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): 48
Minimal number of generators: 9
Number of equivalence classes of cusps: 10
Genus: 0

REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES
 -1/1   0/1   1/2   2/3   3/4   1/1   4/3   3/2   2/1   1/0 

CUSPS AT THE FUNDAMENTAL DOMAIN AND
THEIR IMAGES UNDER THE PULLBACK MAP
  CUSP  IMAGE  PSEUDOIMAGES
  -2/1  -1/1   
  -3/2          -1/1  -1/2  
  -4/3  -1/2   
  -1/1          -1/2   1/0  
  -4/5  -1/2   
  -3/4          -1/2   0/1  
  -2/3   0/1   
  -1/2          -1/1   0/1  
   0/1          -1/2   1/0  
   1/2          -1/1   0/1  
   2/3   0/1   
   3/4           0/1   1/0  
   1/1          -1/2   1/0  
   5/4           0/1   1/0  
   4/3   1/0   
   3/2          -1/1   1/0  
   2/1  -1/1   
   1/0          -1/1   0/1  

GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN
     GENERATOR             EDGE PAIRING           TYPE
Matrix(1,4,0,1)     (-2/1,1/0)  -> (2/1,1/0)   Parabolic
Matrix(7,12,4,7)    (-2/1,-3/2) -> (3/2,2/1)   Hyperbolic
Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2)   Hyperbolic
Matrix(7,8,-8,-9)   (-4/3,-1/1) -> (-1/1,-4/5) Parabolic
Matrix(41,32,32,25) (-4/5,-3/4) -> (5/4,4/3)   Hyperbolic
Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4)   Hyperbolic
Matrix(7,4,12,7)    (-2/3,-1/2) -> (1/2,2/3)   Hyperbolic
Matrix(1,0,4,1)     (-1/2,0/1)  -> (0/1,1/2)   Parabolic
Matrix(9,-8,8,-7)   (3/4,1/1)   -> (1/1,5/4)   Parabolic

IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM
Matrix(1,4,0,1)     -> Matrix(1,0,0,1)
Matrix(7,12,4,7)    -> Matrix(3,2,-2,-1)
Matrix(17,24,12,17) -> Matrix(3,2,-2,-1)
Matrix(7,8,-8,-9)   -> Matrix(1,0,0,1)
Matrix(41,32,32,25) -> Matrix(1,0,2,1)
Matrix(17,12,24,17) -> Matrix(1,0,2,1)
Matrix(7,4,12,7)    -> Matrix(1,0,0,1)
Matrix(1,0,4,1)     -> Matrix(1,0,0,1)
Matrix(9,-8,8,-7)   -> 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): 1
Degree of the the map X: 1
Degree of the the map Y: 8
Permutation triple for Y: 
     ((1,2)(3,4)(5,6)(7,8);
     (1,4,8,5)(2,6,7,3);
     (1,3)(2,5)(4,7)(6,8))

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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.

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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): 24
Minimal number of generators: 5
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: 6
Genus: 0

REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES
  0/1   1/1   4/3   3/2   2/1   1/0 

CUSPS AT THE FUNDAMENTAL DOMAIN AND
THEIR IMAGES UNDER THE PULLBACK MAP
  CUSP  IMAGE  PSEUDOIMAGES
  -2/1  -1/1   
  -3/2          -1/1  -1/2  
  -4/3  -1/2   
  -1/1          -1/2   1/0  
   0/1          -1/2   1/0  
   1/1          -1/2   1/0  
   4/3   1/0   
   3/2          -1/1   1/0  
   2/1  -1/1   
   1/0          -1/1   0/1  

GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN
     GENERATOR            EDGE PAIRING          TYPE
Matrix(1,4,0,1)     (-2/1,1/0)  -> (2/1,1/0) Parabolic
Matrix(7,12,4,7)    (-2/1,-3/2) -> (3/2,2/1) Hyperbolic
Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic
Matrix(7,8,6,7)     (-4/3,-1/1) -> (1/1,4/3) Hyperbolic
Matrix(1,0,2,1)     (-1/1,0/1)  -> (0/1,1/1) Parabolic

IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM
Matrix(1,4,0,1)     -> Matrix(1,0,0,1)
Matrix(7,12,4,7)    -> Matrix(3,2,-2,-1)
Matrix(17,24,12,17) -> Matrix(3,2,-2,-1)
Matrix(7,8,6,7)     -> Matrix(1,1,-2,-1)
Matrix(1,0,2,1)     -> Matrix(1,1,-2,-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): 1

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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

This is a reflection group.

CUSPS AT THE FUNDAMENTAL DOMAIN AND
THEIR IMAGES UNDER THE PULLBACK MAP
 CUSP          IMAGE             c    d
  0/1  (-1/1,0/1).(-1/2,1/0)     0    2
  1/1       (-1/2,1/0)           0    4
  4/3           1/0              1    2
  3/2       (-1/1,1/0)           0    4
  2/1          -1/1              1    2
  1/0       (-1/1,0/1)           0    4

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(7,-8,6,-7)     (1/1,4/3) -> (1/1,4/3) Reflection
Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection
Matrix(7,-12,4,-7)    (3/2,2/1) -> (3/2,2/1) Reflection
Matrix(-1,4,0,1)      (2/1,1/0) -> (2/1,1/0) Reflection

IMAGES OF THE GENERATORS                  MAP ON REFLECTION AXES OR
UNDER THE VIRTUAL ENDOMORPHISM            FIXED POINT OF IMAGE
Matrix(1,0,0,-1)      -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1)
Matrix(1,0,2,-1)      -> Matrix(1,1,0,-1) (0/1,1/1) -> (-1/2,1/0)
Matrix(7,-8,6,-7)     -> Matrix(1,1,0,-1) (1/1,4/3) -> (-1/2,1/0)
Matrix(17,-24,12,-17) -> Matrix(1,2,0,-1) (4/3,3/2) -> (-1/1,1/0)
Matrix(7,-12,4,-7)    -> Matrix(1,2,0,-1) (3/2,2/1) -> (-1/1,1/0)
Matrix(-1,4,0,1)      -> Matrix(-1,0,2,1) (2/1,1/0) -> (-1/1,0/1)

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The pullback map has no extra symmetries.