matrix(335,576,-196,-337) (-19/11,-12/7) -> (-12/7,-29/17) Parabolic Matrix(319,544,112,191) (-29/17,-17/10) -> (17/6,3/1) Hyperbolic Matrix(321,544,-226,-383) (-17/10,-5/3) -> (-27/19,-17/12) Hyperbolic Matrix(79,128,-50,-81) (-5/3,-8/5) -> (-8/5,-11/7) Parabolic Matrix(143,224,30,47) (-11/7,-14/9) -> (14/3,5/1) Hyperbolic Matrix(1487,2304,526,815) (-31/20,-48/31) -> (48/17,17/6) Hyperbolic Matrix(1489,2304,528,817) (-48/31,-17/11) -> (31/11,48/17) Hyperbolic Matrix(353,544,146,225) (-17/11,-3/2) -> (29/12,17/7) Hyperbolic Matrix(175,256,54,79) (-3/2,-16/11) -> (16/5,13/4) Hyperbolic Matrix(177,256,56,81) (-16/11,-13/9) -> (3/1,16/5) Hyperbolic Matrix(2879,4096,854,1215) (-37/26,-64/45) -> (64/19,27/8) Hyperbolic Matrix(2881,4096,856,1217) (-64/45,-27/19) -> (37/11,64/19) Hyperbolic Matrix(113,160,12,17) (-17/12,-24/17) -> (8/1,1/0) Hyperbolic Matrix(159,224,22,31) (-24/17,-7/5) -> (7/1,8/1) Hyperbolic Matrix(47,64,-36,-49) (-7/5,-4/3) -> (-4/3,-9/7) Parabolic Matrix(799,1024,174,223) (-9/7,-32/25) -> (32/7,23/5) Hyperbolic Matrix(801,1024,176,225) (-32/25,-23/18) -> (9/2,32/7) Hyperbolic Matrix(177,224,64,81) (-14/11,-5/4) -> (11/4,14/5) Hyperbolic Matrix(207,256,38,47) (-5/4,-16/13) -> (16/3,11/2) Hyperbolic Matrix(209,256,40,49) (-16/13,-11/9) -> (5/1,16/3) Hyperbolic Matrix(863,1024,134,159) (-19/16,-32/27) -> (32/5,13/2) Hyperbolic Matrix(865,1024,136,161) (-32/27,-13/11) -> (19/3,32/5) Hyperbolic Matrix(193,224,56,65) (-7/6,-8/7) -> (24/7,7/2) Hyperbolic Matrix(143,160,42,47) (-8/7,-1/1) -> (17/5,24/7) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(191,-224,110,-129) (7/6,6/5) -> (26/15,7/4) Hyperbolic Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(369,-512,80,-111) (18/13,7/5) -> (23/5,14/3) Hyperbolic Matrix(383,-544,226,-321) (17/12,10/7) -> (22/13,17/10) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(81,-128,50,-79) (3/2,8/5) -> (8/5,13/8) Parabolic Matrix(607,-992,216,-353) (31/19,18/11) -> (14/5,31/11) Hyperbolic Matrix(511,-864,152,-257) (5/3,22/13) -> (10/3,37/11) Hyperbolic Matrix(337,-576,196,-335) (17/10,12/7) -> (12/7,31/18) Parabolic Matrix(351,-608,56,-97) (19/11,26/15) -> (6/1,19/3) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(95,-224,14,-33) (7/3,19/8) -> (13/2,7/1) Hyperbolic Matrix(241,-576,100,-239) (19/8,12/5) -> (12/5,29/12) Parabolic Matrix(223,-544,66,-161) (17/7,5/2) -> (27/8,17/5) Hyperbolic