# Linear representation theory of projective general linear group:PGL(2,9)

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## Contents

This article gives specific information, namely, linear representation theory, about a particular group, namely: projective general linear group:PGL(2,9).
View linear representation theory of particular groups | View other specific information about projective general linear group:PGL(2,9)

## Summary

Item Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 1,1,8,8,8,8,9,9,10,10,10
number: 11, maximum: 10, lcm: 360, sum of squares: 720

## Family contexts

Family Parameter values General discussion of linear representation theory of family
projective general linear group of degree two over a finite field. $PGL(2,q)$ denotes this group for a field of size $q$. $q = 9$, i.e., field:F9. The group is $PGL(2,9)$. linear representation theory of projective general linear group of degree two over a finite field

## GAP implementation

The degrees of irreducible representations can be determined using GAP's CharacterDegrees function:

gap> CharacterDegrees(PGL(2,9));
[ [ 1, 2 ], [ 8, 4 ], [ 9, 2 ], [ 10, 3 ] ]

The character table can be determined using the Irr and CharacterTable functions:

gap> Irr(CharacterTable(PGL(2,9)));
[ Character( CharacterTable( Group(
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ]) ),
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Group(
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ]) ),
[ 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, -1 ] ),
Character( CharacterTable( Group(
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ]) ),
[ 8, 0, 0, 0, 0, -1, -2, -E(5)-E(5)^4, -E(5)^2-E(5)^3, -E(5)^2-E(5)^3,
-E(5)-E(5)^4 ] ), Character( CharacterTable( Group(
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ]) ),
[ 8, 0, 0, 0, 0, -1, -2, -E(5)^2-E(5)^3, -E(5)-E(5)^4, -E(5)-E(5)^4,
-E(5)^2-E(5)^3 ] ), Character( CharacterTable( Group(
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ]) ),
[ 8, 0, 0, 0, 0, -1, 2, -E(5)-E(5)^4, -E(5)^2-E(5)^3, E(5)^2+E(5)^3,
E(5)+E(5)^4 ] ), Character( CharacterTable( Group(
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ]) ),
[ 8, 0, 0, 0, 0, -1, 2, -E(5)^2-E(5)^3, -E(5)-E(5)^4, E(5)+E(5)^4,
E(5)^2+E(5)^3 ] ), Character( CharacterTable( Group(
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ]) ),
[ 9, 1, 1, 1, 1, 0, -1, -1, -1, -1, -1 ] ),
Character( CharacterTable( Group(
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ]) ),
[ 9, 1, -1, 1, -1, 0, 1, -1, -1, 1, 1 ] ),
Character( CharacterTable( Group(
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ]) ),
[ 10, 2, 0, -2, 0, 1, 0, 0, 0, 0, 0 ] ),
Character( CharacterTable( Group(
