A340519 Smallest order of a non-abelian group with a center of order n.

6, 8, 18, 16, 30, 24, 42, 32, 54, 40, 66, 48, 78, 56, 90, 64, 102, 72, 114, 80, 126, 88, 138, 96, 150, 104, 162, 112, 174, 120, 186, 128, 198, 136, 210, 144, 222, 152, 234, 160, 246, 168, 258, 176, 270, 184, 282, 192, 294, 200, 306, 208, 318, 216, 330, 224, 342, 232, 354, 240, 366, 248

Offset: 1

Author

Bob Heffernan and Des MacHale, Jan 24 2021; corrected Feb 14 2021

Keywords

• nonn

Comments

a(n) is 6n if n is odd and 4n if n is even. This is because the groups involved are C(n) X S3 if n is odd, where S3 is the symmetric group of order 6, and C(n/2) X D8 if n is even, where D8 is the dihedral group of order 8 and C(m) is the cyclic group of order m.

By Lagrange's Theorem a(n) is a multiple of n.

Crossrefs

Equals 2*A106833.

Cf. A059806, A060702, A340518.

Programs

Mathematica

Table[If[OddQ[n],6n,4n],{n,100}] (* Harvey P. Dale, Mar 03 2023 *)