Matrix(49,-128,18,-47) (5/2,8/3) -> (8/3,11/4) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(17,160,12,113) -> Matrix(1,-2,12,-23) Matrix(31,224,22,159) -> Matrix(1,0,12,1) Matrix(33,224,-14,-95) -> Matrix(5,4,-24,-19) Matrix(17,96,-14,-79) -> Matrix(1,2,-12,-23) Matrix(15,64,-4,-17) -> Matrix(3,2,-8,-5) Matrix(79,288,48,175) -> Matrix(3,2,28,19) Matrix(143,512,-112,-401) -> Matrix(5,2,-48,-19) Matrix(65,224,56,193) -> Matrix(1,0,16,1) Matrix(47,160,42,143) -> Matrix(5,2,72,29) Matrix(161,544,-66,-223) -> Matrix(19,8,-88,-37) Matrix(49,160,-34,-111) -> Matrix(17,6,-156,-55) Matrix(47,128,-18,-49) -> Matrix(23,6,-96,-25) Matrix(383,992,222,575) -> Matrix(35,8,328,75) Matrix(385,992,-248,-639) -> Matrix(1,0,-4,1) Matrix(113,288,82,209) -> Matrix(9,2,112,25) Matrix(353,864,-248,-607) -> Matrix(91,20,-860,-189) Matrix(239,576,-100,-241) -> Matrix(83,18,-392,-85) Matrix(417,992,256,609) -> Matrix(37,8,356,77) Matrix(257,608,-216,-511) -> Matrix(19,4,-252,-53) Matrix(15,32,-8,-17) -> Matrix(11,2,-72,-13) Matrix(143,256,62,111) -> Matrix(97,14,672,97) Matrix(145,256,64,113) -> Matrix(99,14,700,99) Matrix(129,224,-110,-191) -> Matrix(29,4,-312,-43) Matrix(335,576,-196,-337) -> Matrix(131,18,-968,-133) Matrix(319,544,112,191) -> Matrix(29,4,152,21) Matrix(321,544,-226,-383) -> Matrix(59,8,-568,-77) Matrix(79,128,-50,-81) -> Matrix(47,6,-384,-49) Matrix(143,224,30,47) -> Matrix(17,2,76,9) Matrix(1487,2304,526,815) -> Matrix(149,18,836,101) Matrix(1489,2304,528,817) -> Matrix(151,18,864,103) Matrix(353,544,146,225) -> Matrix(35,4,236,27) Matrix(175,256,54,79) -> Matrix(89,10,436,49) Matrix(177,256,56,81) -> Matrix(91,10,464,51) Matrix(2879,4096,854,1215) -> Matrix(379,40,1696,179) Matrix(2881,4096,856,1217) -> Matrix(381,40,1724,181) Matrix(113,160,12,17) -> Matrix(19,2,28,3) Matrix(159,224,22,31) -> Matrix(1,0,12,1) Matrix(47,64,-36,-49) -> Matrix(19,2,-200,-21) Matrix(799,1024,174,223) -> Matrix(1,0,16,1) Matrix(801,1024,176,225) -> Matrix(1,0,12,1) Matrix(177,224,64,81) -> Matrix(19,2,104,11) Matrix(207,256,38,47) -> Matrix(65,6,184,17) Matrix(209,256,40,49) -> Matrix(67,6,212,19) Matrix(863,1024,134,159) -> Matrix(1,0,20,1) Matrix(865,1024,136,161) -> Matrix(1,0,8,1) Matrix(193,224,56,65) -> Matrix(1,0,16,1) Matrix(143,160,42,47) -> Matrix(25,2,112,9) Matrix(1,0,2,1) -> Matrix(1,0,28,1) Matrix(191,-224,110,-129) -> Matrix(51,-4,472,-37) Matrix(79,-96,14,-17) -> Matrix(27,-2,68,-5) Matrix(49,-64,36,-47) -> Matrix(25,-2,288,-23) Matrix(369,-512,80,-111) -> Matrix(23,-2,104,-9) Matrix(383,-544,226,-321) -> Matrix(93,-8,872,-75) Matrix(111,-160,34,-49) -> Matrix(67,-6,324,-29) Matrix(81,-128,50,-79) -> Matrix(61,-6,600,-59) Matrix(607,-992,216,-353) -> Matrix(1,0,-4,1) Matrix(511,-864,152,-257) -> Matrix(189,-20,860,-91) Matrix(337,-576,196,-335) -> Matrix(169,-18,1568,-167) Matrix(351,-608,56,-97) -> Matrix(37,-4,28,-3) Matrix(17,-32,8,-15) -> Matrix(17,-2,128,-15) Matrix(95,-224,14,-33) -> Matrix(27,-4,88,-13) Matrix(241,-576,100,-239) -> Matrix(121,-18,800,-119) Matrix(223,-544,66,-161) -> Matrix(53,-8,232,-35) Matrix(49,-128,18,-47) -> Matrix(37,-6,216,-35) Matrix(17,-64,4,-15) -> Matrix(9,-2,32,-7) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 