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): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 30 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -4/1 -7/2 -13/4 -2/1 -7/4 -7/6 0/1 1/1 14/11 7/5 3/2 14/9 7/4 2/1 7/3 5/2 21/8 14/5 3/1 7/2 56/15 4/1 9/2 14/3 5/1 11/2 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 1/0 -6/1 -1/1 0/1 1/0 -5/1 -1/1 0/1 -14/3 0/1 -9/2 0/1 1/0 -13/3 -2/1 -1/1 -4/1 -1/1 0/1 1/0 -7/2 -1/2 1/0 -10/3 -1/1 0/1 1/0 -13/4 -1/1 1/0 -3/1 -1/1 0/1 -14/5 -1/1 -11/4 -1/1 -2/3 -19/7 -1/1 -2/3 -8/3 -1/1 -2/3 -1/2 -5/2 -1/1 -1/2 -7/3 -1/2 -9/4 -1/2 -2/5 -11/5 -2/5 -1/3 -13/6 -3/8 -1/3 -2/1 -1/2 -1/3 0/1 -7/4 -1/2 -1/4 -12/7 -1/2 -1/3 0/1 -17/10 -1/3 0/1 -5/3 -1/3 0/1 -13/8 -1/3 -1/4 -21/13 -1/4 -8/5 -1/3 -1/4 0/1 -27/17 -2/9 -1/5 -19/12 -1/4 0/1 -11/7 -1/5 0/1 -14/9 0/1 -3/2 -1/3 0/1 -7/5 -1/2 -1/4 -11/8 -1/3 0/1 -26/19 -1/2 -1/3 0/1 -41/30 -3/8 -1/3 -56/41 -1/3 -15/11 -1/3 -2/7 -4/3 -1/3 -1/4 0/1 -13/10 -1/3 -1/4 -9/7 -1/3 0/1 -14/11 -1/3 -5/4 -1/3 -1/4 -6/5 -1/3 -2/7 -1/4 -7/6 -1/4 -8/7 -1/4 -3/13 -2/9 -1/1 -1/5 0/1 0/1 0/1 1/1 0/1 1/5 5/4 1/4 1/3 14/11 1/3 9/7 0/1 1/3 4/3 0/1 1/4 1/3 15/11 2/7 1/3 11/8 0/1 1/3 7/5 1/4 1/2 3/2 0/1 1/3 14/9 0/1 11/7 0/1 1/5 19/12 0/1 1/4 8/5 0/1 1/4 1/3 21/13 1/4 13/8 1/4 1/3 18/11 0/1 1/4 1/3 5/3 0/1 1/3 7/4 1/4 1/2 9/5 0/1 1/3 11/6 0/1 1/3 13/7 2/7 1/3 2/1 0/1 1/3 1/2 11/5 1/3 2/5 9/4 2/5 1/2 7/3 1/2 5/2 1/2 1/1 13/5 0/1 1/3 21/8 1/2 29/11 4/7 3/5 8/3 1/2 2/3 1/1 11/4 2/3 1/1 14/5 1/1 3/1 0/1 1/1 7/2 1/2 1/0 11/3 0/1 1/1 26/7 1/2 2/3 1/1 41/11 4/5 1/1 56/15 1/1 15/4 1/1 1/0 4/1 0/1 1/1 1/0 13/3 1/1 2/1 22/5 -1/1 0/1 1/0 9/2 0/1 1/0 14/3 0/1 5/1 0/1 1/1 11/2 0/1 1/1 6/1 0/1 1/1 1/0 7/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,14,0,1) (-7/1,1/0) -> (7/1,1/0) Parabolic Matrix(29,182,18,113) (-7/1,-6/1) -> (8/5,21/13) Hyperbolic Matrix(27,154,-10,-57) (-6/1,-5/1) -> (-19/7,-8/3) Hyperbolic Matrix(29,140,6,29) (-5/1,-14/3) -> (14/3,5/1) Hyperbolic Matrix(55,252,12,55) (-14/3,-9/2) -> (9/2,14/3) Hyperbolic Matrix(111,490,-70,-309) (-9/2,-13/3) -> (-27/17,-19/12) Hyperbolic Matrix(27,112,20,83) (-13/3,-4/1) -> (4/3,15/11) Hyperbolic Matrix(27,98,-8,-29) (-4/1,-7/2) -> (-7/2,-10/3) Parabolic Matrix(111,364,68,223) (-10/3,-13/4) -> (13/8,18/11) Hyperbolic Matrix(57,182,-26,-83) (-13/4,-3/1) -> (-11/5,-13/6) Hyperbolic