Example GrpBB_standard-gens (H70E1)

The following function takes a black box group isomorphic to M₂₄ and finds standard generators. It is taken from the ATLAS of Finite Group Representations page on M₂₄.

> m24_standard := function(B)
> repeat a := PseudoRandom(B); until Order(a) eq 10;
> a := a ^ 5;
> repeat b := PseudoRandom(B); until Order(b) eq 15;
> b := b ^ 5;
> repeat b := b ^ PseudoRandom(B); ab := a*b;
> until Order(ab) eq 23;
> x := ab*(ab^2*b)^2*ab*b;
> if Order(x) eq 5 then b := b^-1; end if;
> return a,b;
> end function;

We take a group which must be M₂₄ and find these generators.

> G := PermutationGroup<24 |
> [ 20, 4, 10, 3, 15, 9, 7, 1, 11, 22, 21, 19, 8, 2, 24, 5,
> 12, 18, 13, 16, 14, 23, 6, 17 ],
> [ 12, 18, 3, 2, 7, 11, 5, 21, 19, 22, 23, 1, 14, 17, 10,
> 8, 4, 13, 24, 20, 9, 15, 6, 16 ]>;
> #G;
244823040
> Transitivity(G);
5
> B := NaturalBlackBoxGroup(G);
> a,b := m24_standard(B); a,b;
GrpBBElt (1, 16)(2, 22)(3, 14)(4, 15)(5, 11)(6, 24)(7,
10)(8, 18)(9, 19)(12, 17)(13, 20)(21, 23)
GrpBBElt (1, 14, 17)(2, 18, 13)(5, 16, 20)(7, 22, 9)(8, 24,
15)(19, 23, 21)

The printing of the GrpBBElts shows the underlying concrete group elements. These may be extracted using the UnderlyingElement intrinsic for use within G.