gap_directly.sagews

%gap
(9 + 5)*(3+3);

84

%gap
F:= GF( 16 );
LeftActingDomain( F );

GF(2^4) GF(2)

%gap
G:= AsVectorSpace( GF( 4 ), F );
F = G;
LeftActingDomain( G );

AsField( GF(2^2), GF(2^4) ) true GF(2^2)

%gap
s8 := Group( (1,2), (1,2,3,4,5,6,7,8) );
a8 := DerivedSubgroup( s8 );

Group([ (1,2), (1,2,3,4,5,6,7,8) ]) Group([ (1,2,3), (2,3,4), (2,4)(3,5), (2,6,4), (3,7,5), (4,8,6) ])

%gap
Size(a8);
IsAbelian(a8);
IsPerfect(a8);

20160 false true

%gap
syl2 := SylowSubgroup( a8, 2 );
Size( syl2 );

Group([ (1,3)(7,8), (2,4)(7,8), (5,6)(7,8), (5,7)(6,8), (1,2)(3,4)(5,7) (6,8), (1,5)(2,7)(3,6)(4,8) ]) 64

%gap
Normalizer( a8, syl2 ) = syl2;

true

%gap
cent := Centralizer( a8, Centre( syl2 ) );
Size( cent );

Group([ (1,3)(2,4)(5,6)(7,8), (5,6)(7,8), (5,7)(6,8), (2,4)(7,8), (2,5) (4,6), (1,2)(3,4) ]) 192