66.13 Generic Construction of Tables of Marks

TableOfMarksCyclic returns the table of marks of the cyclic group of order n.

A cyclic group of order n has as its subgroups for each divisor d of n a cyclic subgroup of order d.

TableOfMarksDihedral returns 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 the example below). Otherwise (i.e., if d does not divide n) there is just one class of dihedral subgroups of order d.

TableOfMarksFrobenius computes the table of marks of a Frobenius group of order p q, where p is a prime and q divides p-1.

gap> Display( TableOfMarksCyclic( 6 ) );
1:  6
2:  3 3
3:  2 . 2
4:  1 1 1 1

gap> Display( TableOfMarksDihedral( 12 ) );
 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

gap> Display( TableOfMarksFrobenius( 5, 4 ) );
1:  20
2:  10 2
3:   5 1 1
4:   4 . . 4
5:   2 2 . 2 2
6:   1 1 1 1 1 1

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