47.24 TomDihedral

TomDihedral( m )

TomDihedral constructs the table of marks of the dihedral group of order m. For each divisor d of m, a dihedral group of order m = 2n contains subgroups of order d according to the following rule. If d is odd and divides n then there is only one cyclic subgroup of order d. If d is even and divides n then there are a cyclic subgroup of order d and two classes of dihedral subgroups of order d which are cyclic, too, in the case d = 2, see example below). Otherwise, (i.e. if d does not divide n, there is just one class of dihedral subgroups of order d.

    gap> d12 := TomDihedral( 12 );
    rec(
      name := [ "1", "2", "D_{2}a", "D_{2}b", "3", "D_{4}", "6", 
          "D_{6}a", "D_{6}b", "D_{12}" ],
      subs := [ [ 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], 
          [ 1, 2, 3, 4, 6 ], [ 1, 2, 5, 7 ], [ 1, 3, 5, 8 ], 
          [ 1, 4, 5, 9 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] ],
      marks := [ [ 12 ], [ 6, 6 ], [ 6, 2 ], [ 6, 2 ], [ 4, 4 ], 
          [ 3, 3, 1, 1, 1 ], [ 2, 2, 2, 2 ], [ 2, 2, 2, 2 ], 
          [ 2, 2, 2, 2 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ] )
    gap> DisplayTom( d12 );
     1:  12
     2:   6 6
     3:   6 . 2
     4:   6 . . 2
     5:   4 . . . 4
     6:   3 3 1 1 . 1
     7:   2 2 . . 2 . 2
     8:   2 . 2 . 2 . . 2
     9:   2 . . 2 2 . . . 2
    10:   1 1 1 1 1 1 1 1 1 1 

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