A340518 -id:A340518 - cp's OEIS frontend

A341293 Smallest order of a non-abelian group with a commutator subgroup of index n.

60, 6, 12, 8, 55, 18, 56, 16, 27, 30, 253, 24, 351, 42, 60, 32, 1020, 54, 1140, 40, 84, 66, 1081, 48, 125, 78, 81, 56, 1711, 90, 992, 64, 132, 102, 280, 72, 2220, 114, 156, 80, 2460, 126, 2580, 88, 135, 138, 2820, 96, 343, 150, 204, 104, 3180, 162, 605, 112, 228, 174, 3540, 120, 3660

Offset: 1

Bob Heffernan and Des MacHale, Feb 05 2021

nonn

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

			Examples for small n:
n a(n) group
1  60  A5
2  6  S3
3  12  A4
4  8  D8
5  55  C11 : C5
6  18  C3 x S3
7  56  (C2 x C2 x C2) : C7
8  16  (C4 x C2) : C2
9  27  (C3 x C3) : C3
10  30  C5 x S3
11  253  C23 : C11
12  24  C3 x D8

Bob Hefferman, Table of n, a(n) for n = 1..255

Cf. A001034, A340518.

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

Original entry on oeis.org

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

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

nonn

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.

Equals 2*A106833.

Cf. A059806, A060702, A340518.

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

cp's OEIS Frontend

A341293 Smallest order of a non-abelian group with a commutator subgroup of index n.

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