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ]) ),
[ 10, -2, -E(8)+E(8)^3, 0, E(8)-E(8)^3, 1, 0, 0, 0, 0, 0 ] ),
Character( CharacterTable( Group(
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ]) ),
[ 10, -2, E(8)-E(8)^3, 0, -E(8)+E(8)^3, 1, 0, 0, 0, 0, 0 ] ) ]

The irreducible representations can be determined using the IrreducibleRepresentations function:

gap> IrreducibleRepresentations(PGL(2,9));
[ [ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ], [ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ] -> [ [ [ -1 ] ], [ [ 1 ] ] ],
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ] ->
[ [ [ 78/5*E(5)+32/5*E(5)^2+32/5*E(5)^3+78/5*E(5)^4, -38/5*E(5)-12/5*E(5)^2-12/5*E(5)^3-38/5*E(5)^4, 28/5*E(5)+12/5*E(5)^2+12/5*E(5)^3+28/5*E(5)^4,
62/5*E(5)+28/5*E(5)^2+28/5*E(5)^3+62/5*E(5)^4, -3*E(5)-E(5)^2-E(5)^3-3*E(5)^4, 8/5*E(5)+2/5*E(5)^2+2/5*E(5)^3+8/5*E(5)^4,
16*E(5)+6*E(5)^2+6*E(5)^3+16*E(5)^4, 19*E(5)+7*E(5)^2+7*E(5)^3+19*E(5)^4 ],
[ 21/5*E(5)+14/5*E(5)^2+14/5*E(5)^3+21/5*E(5)^4, -6/5*E(5)+6/5*E(5)^2+6/5*E(5)^3-6/5*E(5)^4, 11/5*E(5)+14/5*E(5)^2+14/5*E(5)^3+11/5*E(5)^4,
14/5*E(5)+6/5*E(5)^2+6/5*E(5)^3+14/5*E(5)^4, 1, -9/5*E(5)-16/5*E(5)^2-16/5*E(5)^3-9/5*E(5)^4, -4, 4*E(5)+2*E(5)^2+2*E(5)^3+4*E(5)^4 ],
[ -3*E(5)-2*E(5)^2-2*E(5)^3-3*E(5)^4, E(5)-E(5)^2-E(5)^3+E(5)^4, 1, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4, 0, E(5)+2*E(5)^2+2*E(5)^3+E(5)^4,
-3*E(5)-2*E(5)^2-2*E(5)^3-3*E(5)^4, -3*E(5)-2*E(5)^2-2*E(5)^3-3*E(5)^4 ],
[ 2/5*E(5)-2/5*E(5)^2-2/5*E(5)^3+2/5*E(5)^4, -2/5*E(5)-8/5*E(5)^2-8/5*E(5)^3-2/5*E(5)^4, -3/5*E(5)-7/5*E(5)^2-7/5*E(5)^3-3/5*E(5)^4,
-2/5*E(5)-3/5*E(5)^2-3/5*E(5)^3-2/5*E(5)^4, 0, 2/5*E(5)+8/5*E(5)^2+8/5*E(5)^3+2/5*E(5)^4, -E(5)^2-E(5)^3, 0 ],
[ -54/5*E(5)-26/5*E(5)^2-26/5*E(5)^3-54/5*E(5)^4, 24/5*E(5)+1/5*E(5)^2+1/5*E(5)^3+24/5*E(5)^4, -24/5*E(5)-16/5*E(5)^2-16/5*E(5)^3-24/5*E(5)^4,
-46/5*E(5)-19/5*E(5)^2-19/5*E(5)^3-46/5*E(5)^4, 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4, 1/5*E(5)+9/5*E(5)^2+9/5*E(5)^3+1/5*E(5)^4,
-11*E(5)-6*E(5)^2-6*E(5)^3-11*E(5)^4, -13*E(5)-6*E(5)^2-6*E(5)^3-13*E(5)^4 ],
[ 9*E(5)+4*E(5)^2+4*E(5)^3+9*E(5)^4, -4*E(5)-E(5)^2-E(5)^3-4*E(5)^4, 3*E(5)+2*E(5)^2+2*E(5)^3+3*E(5)^4, 7*E(5)+3*E(5)^2+3*E(5)^3+7*E(5)^4,
-2*E(5)-E(5)^2-E(5)^3-2*E(5)^4, E(5)-E(5)^2-E(5)^3+E(5)^4, 10*E(5)+5*E(5)^2+5*E(5)^3+10*E(5)^4, 11*E(5)+4*E(5)^2+4*E(5)^3+11*E(5)^4 ],
[ 0, 0, 0, 0, 0, 1, 0, 0 ], [ -13*E(5)-5*E(5)^2-5*E(5)^3-13*E(5)^4, 6*E(5)+2*E(5)^2+2*E(5)^3+6*E(5)^4, -5*E(5)-3*E(5)^2-3*E(5)^3-5*E(5)^4,
-10*E(5)-4*E(5)^2-4*E(5)^3-10*E(5)^4, 3*E(5)+2*E(5)^2+2*E(5)^3+3*E(5)^4, -E(5)-E(5)^4, -14*E(5)-6*E(5)^2-6*E(5)^3-14*E(5)^4,
-16*E(5)-6*E(5)^2-6*E(5)^3-16*E(5)^4 ] ],
[ [ -54/5*E(5)-16/5*E(5)^2-16/5*E(5)^3-54/5*E(5)^4, 34/5*E(5)+21/5*E(5)^2+21/5*E(5)^3+34/5*E(5)^4, -19/5*E(5)-1/5*E(5)^2-1/5*E(5)^3-19/5*E(5)^4,
