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 -7/1 -6/1 -5/1 -13/3 -4/1 -3/1 -7/3 -2/1 -1/1 -1/2 -3/7 -1/3 -1/4 -1/5 0/1 1/4 1/3 1/2 3/5 5/7 3/4 1/1 9/7 7/5 3/2 5/3 2/1 11/5 7/3 17/7 5/2 3/1 11/3 4/1 13/3 5/1 6/1 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -1/1 -6/1 -1/2 1/0 -5/1 -1/1 -9/2 -1/2 -13/3 -1/1 -1/3 -4/1 -1/2 1/0 -7/2 -1/2 -10/3 -1/2 1/0 -3/1 -1/1 -14/5 -3/4 -1/2 -11/4 -2/3 -1/2 -8/3 -1/2 1/0 -5/2 -1/2 -17/7 -1/1 -29/12 -1/1 0/1 -12/5 -1/2 1/0 -7/3 -1/1 -9/4 -1/1 -2/3 -11/5 -1/1 -3/5 -13/6 -1/2 -2/1 -1/2 1/0 -1/1 0/1 -2/3 1/2 1/0 -7/11 0/1 -5/8 0/1 1/1 -8/13 1/2 1/0 -3/5 -1/1 1/1 -7/12 0/1 1/1 -4/7 3/2 1/0 -5/9 1/0 -1/2 1/0 -5/11 -1/1 1/1 -9/20 -1/1 0/1 -4/9 1/2 1/0 -3/7 1/0 -8/19 -7/2 1/0 -5/12 -2/1 1/0 -7/17 -3/1 -9/22 -5/2 -2/5 -3/2 1/0 -3/8 -1/1 0/1 -7/19 -1/1 -4/11 -1/2 1/0 -1/3 -1/1 1/1 -4/13 -1/2 1/0 -3/10 1/0 -5/17 -1/1 -2/7 -1/2 1/0 -1/4 -1/1 0/1 -4/17 1/2 1/0 -3/13 -1/1 1/1 -5/22 1/0 -2/9 -1/2 1/0 -1/5 0/1 -2/11 1/2 1/0 -1/6 1/0 -2/13 1/2 1/0 -1/7 1/1 -1/8 2/1 1/0 0/1 -1/2 1/0 1/5 -3/1 -1/1 2/9 -5/2 1/0 3/13 -2/1 1/4 -2/1 -1/1 1/3 -1/1 3/8 -1/1 0/1 5/13 -1/1 -1/3 2/5 -1/2 1/0 3/7 -1/1 4/9 -1/2 -1/4 5/11 0/1 1/2 1/0 4/7 -3/2 1/0 3/5 -1/1 8/13 -1/2 1/0 5/8 -1/1 0/1 2/3 -1/2 1/0 9/13 -1/1 1/1 7/10 1/0 5/7 1/0 8/11 -5/2 1/0 11/15 -1/1 3/4 -2/1 1/0 1/1 -1/1 5/4 -1/2 0/1 14/11 -1/2 -1/4 9/7 0/1 13/10 1/0 4/3 -1/2 1/0 11/8 -1/1 0/1 7/5 0/1 10/7 1/2 1/0 3/2 1/0 8/5 -3/2 1/0 5/3 -1/1 12/7 -5/6 -3/4 7/4 -1/1 -2/3 2/1 -1/2 1/0 13/6 -1/4 11/5 0/1 20/9 1/4 1/2 9/4 0/1 1/1 16/7 1/2 1/0 23/10 1/0 7/3 -1/1 1/1 12/5 3/2 1/0 29/12 4/1 1/0 17/7 1/0 22/9 -7/2 1/0 5/2 1/0 3/1 -1/1 7/2 -1/2 25/7 0/1 18/5 -1/2 1/0 11/3 -1/1 -1/3 4/1 -1/2 1/0 17/4 -1/3 0/1 13/3 0/1 22/5 1/2 1/0 9/2 1/0 14/3 1/2 1/0 33/7 1/0 19/4 -2/1 1/0 5/1 -1/1 6/1 -1/2 1/0 13/2 1/0 7/1 -1/1 15/2 -1/2 8/1 -1/2 1/0 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(29,220,-12,-91) (-7/1,1/0) -> (-17/7,-29/12) Hyperbolic Matrix(3,20,-20,-133) (-7/1,-6/1) -> (-2/13,-1/7) Hyperbolic Matrix(7,40,-24,-137) (-6/1,-5/1) -> (-5/17,-2/7) Hyperbolic Matrix(13,60,-44,-203) (-5/1,-9/2) -> (-3/10,-5/17) Hyperbolic Matrix(61,268,-28,-123) (-9/2,-13/3) -> (-11/5,-13/6) Hyperbolic Matrix(15,64,-64,-273) (-13/3,-4/1) -> (-4/17,-3/13) Hyperbolic Matrix(19,68,12,43) (-4/1,-7/2) -> (3/2,8/5) Hyperbolic Matrix(7,24,-40,-137) (-7/2,-10/3) -> (-2/11,-1/6) Hyperbolic Matrix(23,72,-8,-25) (-10/3,-3/1) -> (-3/1,-14/5) Parabolic Matrix(81,224,64,177) (-14/5,-11/4) -> (5/4,14/11) Hyperbolic