Matrix(29,84,10,29) (-3/1,-14/5) -> (14/5,3/1) Hyperbolic Matrix(111,308,40,111) (-14/5,-11/4) -> (11/4,14/5) Hyperbolic Matrix(113,308,62,169) (-11/4,-19/7) -> (9/5,11/6) Hyperbolic Matrix(27,70,-22,-57) (-8/3,-5/2) -> (-5/4,-6/5) Hyperbolic Matrix(29,70,12,29) (-5/2,-7/3) -> (7/3,5/2) Hyperbolic Matrix(55,126,24,55) (-7/3,-9/4) -> (9/4,7/3) Hyperbolic Matrix(139,308,88,195) (-9/4,-11/5) -> (11/7,19/12) Hyperbolic Matrix(197,420,-144,-307) (-13/6,-2/1) -> (-26/19,-41/30) Hyperbolic Matrix(55,98,-32,-57) (-2/1,-7/4) -> (-7/4,-12/7) Parabolic Matrix(337,574,-246,-419) (-12/7,-17/10) -> (-11/8,-26/19) Hyperbolic Matrix(83,140,16,27) (-17/10,-5/3) -> (5/1,11/2) Hyperbolic Matrix(111,182,-86,-141) (-5/3,-13/8) -> (-13/10,-9/7) Hyperbolic Matrix(337,546,208,337) (-13/8,-21/13) -> (21/13,13/8) Hyperbolic Matrix(113,182,18,29) (-21/13,-8/5) -> (6/1,7/1) Hyperbolic Matrix(281,448,106,169) (-8/5,-27/17) -> (29/11,8/3) Hyperbolic Matrix(195,308,88,139) (-19/12,-11/7) -> (11/5,9/4) Hyperbolic Matrix(197,308,126,197) (-11/7,-14/9) -> (14/9,11/7) Hyperbolic Matrix(55,84,36,55) (-14/9,-3/2) -> (3/2,14/9) Hyperbolic Matrix(29,42,20,29) (-3/2,-7/5) -> (7/5,3/2) Hyperbolic Matrix(111,154,80,111) (-7/5,-11/8) -> (11/8,7/5) Hyperbolic Matrix(2295,3136,614,839) (-41/30,-56/41) -> (56/15,15/4) Hyperbolic Matrix(2297,3136,616,841) (-56/41,-15/11) -> (41/11,56/15) Hyperbolic Matrix(83,112,20,27) (-15/11,-4/3) -> (4/1,13/3) Hyperbolic Matrix(85,112,22,29) (-4/3,-13/10) -> (15/4,4/1) Hyperbolic Matrix(197,252,154,197) (-9/7,-14/11) -> (14/11,9/7) Hyperbolic Matrix(111,140,88,111) (-14/11,-5/4) -> (5/4,14/11) Hyperbolic Matrix(83,98,-72,-85) (-6/5,-7/6) -> (-7/6,-8/7) Parabolic Matrix(113,126,26,29) (-8/7,-1/1) -> (13/3,22/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic Matrix(141,-182,86,-111) (9/7,4/3) -> (18/11,5/3) Hyperbolic Matrix(225,-308,122,-167) (15/11,11/8) -> (11/6,13/7) Hyperbolic Matrix(309,-490,70,-111) (19/12,8/5) -> (22/5,9/2) Hyperbolic Matrix(57,-98,32,-55) (5/3,7/4) -> (7/4,9/5) Parabolic Matrix(223,-420,60,-113) (13/7,2/1) -> (26/7,41/11) Hyperbolic Matrix(141,-308,38,-83) (2/1,11/5) -> (11/3,26/7) Hyperbolic Matrix(337,-882,128,-335) (13/5,21/8) -> (21/8,29/11) Parabolic Matrix(57,-154,10,-27) (8/3,11/4) -> (11/2,6/1) Hyperbolic