-46/5*E(5)-14/5*E(5)^2-14/5*E(5)^3-46/5*E(5)^4, 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4, -9/5*E(5)-11/5*E(5)^2-11/5*E(5)^3-9/5*E(5)^4,
-11*E(5)-3*E(5)^2-3*E(5)^3-11*E(5)^4, -14*E(5)-5*E(5)^2-5*E(5)^3-14*E(5)^4 ],
[ 91/5*E(5)+39/5*E(5)^2+39/5*E(5)^3+91/5*E(5)^4, -46/5*E(5)-14/5*E(5)^2-14/5*E(5)^3-46/5*E(5)^4, 31/5*E(5)+14/5*E(5)^2+14/5*E(5)^3+31/5*E(5)^4,
74/5*E(5)+26/5*E(5)^2+26/5*E(5)^3+74/5*E(5)^4, -3*E(5)-E(5)^2-E(5)^3-3*E(5)^4, 1/5*E(5)-6/5*E(5)^2-6/5*E(5)^3+1/5*E(5)^4,
18*E(5)+8*E(5)^2+8*E(5)^3+18*E(5)^4, 22*E(5)+9*E(5)^2+9*E(5)^3+22*E(5)^4 ],
[ -99/5*E(5)-36/5*E(5)^2-36/5*E(5)^3-99/5*E(5)^4, 54/5*E(5)+26/5*E(5)^2+26/5*E(5)^3+54/5*E(5)^4, -34/5*E(5)-11/5*E(5)^2-11/5*E(5)^3-34/5*E(5)^4,
-81/5*E(5)-29/5*E(5)^2-29/5*E(5)^3-81/5*E(5)^4, 4*E(5)+2*E(5)^2+2*E(5)^3+4*E(5)^4, -9/5*E(5)-11/5*E(5)^2-11/5*E(5)^3-9/5*E(5)^4,
-20*E(5)-7*E(5)^2-7*E(5)^3-20*E(5)^4, -25*E(5)-9*E(5)^2-9*E(5)^3-25*E(5)^4 ],
[ 21*E(5)+7*E(5)^2+7*E(5)^3+21*E(5)^4, -12*E(5)-6*E(5)^2-6*E(5)^3-12*E(5)^4, 7*E(5)+E(5)^2+E(5)^3+7*E(5)^4, 18*E(5)+7*E(5)^2+7*E(5)^3+18*E(5)^4,
-4*E(5)-E(5)^2-E(5)^3-4*E(5)^4, 2*E(5)+3*E(5)^2+3*E(5)^3+2*E(5)^4, 20*E(5)+5*E(5)^2+5*E(5)^3+20*E(5)^4, 26*E(5)+9*E(5)^2+9*E(5)^3+26*E(5)^4
], [ -49/5*E(5)-21/5*E(5)^2-21/5*E(5)^3-49/5*E(5)^4, 24/5*E(5)+6/5*E(5)^2+6/5*E(5)^3+24/5*E(5)^4,
-14/5*E(5)-6/5*E(5)^2-6/5*E(5)^3-14/5*E(5)^4, -36/5*E(5)-14/5*E(5)^2-14/5*E(5)^3-36/5*E(5)^4, 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4,
-4/5*E(5)-1/5*E(5)^2-1/5*E(5)^3-4/5*E(5)^4, -10*E(5)-4*E(5)^2-4*E(5)^3-10*E(5)^4, -12*E(5)-5*E(5)^2-5*E(5)^3-12*E(5)^4 ],
[ -31/5*E(5)-9/5*E(5)^2-9/5*E(5)^3-31/5*E(5)^4, 21/5*E(5)+9/5*E(5)^2+9/5*E(5)^3+21/5*E(5)^4, -11/5*E(5)-4/5*E(5)^2-4/5*E(5)^3-11/5*E(5)^4,
-29/5*E(5)-11/5*E(5)^2-11/5*E(5)^3-29/5*E(5)^4, -1, -1/5*E(5)-4/5*E(5)^2-4/5*E(5)^3-1/5*E(5)^4, -6*E(5)-2*E(5)^2-2*E(5)^3-6*E(5)^4,
-8*E(5)-3*E(5)^2-3*E(5)^3-8*E(5)^4 ], [ 63/5*E(5)+22/5*E(5)^2+22/5*E(5)^3+63/5*E(5)^4, -38/5*E(5)-22/5*E(5)^2-22/5*E(5)^3-38/5*E(5)^4,
23/5*E(5)+7/5*E(5)^2+7/5*E(5)^3+23/5*E(5)^4, 52/5*E(5)+18/5*E(5)^2+18/5*E(5)^3+52/5*E(5)^4, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4,
3/5*E(5)+7/5*E(5)^2+7/5*E(5)^3+3/5*E(5)^4, 12*E(5)+4*E(5)^2+4*E(5)^3+12*E(5)^4, 15*E(5)+5*E(5)^2+5*E(5)^3+15*E(5)^4 ],
[ -21/5*E(5)-14/5*E(5)^2-14/5*E(5)^3-21/5*E(5)^4, 6/5*E(5)-6/5*E(5)^2-6/5*E(5)^3+6/5*E(5)^4, -11/5*E(5)-14/5*E(5)^2-14/5*E(5)^3-11/5*E(5)^4,
-14/5*E(5)-6/5*E(5)^2-6/5*E(5)^3-14/5*E(5)^4, -1, 9/5*E(5)+16/5*E(5)^2+16/5*E(5)^3+9/5*E(5)^4, 4, -4*E(5)-2*E(5)^2-2*E(5)^3-4*E(5)^4 ] ] ],
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ] ->
[ [ [ 4/5*E(5)+1/5*E(5)^2+1/5*E(5)^3+4/5*E(5)^4, 3/10*E(5)+1/5*E(5)^2+1/5*E(5)^3+3/10*E(5)^4, 0, 1/2*E(5)+1/2*E(5)^4,