Matrix(47,128,-112,-305) (-11/4,-8/3) -> (-8/19,-5/12) Hyperbolic Matrix(11,28,20,51) (-8/3,-5/2) -> (1/2,4/7) Hyperbolic Matrix(77,188,-188,-459) (-5/2,-17/7) -> (-7/17,-9/22) Hyperbolic Matrix(101,244,12,29) (-29/12,-12/5) -> (8/1,1/0) Hyperbolic Matrix(47,112,-128,-305) (-12/5,-7/3) -> (-7/19,-4/11) Hyperbolic Matrix(37,84,-100,-227) (-7/3,-9/4) -> (-3/8,-7/19) Hyperbolic Matrix(65,144,-144,-319) (-9/4,-11/5) -> (-5/11,-9/20) Hyperbolic Matrix(137,296,56,121) (-13/6,-2/1) -> (22/9,5/2) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(131,84,92,59) (-2/3,-7/11) -> (7/5,10/7) Hyperbolic Matrix(177,112,128,81) (-7/11,-5/8) -> (11/8,7/5) Hyperbolic Matrix(129,80,208,129) (-5/8,-8/13) -> (8/13,5/8) Hyperbolic Matrix(163,100,44,27) (-8/13,-3/5) -> (11/3,4/1) Hyperbolic Matrix(75,44,196,115) (-3/5,-7/12) -> (3/8,5/13) Hyperbolic Matrix(193,112,112,65) (-7/12,-4/7) -> (12/7,7/4) Hyperbolic Matrix(107,60,148,83) (-4/7,-5/9) -> (5/7,8/11) Hyperbolic Matrix(73,40,104,57) (-5/9,-1/2) -> (7/10,5/7) Hyperbolic Matrix(61,28,-268,-123) (-1/2,-5/11) -> (-3/13,-5/22) Hyperbolic Matrix(233,104,56,25) (-9/20,-4/9) -> (4/1,17/4) Hyperbolic Matrix(83,36,-196,-85) (-4/9,-3/7) -> (-3/7,-8/19) Parabolic Matrix(29,12,-220,-91) (-5/12,-7/17) -> (-1/7,-1/8) Hyperbolic Matrix(109,44,52,21) (-9/22,-2/5) -> (2/1,13/6) Hyperbolic Matrix(51,20,28,11) (-2/5,-3/8) -> (7/4,2/1) Hyperbolic Matrix(23,8,-72,-25) (-4/11,-1/3) -> (-1/3,-4/13) Parabolic Matrix(209,64,160,49) (-4/13,-3/10) -> (13/10,4/3) Hyperbolic Matrix(43,12,68,19) (-2/7,-1/4) -> (5/8,2/3) Hyperbolic Matrix(233,56,104,25) (-1/4,-4/17) -> (20/9,9/4) Hyperbolic Matrix(249,56,40,9) (-5/22,-2/9) -> (6/1,13/2) Hyperbolic Matrix(19,4,-100,-21) (-2/9,-1/5) -> (-1/5,-2/11) Parabolic Matrix(249,40,56,9) (-1/6,-2/13) -> (22/5,9/2) Hyperbolic Matrix(125,12,52,5) (-1/8,0/1) -> (12/5,29/12) Hyperbolic Matrix(67,-12,28,-5) (0/1,1/5) -> (7/3,12/5) Hyperbolic Matrix(91,-20,132,-29) (1/5,2/9) -> (2/3,9/13) Hyperbolic Matrix(403,-92,92,-21) (2/9,3/13) -> (13/3,22/5) Hyperbolic Matrix(273,-64,64,-15) (3/13,1/4) -> (17/4,13/3) Hyperbolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(289,-112,80,-31) (5/13,2/5) -> (18/5,11/3) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(155,-68,212,-93) (3/7,4/9) -> (8/11,11/15) Hyperbolic Matrix(319,-144,144,-65) (4/9,5/11) -> (11/5,20/9) Hyperbolic Matrix(165,-76,76,-35) (5/11,1/2) -> (13/6,11/5) Hyperbolic Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(369,-256,160,-111) (9/13,7/10) -> (23/10,7/3) Hyperbolic