16 Degree of the the map X: 32 Degree of the the map Y: 64 Permutation triple for Y: ((2,6,22,7)(3,12,13,4)(5,18)(8,26)(9,10)(15,16)(17,44,52,27)(19,48,54,25)(21,51)(23,53)(28,58,42,29)(30,50,55,41)(31,60,40,32)(33,61,39,46)(34,35)(37,38); (1,4,16,42,56,54,53,61,64,50,34,31,43,17,5,2)(3,10,32,49,25,8,7,24,55,37,58,62,44,51,33,11)(6,20,30,9,29,45,27,26,39,14,13,38,60,63,48,21)(12,35,28,47,19,18,46,59,41,15,40,57,52,23,22,36); (1,2,8,27,57,40,38,55,64,61,51,48,47,28,9,3)(4,14,39,53,52,62,58,35,50,20,6,5,19,49,32,15)(7,23,54,63,60,34,12,11,33,18,17,45,29,16,41,24)(10,30,59,46,26,25,56,42,37,13,36,22,21,44,43,31)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 96 Minimal number of generators: 17 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 14 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 4/3 2/1 8/3 3/1 16/5 10/3 4/1 32/7 14/3 16/3 6/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 0/1 1/15 6/5 1/14 5/4 1/13 2/25 4/3 1/12 11/8 0/1 1/11 18/13 1/12 7/5 2/23 1/11 10/7 3/34 3/2 1/11 2/21 2/1 1/8 5/2 2/13 3/19 8/3 1/6 11/4 4/23 3/17 14/5 1/6 3/1 2/11 1/5 16/5 1/5 13/4 1/5 8/39 10/3 3/14 17/5 1/5 2/9 24/7 1/4 7/2 1/5 2/9 4/1 1/4 9/2 0/1 1/3 32/7 0/1 23/5 0/1 1/5 14/3 1/4 5/1 2/7 1/3 16/3 1/3 11/2 1/3 4/11 6/1 1/2 7/1 1/3 2/5 8/1 1/2 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(41,-48,6,-7) (1/1,6/5) -> (6/1,7/1) Hyperbolic Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(151,-208,53,-73) (11/8,18/13) -> (14/5,3/1) Hyperbolic Matrix(369,-512,80,-111) (18/13,7/5) -> (23/5,14/3) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(49,-128,18,-47) (5/2,8/3) -> (8/3,11/4) Parabolic Matrix(81,-224,17,-47) (11/4,14/5) -> (14/3,5/1) Hyperbolic Matrix(81,-256,25,-79) (3/1,16/5) -> (16/5,13/4) Parabolic Matrix(47,-160,5,-17) (17/5,24/7) -> (8/1,1/0) Hyperbolic Matrix(65,-224,9,-31) (24/7,7/2) -> (7/1,8/1) Hyperbolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic Matrix(225,-1024,49,-223) (9/2,32/7) -> (32/7,23/5) Parabolic Matrix(49,-256,9,-47) (5/1,16/3) -> (16/3,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,14,1) Matrix(41,-48,6,-7) -> Matrix(13,-1,40,-3) Matrix(79,-96,14,-17) -> Matrix(27,-2,68,-5) Matrix(49,-64,36,-47) -> Matrix(25,-2,288,-23) Matrix(151,-208,53,-73) -> Matrix(13,-1,66,-5) Matrix(369,-512,80,-111) -> Matrix(23,-2,104,-9) Matrix(169,-240,50,-71) -> Matrix(35,-3,152,-13) Matrix(111,-160,34,-49) -> Matrix(67,-6,324,-29) Matrix(9,-16,4,-7) -> Matrix(9,-1