-1/5*E(5)+1/5*E(5)^2+1/5*E(5)^3-1/5*E(5)^4, -1, 1/2*E(5)+E(5)^2+E(5)^3+1/2*E(5)^4, 1/2*E(5)+E(5)^2+E(5)^3+1/2*E(5)^4 ],
[ -6/5*E(5)-4/5*E(5)^2-4/5*E(5)^3-6/5*E(5)^4, -17/10*E(5)-13/10*E(5)^2-13/10*E(5)^3-17/10*E(5)^4, -1, 1/2,
9/5*E(5)+6/5*E(5)^2+6/5*E(5)^3+9/5*E(5)^4, -2*E(5)-2*E(5)^4, 1/2*E(5)+3/2*E(5)^2+3/2*E(5)^3+1/2*E(5)^4,
-1/2*E(5)+1/2*E(5)^2+1/2*E(5)^3-1/2*E(5)^4 ], [ 4/5*E(5)+1/5*E(5)^2+1/5*E(5)^3+4/5*E(5)^4, 13/10*E(5)+7/10*E(5)^2+7/10*E(5)^3+13/10*E(5)^4,
0, 1/2, -6/5*E(5)-4/5*E(5)^2-4/5*E(5)^3-6/5*E(5)^4, 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4, 3/2*E(5)+1/2*E(5)^2+1/2*E(5)^3+3/2*E(5)^4,
-1/2*E(5)+1/2*E(5)^2+1/2*E(5)^3-1/2*E(5)^4 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ],
[ -3/5*E(5)-2/5*E(5)^2-2/5*E(5)^3-3/5*E(5)^4, -3/5*E(5)+11/10*E(5)^2+11/10*E(5)^3-3/5*E(5)^4, -2*E(5)^2-2*E(5)^3,
-2*E(5)-5/2*E(5)^2-5/2*E(5)^3-2*E(5)^4, 2/5*E(5)-2/5*E(5)^2-2/5*E(5)^3+2/5*E(5)^4, -2*E(5)-3*E(5)^2-3*E(5)^3-2*E(5)^4,
-E(5)-5/2*E(5)^2-5/2*E(5)^3-E(5)^4, -E(5)-1/2*E(5)^2-1/2*E(5)^3-E(5)^4 ],
[ 2/5*E(5)+3/5*E(5)^2+3/5*E(5)^3+2/5*E(5)^4, -1/10*E(5)-19/10*E(5)^2-19/10*E(5)^3-1/10*E(5)^4, E(5)+2*E(5)^2+2*E(5)^3+E(5)^4,
1/2*E(5)+5/2*E(5)^2+5/2*E(5)^3+1/2*E(5)^4, -3/5*E(5)+3/5*E(5)^2+3/5*E(5)^3-3/5*E(5)^4, 2*E(5)+3*E(5)^2+3*E(5)^3+2*E(5)^4,
3/2*E(5)+5/2*E(5)^2+5/2*E(5)^3+3/2*E(5)^4, 1/2 ],
[ 2/5*E(5)-2/5*E(5)^2-2/5*E(5)^3+2/5*E(5)^4, 9/10*E(5)+11/10*E(5)^2+11/10*E(5)^3+9/10*E(5)^4, 1, -1/2*E(5)-3/2*E(5)^2-3/2*E(5)^3-1/2*E(5)^4,
-3/5*E(5)-2/5*E(5)^2-2/5*E(5)^3-3/5*E(5)^4, E(5)-E(5)^2-E(5)^3+E(5)^4, 1/2, -1/2 ], [ 0, 0, 0, 0, 0, -1, 0, 0 ] ],
[ [ 0, -1/2*E(5)+1/2*E(5)^2+1/2*E(5)^3-1/2*E(5)^4, 0, 1/2*E(5)-1/2*E(5)^2-1/2*E(5)^3+1/2*E(5)^4, 0, 1, 3/2,
1/2*E(5)-1/2*E(5)^2-1/2*E(5)^3+1/2*E(5)^4 ], [ -4/5*E(5)-6/5*E(5)^2-6/5*E(5)^3-4/5*E(5)^4, 6/5*E(5)+14/5*E(5)^2+14/5*E(5)^3+6/5*E(5)^4,
-2*E(5)^2-2*E(5)^3, -E(5)-3*E(5)^2-3*E(5)^3-E(5)^4, -4/5*E(5)-6/5*E(5)^2-6/5*E(5)^3-4/5*E(5)^4, -2*E(5)-4*E(5)^2-4*E(5)^3-2*E(5)^4,
-E(5)-3*E(5)^2-3*E(5)^3-E(5)^4, 0 ], [ 0, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, E(5)+2*E(5)^2+2*E(5)^3+E(5)^4, E(5)+2*E(5)^2+2*E(5)^3+E(5)^4, -1,
2*E(5)^2+2*E(5)^3, E(5)+2*E(5)^2+2*E(5)^3+E(5)^4, 0 ], [ 1, E(5)^2+E(5)^3, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4,
E(5)+E(5)^4, -2*E(5)-3*E(5)^2-3*E(5)^3-2*E(5)^4, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, 0 ],
[ -6/5*E(5)-4/5*E(5)^2-4/5*E(5)^3-6/5*E(5)^4, -17/10*E(5)-13/10*E(5)^2-13/10*E(5)^3-17/10*E(5)^4, -1, 1/2,
9/5*E(5)+6/5*E(5)^2+6/5*E(5)^3+9/5*E(5)^4, -2*E(5)-2*E(5)^4, 1/2*E(5)+3/2*E(5)^2+3/2*E(5)^3+1/2*E(5)^4,
-1/2*E(5)+1/2*E(5)^2+1/2*E(5)^3-1/2*E(5)^4 ], [ 2/5*E(5)+3/5*E(5)^2+3/5*E(5)^3+2/5*E(5)^4, 2/5*E(5)+1/10*E(5)^2+1/10*E(5)^3+2/5*E(5)^4, 0,