Matrix(305,-224,64,-47) (11/15,3/4) -> (19/4,5/1) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(401,-512,112,-143) (14/11,9/7) -> (25/7,18/5) Hyperbolic Matrix(299,-388,84,-109) (9/7,13/10) -> (7/2,25/7) Hyperbolic Matrix(155,-212,68,-93) (4/3,11/8) -> (9/4,16/7) Hyperbolic Matrix(91,-132,20,-29) (10/7,3/2) -> (9/2,14/3) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(185,-424,24,-55) (16/7,23/10) -> (15/2,8/1) Hyperbolic Matrix(529,-1280,112,-271) (29/12,17/7) -> (33/7,19/4) Hyperbolic Matrix(395,-964,84,-205) (17/7,22/9) -> (14/3,33/7) Hyperbolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(29,-196,4,-27) (13/2,7/1) -> (7/1,15/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(29,220,-12,-91) -> Matrix(1,0,0,1) Matrix(3,20,-20,-133) -> Matrix(1,0,2,1) Matrix(7,40,-24,-137) -> Matrix(1,0,0,1) Matrix(13,60,-44,-203) -> Matrix(3,2,-2,-1) Matrix(61,268,-28,-123) -> Matrix(5,2,-8,-3) Matrix(15,64,-64,-273) -> Matrix(1,0,2,1) Matrix(19,68,12,43) -> Matrix(3,2,-2,-1) Matrix(7,24,-40,-137) -> Matrix(1,0,2,1) Matrix(23,72,-8,-25) -> Matrix(1,2,-2,-3) Matrix(81,224,64,177) -> Matrix(3,2,-8,-5) Matrix(47,128,-112,-305) -> Matrix(7,4,-2,-1) Matrix(11,28,20,51) -> Matrix(3,2,-2,-1) Matrix(77,188,-188,-459) -> Matrix(1,-2,0,1) Matrix(101,244,12,29) -> Matrix(1,0,0,1) Matrix(47,112,-128,-305) -> Matrix(1,0,0,1) Matrix(37,84,-100,-227) -> Matrix(3,2,-2,-1) Matrix(65,144,-144,-319) -> Matrix(3,2,-2,-1) Matrix(137,296,56,121) -> Matrix(7,4,-2,-1) Matrix(3,4,-4,-5) -> Matrix(1,0,2,1) Matrix(131,84,92,59) -> Matrix(1,0,0,1) Matrix(177,112,128,81) -> Matrix(1,0,-2,1) Matrix(129,80,208,129) -> Matrix(1,0,-2,1) Matrix(163,100,44,27) -> Matrix(1,0,-2,1) Matrix(75,44,196,115) -> Matrix(1,0,-2,1) Matrix(193,112,112,65) -> Matrix(3,-2,-4,3) Matrix(107,60,148,83) -> Matrix(1,-4,0,1) Matrix(73,40,104,57) -> Matrix(1,0,0,1) Matrix(61,28,-268,-123) -> Matrix(1,0,0,1) Matrix(233,104,56,25) -> Matrix(1,0,-2,1) Matrix(83,36,-196,-85) -> Matrix(1,-4,0,1) Matrix(29,12,-220,-91) -> Matrix(1,4,0,1) Matrix(109,44,52,21) -> Matrix(1,2,-2,-3) Matrix(51,20,28,11) -> Matrix(1,2,-2,-3) Matrix(23,8,-72,-25) -> Matrix(1,0,0,1) Matrix(209,64,160,49) -> Matrix(1,0,0,1) Matrix(43,12,68,19) -> Matrix(1,0,0,1) Matrix(233,56,104,25) -> Matrix(1,0,2,1) Matrix(249,56,40,9) -> Matrix(1,0,0,1) Matrix(19,4,-100,-21) -> Matrix(1,0,2,1) Matrix(249,40,56,9) -> Matrix(1,0,0,1) Matrix(125,12,52,5) -> Matrix(1,2,0,1) Matrix(67,-12,28,-5) -> Matrix