,64,-7) Matrix(49,-128,18,-47) -> Matrix(37,-6,216,-35) Matrix(81,-224,17,-47) -> Matrix(11,-2,50,-9) Matrix(81,-256,25,-79) -> Matrix(51,-10,250,-49) Matrix(47,-160,5,-17) -> Matrix(9,-2,14,-3) Matrix(65,-224,9,-31) -> Matrix(1,0,-2,1) Matrix(17,-64,4,-15) -> Matrix(9,-2,32,-7) Matrix(225,-1024,49,-223) -> Matrix(1,0,2,1) Matrix(49,-256,9,-47) -> Matrix(19,-6,54,-17) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 16 ----------------------------------------------------------------------- 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 14 1 2/1 1/8 2 8 8/3 1/6 6 2 14/5 1/6 2 8 3/1 (2/11,1/5) 0 16 16/5 1/5 10 1 10/3 3/14 2 8 24/7 1/4 2 2 7/2 (1/5,2/9) 0 16 4/1 1/4 2 4 9/2 (0/1,1/3) 0 16 32/7 0/1 2 1 14/3 1/4 2 8 5/1 (2/7,1/3) 0 16 16/3 1/3 6 1 6/1 1/2 2 8 8/1 1/2 2 2 1/0 (0/1,1/1) 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(7,-16,3,-7) (2/1,8/3) -> (2/1,8/3) Reflection Matrix(41,-112,15,-41) (8/3,14/5) -> (8/3,14/5) Reflection Matrix(39,-112,8,-23) (14/5,3/1) -> (14/3,5/1) Glide Reflection Matrix(31,-96,10,-31) (3/1,16/5) -> (3/1,16/5) Reflection Matrix(49,-160,15,-49) (16/5,10/3) -> (16/5,10/3) Reflection Matrix(71,-240,21,-71) (10/3,24/7) -> (10/3,24/7) Reflection Matrix(23,-80,2,-7) (24/7,7/2) -> (8/1,1/0) Glide Reflection Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic Matrix(127,-576,28,-127) (9/2,32/7) -> (9/2,32/7) Reflection Matrix(97,-448,21,-97) (32/7,14/3) -> (32/7,14/3) Reflection Matrix(31,-160,6,-31) (5/1,16/3) -> (5/1,16/3) Reflection Matrix(17,-96,3,-17) (16/3,6/1) -> (16/3,6/1) Reflection Matrix(7,-48,1,-7) (6/1,8/1) -> (6/1,8/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,2,-1) (0/1,1/0) -> (0/1,1/1) Matrix(1,0,1,-1) -> Matrix(1,0,16,-1) (0/1,2/1) -> (0/1,1/8) Matrix(7,-16,3,-7) -> Matrix(7,-1,48,-7) (2/1,8/3) -> (1/8,1/6) Matrix(41,-112,15,-41) -> Matrix(29,-5,168,-29) (8/3,14/5) -> (1/6,5/28) Matrix(39,-112,8,-23) -> Matrix(17,-3,62,-11) Matrix(31,-96,10,-31) -> Matrix(21,-4,110,-21) (3/1,16/5) -> (2/11,1/5) Matrix(49,-160,15,-49) -> Matrix(29,-6,140,-29) (16/5,10/3) -> (1/5,3/14) Matrix(71,-240,21,-71) -> Matrix(13,-3,56,-13) (10/3,24/7) -> (3/14,1/4) Matrix(23,-80,2,-7) -> Matrix(5,-1,14,-3) Matrix(17,-64,4,-15) -> Matrix(9,-2,32,-7) 1/4 Matrix(127,-576,28,-127) -> Matrix(1,0,6,-1) (9/2,32/7) -> (0/1,1/3) Matrix(97,-448,21,-97) -> Matrix(1,0,8,-1) (32/7,14/3) -> (0/1,1/4) Matrix(31,-160,6,-31) -> Matrix(13,-4,42,-13) (5/1,16/3) -> (2/7,1/3) Matrix(17,-96,3,-17) -> Matrix(5,-2,12,-5) (16/3,6/1) -> (1/3,1/2) Matrix(7,-48,1,-7) -> Matrix(3,-1,8,-3) (6/1,8/1) -> (1/4,1/2) ----------------------------------------------------------------------- The pullback map has no extra symmetries.