1/2*E(5)^2+1/2*E(5)^3, 2/5*E(5)-2/5*E(5)^2-2/5*E(5)^3+2/5*E(5)^4, 0, E(5)+1/2*E(5)^2+1/2*E(5)^3+E(5)^4, -E(5)-1/2*E(5)^2-1/2*E(5)^3-E(5)^4 ]
, [ 1, -E(5)+2*E(5)^2+2*E(5)^3-E(5)^4, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, -2*E(5)-3*E(5)^2-3*E(5)^3-2*E(5)^4, E(5)+E(5)^4,
-2*E(5)-3*E(5)^2-3*E(5)^3-2*E(5)^4, -2*E(5)-3*E(5)^2-3*E(5)^3-2*E(5)^4, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ] ] ],
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ] ->
[ [ [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ -E(5)^2-E(5)^3, -E(5)^2-E(5)^3, 1, -1, 0, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, E(5)^2+E(5)^3, -E(5)^2-E(5)^3 ],
[ 0, 0, 0, -E(5)^2-E(5)^3, 0, E(5)^2+E(5)^3, 1, 0 ], [ -E(5)^2-E(5)^3, -E(5)^2-E(5)^3, 0, 0, 0, -E(5)^2-E(5)^3, 0, 1 ],
[ 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4, 0, -E(5)-E(5)^4, 1, -1, -E(5)-E(5)^4, -E(5)^2-E(5)^3, 1 ], [ 0, 0, 0, 1, 0, E(5)^2+E(5)^3, -E(5)^2-E(5)^3, 0 ],
[ -3*E(5)-E(5)^2-E(5)^3-3*E(5)^4, -E(5)-E(5)^4, 3*E(5)+E(5)^2+E(5)^3+3*E(5)^4, -E(5)^2-E(5)^3, -E(5)-E(5)^4, -2, 2, E(5)+E(5)^4 ],
[ E(5)^2+E(5)^3, E(5)^2+E(5)^3, -1, 0, 0, E(5)^2+E(5)^3, 0, -1 ] ],
[ [ 0, -1, 0, 0, 0, -1, 0, 0 ], [ E(5)^2+E(5)^3, E(5)^2+E(5)^3, -1, 0, 0, -1, 0, E(5)^2+E(5)^3 ],
[ -E(5)^2-E(5)^3, -E(5)^2-E(5)^3, 0, 0, 0, -E(5)^2-E(5)^3, 0, 1 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ],
[ 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4, E(5)+E(5)^4, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4, E(5)^2+E(5)^3, E(5)+E(5)^4, 1, -2, -E(5)-E(5)^4 ],
[ 1, -E(5)^2-E(5)^3, 0, 0, 0, -E(5)^2-E(5)^3, 0, -E(5)^2-E(5)^3 ], [ -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4, 1, 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4, -1,
-E(5)-E(5)^4, -E(5)^2-E(5)^3, -E(5)-E(5)^4, E(5)+E(5)^4 ], [ -E(5)-E(5)^4, 0, E(5)+E(5)^4, 0, 0, -1, 0, -1 ] ] ],
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ] ->
[ [ [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ -1, -1, 0, 0, E(5)+E(5)^4, E(5)^2+E(5)^3, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, E(5)^2+E(5)^3 ],
[ -E(5)^2-E(5)^3, 0, -E(5)^2-E(5)^3, 1, 0, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, E(5)+2*E(5)^2+2*E(5)^3+E(5)^4, -E(5)^2-E(5)^3 ],
[ 0, 0, -1, 0, 0, E(5)^2+E(5)^3, -E(5)^2-E(5)^3, -1 ], [ E(5)^2+E(5)^3, 1, E(5)+2*E(5)^2+2*E(5)^3+E(5)^4, E(5)^2+E(5)^3, 1,
E(5)+2*E(5)^2+2*E(5)^3+E(5)^4, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, 0 ], [ E(5)+2*E(5)^2+2*E(5)^3+E(5)^4, 0, E(5)+2*E(5)^2+2*E(5)^3+E(5)^4,
E(5)^2+E(5)^3, -E(5)^2-E(5)^3, E(5)+3*E(5)^2+3*E(5)^3+E(5)^4, -E(5)-3*E(5)^2-3*E(5)^3-E(5)^4, E(5)^2+E(5)^3 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ],
[ -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, 1, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, -E(5)^2-E(5)^3, 0, -E(5)-3*E(5)^2-3*E(5)^3-E(5)^4,