(1,2,0,1) Matrix(91,-20,132,-29) -> Matrix(1,2,0,1) Matrix(403,-92,92,-21) -> Matrix(1,2,2,5) Matrix(273,-64,64,-15) -> Matrix(1,2,-4,-7) Matrix(13,-4,36,-11) -> Matrix(1,2,-2,-3) Matrix(289,-112,80,-31) -> Matrix(1,0,0,1) Matrix(67,-28,12,-5) -> Matrix(1,0,0,1) Matrix(155,-68,212,-93) -> Matrix(3,2,-2,-1) Matrix(319,-144,144,-65) -> Matrix(1,0,6,1) Matrix(165,-76,76,-35) -> Matrix(1,0,-4,1) Matrix(61,-36,100,-59) -> Matrix(1,2,-2,-3) Matrix(369,-256,160,-111) -> Matrix(1,0,0,1) Matrix(305,-224,64,-47) -> Matrix(1,0,0,1) Matrix(9,-8,8,-7) -> Matrix(1,2,-2,-3) Matrix(401,-512,112,-143) -> Matrix(1,0,2,1) Matrix(299,-388,84,-109) -> Matrix(1,0,-2,1) Matrix(155,-212,68,-93) -> Matrix(1,0,2,1) Matrix(91,-132,20,-29) -> Matrix(1,0,0,1) Matrix(61,-100,36,-59) -> Matrix(5,6,-6,-7) Matrix(185,-424,24,-55) -> Matrix(1,0,-2,1) Matrix(529,-1280,112,-271) -> Matrix(1,-6,0,1) Matrix(395,-964,84,-205) -> Matrix(1,4,0,1) Matrix(13,-36,4,-11) -> Matrix(1,2,-2,-3) Matrix(29,-196,4,-27) -> Matrix(1,2,-2,-3) 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): 13 Degree of the the map X: 13 Degree of the the map Y: 64 Permutation triple for Y: ((1,7,25,46,55,26,8,2)(3,10,35,43,64,61,36,11)(4,15,48,59,30,9,16,5)(6,20,52,54,56,41,28,21)(12,17,33,32,51,53,58,40)(13,44,60,31,38,22,34,14)(18,47,63,62,37,29,49,19)(23,42,57,27,39,50,45,24); (1,5,19,6)(2,3)(4,14)(7,24)(8,28,29,9)(10,33,45,34)(11,38,39,12)(13,23,51,43)(15,25,52,47)(16,17)(18,35)(20,32)(21,22)(26,27)(30,31)(36,37)(40,41)(42,63)(44,54)(46,64)(48,53)(49,50)(55,59,62,56)(57,60,61,58); (1,3,12,41,62,42,13,4)(2,9,31,57,63,52,32,10)(5,17,39,26,56,44,43,18)(6,22,11,37,59,53,23,7)(8,27,58,48,47,35,34,21)(14,45,49,28,40,61,46,15)(16,29,36,60,54,25,24,33)(19,50,38,30,55,64,51,20)) ----------------------------------------------------------------------- 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 -1/1 0/1 1 2 -1/2 1/0 1 8 -3/7 1/0 2 2 -2/5 0 8 -3/8 (-1/1,0/1) 0 8 -1/3 0 4 -2/7 0 8 -1/4 (-1/1,0/1) 0 8 -1/5 0/1 1 2 0/1 0 8 1/5 0 4 2/9 0 8 3/13 -2/1 3 2 1/4 (-2/1,-1/1) 0 8 1/3 -1/1 1 2 3/8 (-1/1,0/1) 0 8 5/13 0 4 2/5 0 8 3/7 -1/1 1 4 4/9 0 8 5/11 0/1 5 2 1/2 1/0 1 8 3/5 -1/1 1 2 5/8 (-1/1,0/1) 0 8 2/3 0 8 9/13 0 4 7/10 1/0 1 8 5/7 1/0 2 2 8/11 0 8 11/15 -1/1 1 4 3/4 (-2/1,1/0) 0 8 1/1 -1/1 1 4 5/4 (-1/2,0/1) 0 8 9/7 0/1 2 2 13/10 1/0 1 8 4/3 0 8 11/8 (-1/1,0/1) 0 8 7/5 0/1 1 2 10/7 0 8 3/2 1/0 1 8 5/3 -1/1 3 2 7/4 (-1/1,-2/3) 0 8 2/1 0 8 11/5 0/1 5 2 9/4 (0/1,1/1) 0 8 16/7 0 8 23/10 1/0 1 8 7/3 0 4 12/5 0 8 