[GAP Forum] Ikosaeder group

[Thomas Breuer]

Dear GAP Forum,

Rudolf Zlabinger wrote

> I dealt with the finite rotation group for the ikosaeder.
> 
> My task was to find a permutation group acting on the 12 vertices of
> ikosaeder recognizing a special numbering of the vertices as such: look at
> the ikosaeder from a top vertex, numbered as 1, then on the 2 x 5 vertices
> forming 2 somehow  parallel cycles numbered by 2,3,4,5,6 and 7,8,9,10,11,
> followed by the bottom vertex numbered by 12. Thus the permutation group
> looked for should contain the rotation around the axis 1,12 executed by the
> permutation
> (2,3,4,5,6)(7,8,9,10,11).
> 
> Unfortunately the representative isomorphism Groups of symmetric group 12
> itself did not contain the desired rotation, so I had to look for a
> conjugate group of them containing it. I for the first glance found no
> direct method for doing that, so I used the following sequence:
> [...]

Perhaps the most explicit way to deal with the symmetries of a regular
icosahedron is to start from a three dimensional model for it.

Following the description on p. 2 of the Atlas Of Finite groups,
the vertices have the coordinates $(0, \pm b, \pm 1)^C$
where $b = ( 1 + \sqrt{5} ) / 2$ and the superscript $C$ denotes cyclic
permutation of the coordinates.
In GAP,
we can write down symmetries of the isosahedron by the following matrices.

    b:= -E(5)^2-E(5)^3;
    mat1:= [ [ 0, 1, 0 ],                 # permutation of coordinates
             [ 0, 0, 1 ],
             [ 1, 0, 0 ] ];;
    mat2:= 1/2 * [ [ b-1,   1,  -b ],     # order 5 rotation
                   [  -1,   b, b-1 ],
                   [   b, b-1,   1 ] ];;

The permutation action on the twelve vertices can be obtained as follows.

    p1:= [ 0, b, 1 ];                     # one vertex
    g:= Group( mat1, mat2 );
    vertices:= Set( Orbit( g, p1 ) );
    act:= Action( g, vertices );

And now one can play with the groups.

    IsPrimitive( g, vertices );           # no
    bl:= Blocks( g, vertices );;
    bl[1];                                # opposite vertices form a block
    act2:= Action( g, bl, OnSets );       # an action on six points

All the best,
Thomas

P.S.:
Here is a LaTeX picture of the icosahedron.

\documentclass{article}

\usepackage{epic}

\begin{document}

\setlength\unitlength{0.4mm}
\begin{picture}(200,220)(-50,-100)

\thicklines

% foreground
\begin{drawjoin}

% outer circle
\jput(57,-72){}       % P_1
\jput(91,13){}        % P_7
\jput(34.5,85){}      % P_9
\jput(-57,72){}       % P_3
\jput(-91,-13){}      % P_5
\jput(-34.5,-85){}    % P_{11}
\jput(57,-72){}       % P_1

% inner circles
\jput(57,7.5){}       % P_2
\jput(34.5,85){}      % P_9
\jput(-34.5,45){}     % P_{10}
\jput(-91,-13){}      % P_5
\jput(-21,-52){}      % P_6
\jput(57,-72){}       % P_1
\end{drawjoin}

\begin{drawjoin}
\jput(-21,-52){}      % P_6
\jput(57,7.5){}       % P_2
\jput(-34.5,45){}     % P_{10}
\jput(-21,-52){}      % P_6
\end{drawjoin}

% remaining lines
\drawline(-21,-52)(-34.5,-85)   % P_6 to P_{11}
\drawline(57,7.5)(91,13)        % P_2 to P_7
\drawline(-34.5,45)(-57,72)     % P_{10} to P_3

% background
\begin{dottedjoin}{3}

% inner circles
\jput( 34.5, -45){}    % P_{12}
\jput(   91,  13){}    % P_7
\jput(   21,  52){}    % P_8
\jput(  -57,  72){}    % P_3
\jput(  -57,-7.5){}    % P_4
\jput(-34.5, -85){}    % P_{11}
\jput( 34.5, -45){}    % P_{12}
\jput(   21,  52){}    % P_8
\jput(  -57,-7.5){}    % P_4
\jput( 34.5, -45){}    % P_{12}
\jput(   57, -72){}    % P_1
\end{dottedjoin}

% remaining lines
\dottedline{3}(-91,-13)(-57,-7.5)    % P_5 to P_4
\dottedline{3}(21,52)(34.5,85)       % P_8 to P_9

\end{picture}\ \begin{picture}(200,220)(-50,-100)

\thicklines

% axes
\drawline(-87,-50)(87,50)
\put(-92,-55){\makebox(0,0){$x$}}
\put(-87,-50){\vector(-2,-1){0}}

\drawline(-87,50)(87,-50)
\put(92,-55){\makebox(0,0){$y$}}
\put(87,-50){\vector(2,-1){0}}

\put(0,-100){\vector(0,1){200}}
\put(0,105){\makebox(0,0){$z$}}

% circles at points
\put(   57, -72){\circle{3}}
\put(   57, 7.5){\circle{3}}
\put(  -57,  72){\circle{3}}
\put(  -57,-7.5){\circle{3}}
\put(  -91, -13){\circle{3}}
\put(  -21, -52){\circle{3}}
\put(   91,  13){\circle{3}}
\put(   21,  52){\circle{3}}
\put( 34.5,  85){\circle{3}}
\put(-34.5,  45){\circle{3}}
\put(-34.5, -85){\circle{3}}
\put( 34.5, -45){\circle{3}}

% point names
\put(   64,  -79){\makebox(0,0){$1$}}
\put(   64, 12.5){\makebox(0,0){$2$}}
\put(  -64,   79){\makebox(0,0){$3$}}
\put(  -64,-12.5){\makebox(0,0){$4$}}
\put(  -98,  -14){\makebox(0,0){$5$}}
\put(  -20,  -60){\makebox(0,0){$6$}}
\put(   98,   14){\makebox(0,0){$7$}}
\put(   20,   60){\makebox(0,0){$8$}}
\put( 41.5,   91){\makebox(0,0){$9$}}
\put(-43.5,   47){\makebox(0,0){$10$}}
\put(-41.5,  -91){\makebox(0,0){$11$}}
\put( 43.5,  -47){\makebox(0,0){$12$}}

% join points in a plane
\begin{dottedjoin}{3}
\jput(-91,-13){}      % P_5
\jput(-21,-52){}      % P_6
\jput(91,13){}        % P_7
\jput(21,52){}        % P_8
\jput(-91,-13){}      % P_5
\end{dottedjoin}

\begin{dottedjoin}{3}
\jput(57,-72){}       % P_1
\jput(57,7.5){}       % P_2
\jput(-57,72){}       % P_3
\jput(-57,-7.5){}     % P_4
\jput(57,-72){}       % P_1
\end{dottedjoin}

\begin{dottedjoin}{3}
\jput(34.5,85){}      % P_9
\jput(-34.5,45){}     % P_{10}
\jput(-34.5,-85){}    % P_{11}
\jput(34.5,-45){}     % P_{12}
\jput(34.5,85){}      % P_9
\end{dottedjoin}

\end{picture}

\end{document}