Matrix(29,-98,8,-27) (3/1,7/2) -> (7/2,11/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,14,0,1) -> Matrix(1,0,0,1) Matrix(29,182,18,113) -> Matrix(1,0,4,1) Matrix(27,154,-10,-57) -> Matrix(1,2,-2,-3) Matrix(29,140,6,29) -> Matrix(1,0,2,1) Matrix(55,252,12,55) -> Matrix(1,0,0,1) Matrix(111,490,-70,-309) -> Matrix(1,0,-4,1) Matrix(27,112,20,83) -> Matrix(1,0,4,1) Matrix(27,98,-8,-29) -> Matrix(1,0,0,1) Matrix(111,364,68,223) -> Matrix(1,0,4,1) Matrix(57,182,-26,-83) -> Matrix(3,2,-8,-5) Matrix(29,84,10,29) -> Matrix(1,0,2,1) Matrix(111,308,40,111) -> Matrix(5,4,6,5) Matrix(113,308,62,169) -> Matrix(3,2,10,7) Matrix(27,70,-22,-57) -> Matrix(1,0,-2,1) Matrix(29,70,12,29) -> Matrix(3,2,4,3) Matrix(55,126,24,55) -> Matrix(9,4,20,9) Matrix(139,308,88,195) -> Matrix(5,2,22,9) Matrix(197,420,-144,-307) -> Matrix(1,0,0,1) Matrix(55,98,-32,-57) -> Matrix(1,0,0,1) Matrix(337,574,-246,-419) -> Matrix(1,0,0,1) Matrix(83,140,16,27) -> Matrix(1,0,4,1) Matrix(111,182,-86,-141) -> Matrix(1,0,0,1) Matrix(337,546,208,337) -> Matrix(7,2,24,7) Matrix(113,182,18,29) -> Matrix(1,0,4,1) Matrix(281,448,106,169) -> Matrix(7,2,10,3) Matrix(195,308,88,139) -> Matrix(9,2,22,5) Matrix(197,308,126,197) -> Matrix(1,0,10,1) Matrix(55,84,36,55) -> Matrix(1,0,6,1) Matrix(29,42,20,29) -> Matrix(1,0,6,1) Matrix(111,154,80,111) -> Matrix(1,0,6,1) Matrix(2295,3136,614,839) -> Matrix(11,4,8,3) Matrix(2297,3136,616,841) -> Matrix(19,6,22,7) Matrix(83,112,20,27) -> Matrix(1,0,4,1) Matrix(85,112,22,29) -> Matrix(1,0,4,1) Matrix(197,252,154,197) -> Matrix(1,0,6,1) Matrix(111,140,88,111) -> Matrix(7,2,24,7) Matrix(83,98,-72,-85) -> Matrix(15,4,-64,-17) Matrix(113,126,26,29) -> Matrix(9,2,4,1) Matrix(1,0,2,1) -> Matrix(1,0,10,1) Matrix(57,-70,22,-27) -> Matrix(1,0,-2,1) Matrix(141,-182,86,-111) -> Matrix(1,0,0,1) Matrix(225,-308,122,-167) -> Matrix(1,0,0,1) Matrix(309,-490,70,-111) -> Matrix(1,0,-4,1) Matrix(57,-98,32,-55) -> Matrix(1,0,0,1) Matrix(223,-420,60,-113) -> Matrix(5,-2,8,-3) Matrix(141,-308,38,-83) -> Matrix(5,-2,8,-3) Matrix(337,-882,128,-335) -> Matrix(9,-4,16,-7) Matrix(57,-154,10,-27) -> Matrix(3,-2,2,-1) Matrix(29,-98,8,-27) -> Matrix(1,0,0,1) 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): 8 Degree of the the map X: 8 Degree of the the map Y: 48 Permutation triple for Y: ((2,6,22,44,46,23,7)(3,12,35,47,36,13,4)(5,18,32,10,9,31,19)(8,25,37,16,15,41,26)(11,29,28,21,20,45,34)(14,38,27,24,17,43,39); (1,4,16,34,45,41,47,48,44,31,43,17,5,2)(3,10,32,35,20,40,39,46,25,8,7,24,33,11)(6,21,18,42,37,14,13,36,27,26,30,9,29,22)(12,23)(15,19)(28,38); (1,2,8,27,28,9,3)(4,14,40,20,6,5,15)(7,12,11,16,42,18,17)(19,44,29,33,24,36,41)(21,38,37,46,48,47,32)(23,39,31,30,26,45,35)) ----------------------------------------------------------------------- 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): 144 Minimal number of generators: 25 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: 18 Genus: 4 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -7/6 0/1 1/1 7/5 3/2 7/4 2/1 7/3 5/2 14/5 3/1 7/2 4/1 14/3 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -1/1 0/1 -4/1 -1/1 0/1 1/0 -7/2 -1/2 1/0 -3/1 -1/1 0/1 -8/3 -1/1 -2/3 -1/2 -5/2 -1/1 -1/2 -7/3 -1/2 -2/1 -1/2 -1/3 0/1 -7/4 -1/2 -1/4 -5/3 -1/3 0/1 -13/8 -1/3 -1/4 -21/13 -1/4 -8/5 -1/3 -1/4 0/1 -11/7 -1/5 0/1 -14/9 0/1 -3/2 -1/3 0/1 -7/5 -1/2 -1/4 -4/3 -1/3 -1/4 0/1 -9/7 -1/3 0/1 -14/11 -1/3 -5/4 -1/3 -1/4 -6/5 -1/3 -2/7 -1/4 -7/6 -1/4 -1/1 -1/5 0/1 0/1 0/1 1/1 0/1 1/5 5/4 1/4 1/3 4/3 0/1 1/4 1/3 7/5 1/4 1/2 3/2 0/1 1/3 8/5 0/1 1/4 1/3 5/3 0/1 1/3 7/4 1/4 1/2 2/1 0/1 1/3 1/2 7/3 1/2 5/2 1/2 1/1 13/5 0/1 1/3 21/8 1/2 8/3 1/2 2/3 1/1 11/4 2/3 1/1 14/5 1/1 3/1 0/1 1/1 7/2 1/2 1/0 4/1 0/1 1/1 1/0 9/2 0/1 1/0 14/3 0/1 5/1 0/1 1/1 6/1 0/1 1/1 1/0 7/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(13,70,-8,-43) (-5/1,1/0) -> (-5/3,-13/8) Hyperbolic Matrix(13,56,-10,-43) (-5/1,-4/1) -> (-4/3,-9/7) Hyperbolic Matrix(15,56,4,15) (-4/1,-7/2) -> (7/2,4/1) Hyperbolic Matrix(13,42,4,13) (-7/2,-3/1) -> (3/1,7/2) Hyperbolic Matrix(41,112,-26,-71) (-3/1,-8/3) -> (-8/5,-11/7) Hyperbolic Matrix(27,70,-22,-57) (-8/3,-5/2) -> (-5/4,-6/5) Hyperbolic Matrix(29,70,12,29) (-5/2,-7/3) -> (7/3,5/2) Hyperbolic Matrix(13,28,6,13) (-7/3,-2/1) -> (2/1,7/3) Hyperbolic Matrix(15,28,8,15) (-2/1,-7/4) -> (7/4,2/1) Hyperbolic Matrix(41,70,24,41) (-7/4,-5/3) -> (5/3,7/4) Hyperbolic Matrix(69,112,8,13) (-13/8,-21/13) -> (7/1,1/0) Hyperbolic Matrix(113,182,18,29) (-21/13,-8/5) -> (6/1,7/1) Hyperbolic Matrix(125,196,44,69) (-11/7,-14/9) -> (14/5,3/1) Hyperbolic Matrix(127,196,46,71) (-14/9,-3/2) -> (11/4,14/5) Hyperbolic Matrix(29,42,20,29) (-3/2,-7/5) -> (7/5,3/2) Hyperbolic Matrix(41,56,30,41) (-7/5,-4/3) -> (4/3,7/5) Hyperbolic