E(5)+4*E(5)^2+4*E(5)^3+E(5)^4, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4 ] ],
[ [ 0, 1, 0, 0, 1, -1, 0, 0 ], [ -E(5)^2-E(5)^3, -1, -2*E(5)-3*E(5)^2-3*E(5)^3-2*E(5)^4, -E(5)^2-E(5)^3, -1, -E(5)-3*E(5)^2-3*E(5)^3-E(5)^4,
E(5)+3*E(5)^2+3*E(5)^3+E(5)^4, 1 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1 ],
[ 0, 0, -1, 0, 0, E(5)^2+E(5)^3, -E(5)^2-E(5)^3, -1 ], [ -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, 0, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, -E(5)^2-E(5)^3,
E(5)^2+E(5)^3, -E(5)-3*E(5)^2-3*E(5)^3-E(5)^4, 2*E(5)+4*E(5)^2+4*E(5)^3+2*E(5)^4, -E(5)^2-E(5)^3 ],
[ -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, 1, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4, -E(5)^2-E(5)^3, 0, -E(5)-3*E(5)^2-3*E(5)^3-E(5)^4,
E(5)+4*E(5)^2+4*E(5)^3+E(5)^4, -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4 ], [ 0, -E(5)^2-E(5)^3, -1, 0, -E(5)^2-E(5)^3, 0, -1, -E(5)^2-E(5)^3 ] ] ],
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ] ->
[ [ [ -142/11, -139/11, 125/11, 79/11, -38/11, -224/11, -34/11, -342/11, -76/11 ], [ -97/11, -80/11, 74/11, 48/11, -10/11, -140/11, -35/11, -211/11,
-64/11 ], [ 74/11, 76/11, -45/11, -39/11, 26/11, 100/11, 14/11, 146/11, 41/11 ],
[ 3/11, 12/11, 1/11, -5/11, 7/11, -12/11, -3/11, -3/11, 3/11 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
[ 170/11, 152/11, -145/11, -100/11, 30/11, 255/11, 50/11, 391/11, 93/11 ], [ 0, 0, 0, 0, 0, 0, -1, 0, 0 ],
[ -23/11, -15/11, 29/11, 20/11, 5/11, -40/11, -10/11, -65/11, -12/11 ], [ 170/11, 163/11, -145/11, -100/11, 30/11, 255/11, 50/11, 402/11,
104/11 ] ], [ [ -145/11, -140/11, 102/11, 84/11, -34/11, -201/11, -31/11, -306/11, -79/11 ], [ 0, 0, 0, 0, 0, 0, -1, 0, 0 ],
[ 196/11, 168/11, -151/11, -103/11, 32/11, 272/11, 57/11, 420/11, 108/11 ], [ 23/11, 4/11, -18/11, -9/11, -5/11, 40/11, 10/11, 54/11, 12/11 ],
[ 30/11, 32/11, -23/11, -17/11, 15/11, 45/11, 14/11, 69/11, 19/11 ], [ -97/11, -80/11, 74/11, 48/11, -10/11, -140/11, -35/11, -211/11, -64/11 ],
[ -23/11, -15/11, 29/11, 20/11, 5/11, -40/11, -10/11, -65/11, -12/11 ], [ 235/11, 203/11, -182/11, -124/11, 35/11, 336/11, 73/11, 513/11,
136/11 ], [ -128/11, -116/11, 115/11, 63/11, -20/11, -192/11, -37/11, -301/11, -73/11 ] ] ],
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ] ->
[ [ [ 0, 1, 0, 0, 0, 0, 0, 0, 0 ], [ -1/4, -1/4, 1, 3/4, -1, 1, 0, -5/4, 3/4 ], [ 0, 0, 0, 1, 0, 0, 0, 0, 0 ], [ 6, 0, 1, 1, -4, 3, -3, -3, -3 ],
[ 15/4, 3/4, 0, -1/4, -2, 1, -1, -5/4, -9/4 ], [ -31/4, -3/4, 1, 1/4, 2, -1, 5, 5/4, 13/4 ], [ -19/4, 5/4, 1, -3/4, 2, -1, 2, 1/4, 9/4 ],
[ -6, -1, 3, 2, 1, 0, 3, -1, 2 ], [ 1/4, 1/4, -1, 1/4, 0, 0, 0, 5/4, 1/4 ] ],
