54.14 MinimalNonmonomialGroup

MinimalNonmonomialGroup( p, factsize )

returns a minimal nonmonomial group described by the parameters factsize and p if such a group exists, and false otherwise.

Suppose that a required group K exists. factsize is the size of the Fitting factor K / F(K); this value must be 4, 8, an odd prime, twice an odd prime, or four times an odd prime.

In the case that factsize is twice an odd prime the centre Z(K) iscyclic of order 2^{p+1}. In all other cases p denotes the (unique) prime that divides the order of F(K).

The solvable minimal nonmonomial groups were classified by van der Waall (see~vdW76, the construction follows this article).

    gap> MinimalNonmonomialGroup(  2,  3 ); # $SL_2(3)$
    2^(1+2):3
    gap> MinimalNonmonomialGroup(  3,  4 );
    3^(1+2):4
    gap> MinimalNonmonomialGroup(  5,  8 );
    5^(1+2):Q8
    gap> MinimalNonmonomialGroup( 13, 12 );
    13^(1+2):2.D6
    gap> MinimalNonmonomialGroup(  1, 14 );
    2^(1+6):D14
    gap> MinimalNonmonomialGroup(  2, 14 );
    (2^(1+6)Y4):D14 

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