29/12 (4/1,1/0) 0 8 17/7 1/0 5 2 5/2 1/0 1 8 3/1 -1/1 1 2 7/2 -1/2 1 8 18/5 0 8 11/3 0 4 4/1 0 8 13/3 0/1 3 2 9/2 1/0 1 8 14/3 0 8 19/4 (-2/1,1/0) 0 8 5/1 -1/1 1 4 6/1 0 8 13/2 1/0 1 8 7/1 -1/1 1 2 1/0 (-1/1,0/1) 0 8 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(3,2,-4,-3) (-1/1,-1/2) -> (-1/1,-1/2) Reflection Matrix(59,26,84,37) (-1/2,-3/7) -> (7/10,5/7) Glide Reflection Matrix(81,34,112,47) (-3/7,-2/5) -> (5/7,8/11) Glide Reflection Matrix(51,20,28,11) (-2/5,-3/8) -> (7/4,2/1) Hyperbolic Matrix(49,18,128,47) (-3/8,-1/3) -> (3/8,5/13) Glide Reflection Matrix(89,26,24,7) (-1/3,-2/7) -> (11/3,4/1) Glide Reflection Matrix(43,12,68,19) (-2/7,-1/4) -> (5/8,2/3) Hyperbolic Matrix(83,18,60,13) (-1/4,-1/5) -> (11/8,7/5) Glide Reflection Matrix(57,10,40,7) (-1/5,0/1) -> (7/5,10/7) Glide Reflection Matrix(67,-12,28,-5) (0/1,1/5) -> (7/3,12/5) Hyperbolic Matrix(91,-20,132,-29) (1/5,2/9) -> (2/3,9/13) Hyperbolic Matrix(169,-38,40,-9) (2/9,3/13) -> (4/1,13/3) Glide Reflection Matrix(25,-6,104,-25) (3/13,1/4) -> (3/13,1/4) Reflection Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(289,-112,80,-31) (5/13,2/5) -> (18/5,11/3) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(155,-68,212,-93) (3/7,4/9) -> (8/11,11/15) Hyperbolic Matrix(121,-54,56,-25) (4/9,5/11) -> (2/1,11/5) Glide Reflection Matrix(21,-10,44,-21) (5/11,1/2) -> (5/11,1/2) Reflection Matrix(11,-6,20,-11) (1/2,3/5) -> (1/2,3/5) Reflection Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(369,-256,160,-111) (9/13,7/10) -> (23/10,7/3) Hyperbolic Matrix(305,-224,64,-47) (11/15,3/4) -> (19/4,5/1) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(71,-90,56,-71) (5/4,9/7) -> (5/4,9/7) Reflection Matrix(181,-234,140,-181) (9/7,13/10) -> (9/7,13/10) Reflection Matrix(201,-262,56,-73) (13/10,4/3) -> (7/2,18/5) Glide Reflection Matrix(155,-212,68,-93) (4/3,11/8) -> (9/4,16/7) Hyperbolic Matrix(91,-132,20,-29) (10/7,3/2) -> (9/2,14/3) Hyperbolic Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(41,-70,24,-41) (5/3,7/4) -> (5/3,7/4) Reflection Matrix(89,-198,40,-89) (11/5,9/4) -> (11/5,9/4) Reflection Matrix(151,-346,24,-55) (16/7,23/10) -> (6/1,13/2) Glide Reflection Matrix(263,-634,56,-135) (12/5,29/12) -> (14/3,19/4) Glide Reflection Matrix(407,-986,168,-407) (29/12,17/7) -> (29/12,17/7) Reflection Matrix(69,-170,28,-69) (17/7,5/2) -> (17/7,5/2) Reflection Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(53,-234,12,-53) (13/3,9/2) -> (13/3,9/2) Reflection