[ [ 0, 0, 0, 1, 0, 0, 0, 0, 0 ], [ 16, 1, -1, 0, -7, 4, -8, -4, -8 ], [ -6, -1, 3, 2, 1, 0, 3, -1, 2 ],
[ 33/4, 1/4, -1, 1/4, -4, 3, -4, -7/4, -15/4 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
[ -6, -1, 4, 2, 0, 1, 3, -2, 2 ], [ 23/4, -1/4, -2, 3/4, -1, 0, -3, -1/4, -9/4 ] ] ], [ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ] ->
[ [ [ -1, 1, -1, 0, 0, 1, 0, 0, 0, 1 ], [ 1, 0, 1, -1, 0, 0, -1, 1, 0, 0 ], [ 2, 0, 1, 0, 0, -1, -1, 1, 0, -1 ],
[ 0, 1/2, 1, -1/2, 1, 1/2, 1/2, 1, -1/2, 1/2 ], [ -2, 0, 0, -1, 1, 2, 1, 0, -1, 1 ], [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, -1, 0, -1, -1, -1, -1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], [ 1, -3/2, 1, 1/2, 0, -3/2, -1/2, -1, 1/2, -1/2 ],
[ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ], [ [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 1, -1, -1, 0, 0, 1, 0 ],
[ 1, -3/2, 1, 1/2, 0, -3/2, -1/2, -1, 1/2, -1/2 ], [ -2, -1, -1, 1, 0, 0, 1, -2, -1, 0 ], [ -2, 1/2, -1, 1/2, 0, 3/2, 3/2, 0, -1/2, 1/2 ],
[ -1, -1, 0, 0, 0, -1, 0, -2, 0, 0 ], [ 2, 3/2, 1, -1/2, 0, 1/2, -1/2, 2, 1/2, 1/2 ], [ -1, 0, -1, 0, 0, 1, 0, 0, -1, 0 ],
[ 0, -1/2, 0, -1/2, 0, 1/2, 1/2, 0, 1/2, 1/2 ], [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] ] ], [ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ] ->
[ [ [ -1/2*E(8)+1/2*E(8)^3, 0, 1-1/2*E(8)+1/2*E(8)^3, 1, 1/2*E(8)-1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3, 1, -1-E(8)+E(8)^3,
1/2*E(8)-1/2*E(8)^3 ], [ -2, 1-E(8)+E(8)^3, -4+3*E(8)-3*E(8)^3, -1, 2-2*E(8)+2*E(8)^3, -2+2*E(8)-2*E(8)^3, -4+2*E(8)-2*E(8)^3,
-1+E(8)-E(8)^3, -1, 5-3*E(8)+3*E(8)^3 ], [ -2, 1, -2+2*E(8)-2*E(8)^3, -1, 1-E(8)+E(8)^3, -1+E(8)-E(8)^3, -2+E(8)-E(8)^3, E(8)-E(8)^3, -1,
3-E(8)+E(8)^3 ], [ 1-3/2*E(8)+3/2*E(8)^3, -1+E(8)-E(8)^3, 5-7/2*E(8)+7/2*E(8)^3, -E(8)+E(8)^3, -3+3/2*E(8)-3/2*E(8)^3,
3-3/2*E(8)+3/2*E(8)^3, 4-5/2*E(8)+5/2*E(8)^3, 1-E(8)+E(8)^3, 1, -5+7/2*E(8)-7/2*E(8)^3 ],
[ 1/2*E(8)-1/2*E(8)^3, 0, -1+1/2*E(8)-1/2*E(8)^3, 0, 1-1/2*E(8)+1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3, -1+1/2*E(8)-1/2*E(8)^3, 0, 0,
1-1/2*E(8)+1/2*E(8)^3 ], [ -1+1/2*E(8)-1/2*E(8)^3, 2-E(8)+E(8)^3, -5+7/2*E(8)-7/2*E(8)^3, 0, 4-5/2*E(8)+5/2*E(8)^3, -3+5/2*E(8)-5/2*E(8)^3,
-6+7/2*E(8)-7/2*E(8)^3, -2+E(8)-E(8)^3, -1, 6-9/2*E(8)+9/2*E(8)^3 ], [ 1-1/2*E(8)+1/2*E(8)^3, -1+E(8)-E(8)^3, 3-5/2*E(8)+5/2*E(8)^3, 0,
-2+3/2*E(8)-3/2*E(8)^3, 2-3/2*E(8)+3/2*E(8)^3, 3-5/2*E(8)+5/2*E(8)^3, 1-E(8)+E(8)^3, 0, -4+5/2*E(8)-5/2*E(8)^3 ],
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, -1, 0 ] ],
[ [ -1-1/2*E(8)+1/2*E(8)^3, 0, 1/2*E(8)-1/2*E(8)^3, -1, -1-1/2*E(8)+1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3, 0, 0,
1/2*E(8)-1/2*E(8)^3 ], [ -2, 1, -2+2*E(8)-2*E(8)^3, -1, 1-E(8)+E(8)^3, -1+E(8)-E(8)^3, -2+E(8)-E(8)^3, E(8)-E(8)^3, -1, 3-E(8)+E(8)^3 ],