Matrix(153,196,32,41) (-9/7,-14/11) -> (14/3,5/1) Hyperbolic Matrix(155,196,34,43) (-14/11,-5/4) -> (9/2,14/3) Hyperbolic Matrix(153,182,58,69) (-6/5,-7/6) -> (21/8,8/3) Hyperbolic Matrix(99,112,38,43) (-7/6,-1/1) -> (13/5,21/8) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic Matrix(43,-56,10,-13) (5/4,4/3) -> (4/1,9/2) Hyperbolic Matrix(71,-112,26,-41) (3/2,8/5) -> (8/3,11/4) Hyperbolic Matrix(43,-70,8,-13) (8/5,5/3) -> (5/1,6/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(13,70,-8,-43) -> Matrix(1,1,-4,-3) Matrix(13,56,-10,-43) -> Matrix(1,1,-4,-3) Matrix(15,56,4,15) -> Matrix(1,1,0,1) Matrix(13,42,4,13) -> Matrix(1,1,0,1) Matrix(41,112,-26,-71) -> Matrix(1,1,-6,-5) Matrix(27,70,-22,-57) -> Matrix(1,0,-2,1) Matrix(29,70,12,29) -> Matrix(3,2,4,3) Matrix(13,28,6,13) -> Matrix(3,1,8,3) Matrix(15,28,8,15) -> Matrix(3,1,8,3) Matrix(41,70,24,41) -> Matrix(3,1,8,3) Matrix(69,112,8,13) -> Matrix(3,1,-4,-1) Matrix(113,182,18,29) -> Matrix(1,0,4,1) Matrix(125,196,44,69) -> Matrix(5,1,4,1) Matrix(127,196,46,71) -> Matrix(1,1,0,1) Matrix(29,42,20,29) -> Matrix(1,0,6,1) Matrix(41,56,30,41) -> Matrix(3,1,8,3) Matrix(153,196,32,41) -> Matrix(3,1,2,1) Matrix(155,196,34,43) -> Matrix(3,1,-4,-1) Matrix(153,182,58,69) -> Matrix(11,3,18,5) Matrix(99,112,38,43) -> Matrix(5,1,14,3) Matrix(1,0,2,1) -> Matrix(1,0,10,1) Matrix(57,-70,22,-27) -> Matrix(1,0,-2,1) Matrix(43,-56,10,-13) -> Matrix(3,-1,4,-1) Matrix(71,-112,26,-41) -> Matrix(5,-1,6,-1) Matrix(43,-70,8,-13) -> Matrix(3,-1,4,-1) 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): 8 ----------------------------------------------------------------------- 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 5 1 1/1 (0/1,1/5).(1/6,1/4) 0 14 5/4 (1/4,1/3) 0 14 4/3 (1/4,1/2) 0 7 7/5 (0/1,1/3).(1/4,1/2) 0 2 3/2 (0/1,1/3) 0 14 8/5 (1/4,1/2) 0 7 5/3 (0/1,1/3).(1/4,1/2) 0 14 7/4 (1/4,1/2) 0 2 2/1 (1/4,1/2) 0 7 7/3 1/2 3 2 5/2 (1/2,1/1) 0 14 13/5 (0/1,1/3).(1/4,1/2) 0 14 21/8 1/2 4 2 8/3 (1/2,3/4) 0 7 11/4 (2/3,1/1) 0 14 14/5 1/1 2 1 3/1 (0/1,1/1).(1/2,1/0) 0 14 7/2 (1/2,1/0) 0 2 4/1 (1/2,1/0) 0 7 9/2 (0/1,1/0) 0 14 14/3 0/1 1 1 5/1 (0/1,1/1).