Smallest group with a derived series of length 2, 3 and 4

Question

What are the smallest group with a derived series of length 2, 3 and 4?. I know that for n=2 the answer is S3 because that's the smallest metabelian non-abelian group. Could you help me out?

Thanks

Answer 1

For reference, the smallest groups of derived lengths 2, 3, 4, 5, and 6 respectively have orders 6, 24, 48, 432, and 1296 respectively.

The smallest group with derived length 2 is symmetric group on 3 points, $S_3$.

The smallest groups with derived length 3 are the symmetric group on 4 points, $S_4$, and the group of $2\times2$ matrices of determinant 1 with entries from a field with 3 elements, $\operatorname{SL}_2(\mathbb{Z}/3\mathbb{Z})$.

The smallest groups with derived length 4 are the group of $2\times2$ matrices of nonzero determinant with entries from a field with 3 elements, $\operatorname{GL}_2(\mathbb{Z}/3\mathbb{Z})$, and the so-called fake GL(2,3) from this answer.

The smallest group with derived length 5 is the group of affine transformations $$ \operatorname{AGL}_2(\mathbb{Z}/3\mathbb{Z}) = \left\{ \left[ \begin{smallmatrix} a & b & e \\ c & d & f \\ 0 & 0 & 1 \end{smallmatrix}\right] : a,b,c,d,e,f \in \mathbb{Z}/3\mathbb{Z}, ad-bc \neq 0 \right\} $$

The smallest groups with derived length 6 are three non-split extensions of $ \operatorname{AGL}_2(\mathbb{Z}/3\mathbb{Z})$ by a non-central, cyclic, normal subgroup of order 3. One of these is $\Gamma U_3(\mathbb{Z}/2\mathbb{Z})$, the split extension of $\operatorname{GL}_2(\mathbb{Z}/3\mathbb{Z})$ acting on an extraspecial group of order 27 and exponent 3; the other two are nonsplit.

An exploration of a similar problem (where instead of group order, the number of prime factors is consider, so 2, 4, 5, 7, and 8 respectively) is in Glasby (1986, 2005).

• Glasby, S. P. “The composition and derived lengths of a soluble group.” J. Algebra 120 (1989), no. 2, 406–413. MR989905 DOI:10.1016/0021-8693(89)90204-4
• Glasby, S. P. “Solvable groups with a given solvable length, and minimal composition length.” J. Group Theory 8 (2005), no. 3, 339–350. MR2137974 DOI:10.1515/jgth.2005.8.3.339