[ 1/2*E(8)-1/2*E(8)^3, 0, -1+1/2*E(8)-1/2*E(8)^3, 0, 1-1/2*E(8)+1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3, -2+1/2*E(8)-1/2*E(8)^3, -1, 0,
1-3/2*E(8)+3/2*E(8)^3 ], [ 1-1/2*E(8)+1/2*E(8)^3, 0, 2-3/2*E(8)+3/2*E(8)^3, 1, 3/2*E(8)-3/2*E(8)^3, 1-1/2*E(8)+1/2*E(8)^3,
1-3/2*E(8)+3/2*E(8)^3, 1, -1-E(8)+E(8)^3, -2+3/2*E(8)-3/2*E(8)^3 ],
[ 1+1/2*E(8)-1/2*E(8)^3, 0, -1+1/2*E(8)-1/2*E(8)^3, 0, 1-1/2*E(8)+1/2*E(8)^3, -1+1/2*E(8)-1/2*E(8)^3, -1+3/2*E(8)-3/2*E(8)^3, -1,
1+E(8)-E(8)^3, 1-3/2*E(8)+3/2*E(8)^3 ], [ 1/2*E(8)-1/2*E(8)^3, 0, -1+1/2*E(8)-1/2*E(8)^3, E(8)-E(8)^3, 1/2*E(8)-1/2*E(8)^3,
1/2*E(8)-1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3, 0, -1, -1/2*E(8)+1/2*E(8)^3 ], [ 1, 0, 1-E(8)+E(8)^3, 1, 0, 0, 0, 0, 0, -1 ],
[ -1-1/2*E(8)+1/2*E(8)^3, 0, -1+1/2*E(8)-1/2*E(8)^3, -1, -1/2*E(8)+1/2*E(8)^3, 1/2*E(8)-1/2*E(8)^3, -1+1/2*E(8)-1/2*E(8)^3, 0, 0,
1-1/2*E(8)+1/2*E(8)^3 ], [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ] ] ],
[ (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) ] ->
[ [ [ 1, 1+E(8)-E(8)^3, -1+E(8)-E(8)^3, 1, 1, -E(8)+E(8)^3, 1-E(8)+E(8)^3, 0, -1, 0 ], [ -1+E(8)-E(8)^3, 2, 0, 0, -1+E(8)-E(8)^3, -1, -1, 0,
-1-E(8)+E(8)^3, -1+E(8)-E(8)^3 ], [ 2-E(8)+E(8)^3, -1+E(8)-E(8)^3, -2+E(8)-E(8)^3, 1-E(8)+E(8)^3, 2-E(8)+E(8)^3, 1-E(8)+E(8)^3,
1-E(8)+E(8)^3, 1, -1, 1-E(8)+E(8)^3 ], [ 2-E(8)+E(8)^3, -1+E(8)-E(8)^3, -1+E(8)-E(8)^3, 1-E(8)+E(8)^3, 1-E(8)+E(8)^3, 1-E(8)+E(8)^3,
2-E(8)+E(8)^3, 1-E(8)+E(8)^3, 0, 2-E(8)+E(8)^3 ], [ 0, 0, 0, 0, 0, 0, -1, 0, 0, 0 ],
[ 1, 1+E(8)-E(8)^3, 0, 0, 1, -1, -1, 0, -1-E(8)+E(8)^3, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, -1, 0, 1, 0, 0, E(8)-E(8)^3, -1, 0 ],
[ -1+E(8)-E(8)^3, 2, 2-E(8)+E(8)^3, -1+E(8)-E(8)^3, -1+E(8)-E(8)^3, -2, -2+E(8)-E(8)^3, 0, -E(8)+E(8)^3, -1+E(8)-E(8)^3 ],
[ -2+E(8)-E(8)^3, 1-E(8)+E(8)^3, 1-E(8)+E(8)^3, -1, -2+E(8)-E(8)^3, -1+E(8)-E(8)^3, -1+E(8)-E(8)^3, 0, 1, -1+E(8)-E(8)^3 ] ],
[ [ -1, -1-E(8)+E(8)^3, -1, 0, 1, E(8)-E(8)^3, -1+E(8)-E(8)^3, E(8)-E(8)^3, E(8)-E(8)^3, -1+E(8)-E(8)^3 ],
[ 2-E(8)+E(8)^3, -1+E(8)-E(8)^3, -2+E(8)-E(8)^3, 1-E(8)+E(8)^3, 2-E(8)+E(8)^3, 1-E(8)+E(8)^3, 1-E(8)+E(8)^3, 1, -1, 1-E(8)+E(8)^3 ],
[ 0, 0, 1, -1, -1, 0, 0, -1, 0, 0 ], [ -E(8)+E(8)^3, -2, 0, 0, 0, 1, 2, 0, 1+E(8)-E(8)^3, 1-E(8)+E(8)^3 ],
[ 1, 1+E(8)-E(8)^3, 0, 0, 0, -E(8)+E(8)^3, -E(8)+E(8)^3, 0, -1-E(8)+E(8)^3, 1 ], [ 0, 0, 0, -1, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, -1, 0, 0 ], [ -2+E(8)-E(8)^3, 1-2*E(8)+2*E(8)^3, 1-E(8)+E(8)^3, -1, -2+E(8)-E(8)^3, E(8)-E(8)^3, -1+E(8)-E(8)^3, 0, 1,
-2+E(8)-E(8)^3 ], [ 0, -1, -1, 0, 1, 0, 0, 1+E(8)-E(8)^3, 0, 0 ], [ 2-E(8)+E(8)^3, -1+E(8)-E(8)^3, -1+E(8)-E(8)^3, 1-E(8)+E(8)^3,
1-E(8)+E(8)^3, 1-E(8)+E(8)^3, 2-E(8)+E(8)^3, 1-E(8)+E(8)^3, 0, 2-E(8)+E(8)^3 ] ] ] ]