Matrix(27,-182,4,-27) (13/2,7/1) -> (13/2,7/1) Reflection Matrix(-1,14,0,1) (7/1,1/0) -> (7/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(3,2,-4,-3) -> Matrix(1,0,0,-1) (-1/1,-1/2) -> (0/1,1/0) Matrix(59,26,84,37) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(81,34,112,47) -> Matrix(1,4,0,-1) *** -> (-2/1,1/0) Matrix(51,20,28,11) -> Matrix(1,2,-2,-3) -1/1 Matrix(49,18,128,47) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(89,26,24,7) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(43,12,68,19) -> Matrix(1,0,0,1) Matrix(83,18,60,13) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(57,10,40,7) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(67,-12,28,-5) -> Matrix(1,2,0,1) 1/0 Matrix(91,-20,132,-29) -> Matrix(1,2,0,1) 1/0 Matrix(169,-38,40,-9) -> Matrix(1,2,-2,-5) Matrix(25,-6,104,-25) -> Matrix(3,4,-2,-3) (3/13,1/4) -> (-2/1,-1/1) Matrix(13,-4,36,-11) -> Matrix(1,2,-2,-3) -1/1 Matrix(289,-112,80,-31) -> Matrix(1,0,0,1) Matrix(67,-28,12,-5) -> Matrix(1,0,0,1) Matrix(155,-68,212,-93) -> Matrix(3,2,-2,-1) -1/1 Matrix(121,-54,56,-25) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(21,-10,44,-21) -> Matrix(1,0,0,-1) (5/11,1/2) -> (0/1,1/0) Matrix(11,-6,20,-11) -> Matrix(1,2,0,-1) (1/2,3/5) -> (-1/1,1/0) Matrix(49,-30,80,-49) -> Matrix(-1,0,2,1) (3/5,5/8) -> (-1/1,0/1) Matrix(369,-256,160,-111) -> Matrix(1,0,0,1) Matrix(305,-224,64,-47) -> Matrix(1,0,0,1) Matrix(9,-8,8,-7) -> Matrix(1,2,-2,-3) -1/1 Matrix(71,-90,56,-71) -> Matrix(-1,0,4,1) (5/4,9/7) -> (-1/2,0/1) Matrix(181,-234,140,-181) -> Matrix(1,0,0,-1) (9/7,13/10) -> (0/1,1/0) Matrix(201,-262,56,-73) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(155,-212,68,-93) -> Matrix(1,0,2,1) 0/1 Matrix(91,-132,20,-29) -> Matrix(1,0,0,1) Matrix(19,-30,12,-19) -> Matrix(1,2,0,-1) (3/2,5/3) -> (-1/1,1/0) Matrix(41,-70,24,-41) -> Matrix(5,4,-6,-5) (5/3,7/4) -> (-1/1,-2/3) Matrix(89,-198,40,-89) -> Matrix(1,0,2,-1) (11/5,9/4) -> (0/1,1/1) Matrix(151,-346,24,-55) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(263,-634,56,-135) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(407,-986,168,-407) -> Matrix(-1,8,0,1) (29/12,17/7) -> (4/1,1/0) Matrix(69,-170,28,-69) -> Matrix(1,2,0,-1) (17/7,5/2) -> (-1/1,1/0) Matrix(13,-36,4,-11) -> Matrix(1,2,-2,-3) -1/1 Matrix(53,-234,12,-53) -> Matrix(1,0,0,-1) (13/3,9/2) -> (0/1,1/0) Matrix(27,-182,4,-27) -> Matrix(1,2,0,-1) (13/2,7/1) -> (-1/1,1/0) Matrix(-1,14,0,1) -> Matrix(-1,0,2,1) (7/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.