(1/2,1/0) 0 14 6/1 (1/2,1/0) 0 7 7/1 1/0 1 2 1/0 (0/1,1/0) 0 14 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(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic Matrix(43,-56,10,-13) (5/4,4/3) -> (4/1,9/2) Hyperbolic Matrix(41,-56,30,-41) (4/3,7/5) -> (4/3,7/5) Reflection Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) Reflection Matrix(71,-112,26,-41) (3/2,8/5) -> (8/3,11/4) Hyperbolic Matrix(43,-70,8,-13) (8/5,5/3) -> (5/1,6/1) Hyperbolic Matrix(41,-70,24,-41) (5/3,7/4) -> (5/3,7/4) Reflection Matrix(15,-28,8,-15) (7/4,2/1) -> (7/4,2/1) Reflection Matrix(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(209,-546,80,-209) (13/5,21/8) -> (13/5,21/8) Reflection Matrix(127,-336,48,-127) (21/8,8/3) -> (21/8,8/3) Reflection Matrix(111,-308,40,-111) (11/4,14/5) -> (11/4,14/5) Reflection Matrix(29,-84,10,-29) (14/5,3/1) -> (14/5,3/1) Reflection Matrix(13,-42,4,-13) (3/1,7/2) -> (3/1,7/2) Reflection Matrix(15,-56,4,-15) (7/2,4/1) -> (7/2,4/1) Reflection Matrix(55,-252,12,-55) (9/2,14/3) -> (9/2,14/3) Reflection Matrix(29,-140,6,-29) (14/3,5/1) -> (14/3,5/1) Reflection Matrix(13,-84,2,-13) (6/1,7/1) -> (6/1,7/1) 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(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,10,-1) (0/1,1/1) -> (0/1,1/5) Matrix(57,-70,22,-27) -> Matrix(1,0,-2,1) 0/1 Matrix(43,-56,10,-13) -> Matrix(3,-1,4,-1) 1/2 Matrix(41,-56,30,-41) -> Matrix(3,-1,8,-3) (4/3,7/5) -> (1/4,1/2) Matrix(29,-42,20,-29) -> Matrix(1,0,6,-1) (7/5,3/2) -> (0/1,1/3) Matrix(71,-112,26,-41) -> Matrix(5,-1,6,-1) Matrix(43,-70,8,-13) -> Matrix(3,-1,4,-1) 1/2 Matrix(41,-70,24,-41) -> Matrix(3,-1,8,-3) (5/3,7/4) -> (1/4,1/2) Matrix(15,-28,8,-15) -> Matrix(3,-1,8,-3) (7/4,2/1) -> (1/4,1/2) Matrix(13,-28,6,-13) -> Matrix(3,-1,8,-3) (2/1,7/3) -> (1/4,1/2) Matrix(29,-70,12,-29) -> Matrix(3,-2,4,-3) (7/3,5/2) -> (1/2,1/1) Matrix(209,-546,80,-209) -> Matrix(3,-1,8,-3) (13/5,21/8) -> (1/4,1/2) Matrix(127,-336,48,-127) -> Matrix(5,-3,8,-5) (21/8,8/3) -> (1/2,3/4) Matrix(111,-308,40,-111) -> Matrix(5,-4,6,-5) (11/4,14/5) -> (2/3,1/1) Matrix(29,-84,10,-29) -> Matrix(1,0,2,-1) (14/5,3/1) -> (0/1,1/1) Matrix(13,-42,4,-13) -> Matrix(-1,1,0,1) (3/1,7/2) -> (1/2,1/0) Matrix(15,-56,4,-15) -> Matrix(-1,1,0,1) (7/2,4/1) -> (1/2,1/0) Matrix(55,-252,12,-55) -> Matrix(1,0,0,-1) (9/2,14/3) -> (0/1,1/0) Matrix(29,-140,6,-29) -> Matrix(1,0,2,-1) (14/3,5/1) -> (0/1,1/1) Matrix(13,-84,2,-13) -> Matrix(-1,1,0,1) (6/1,7/1) -> (1/2,1/0) Matrix(-1,14,0,1) -> Matrix(1,0,0,-1) (7/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.