A104114 The order of the smallest solvable group with derived length n.

1, 2, 6, 24, 48, 432, 1296, 82944, 165888, 1088391168, 3265173504

Offset: 0

Author

David L. Harden, Mar 31 2005

Keywords

• nice
• nonn
• hard
• more

Comments

At order 24, the two groups which achieve the maximum derived length are S_4 and SL(2,3). At order 48, a group which achieves the maximum derived length is GL(2,3). At order 432, the only group which achieves the maximum derived length is 3^2:GL(2,3).

There are three groups of order 1296 achieving maximal derived length. - Charles R Greathouse IV, May 26 2014

Examples

			a_2=6 because the smallest solvable group with derived length 2 is the symmetric group S_3, which has order 6.
		

Programs

GAP

a := function(arg)
    local n, size, gr;
    n := arg[1];
    if Length(arg) > 1 then
        size := arg[2];
    else
        size := 1;
    fi;
    gr := OneSmallGroup(Size, size, DerivedLength, n);
    if gr <> fail then
        return size;
    else
        return a(n, size + 1);
    fi;
end;  # Charles R Greathouse IV, May 23 2014

Formula

a(n) >= 2*a(n-1). - Charles R Greathouse IV, May 26 2014

Extensions

a(6) from Charles R Greathouse IV, May 26 2014

a(7)-a(10) from Des MacHale, Jun 26 2025