# Non solvable group requiring more than 2 generators

The smallest group requiring more than 2 generators is $$C_2 \times C_2 \times C_2$$ but that group is abelian. The smallest non abelian groups requiring more than 2 generators are groups with quotient $$C_2^3$$, such as $$D_8 \times C_2$$ and $$Q_8 \times C_2$$. But these extensions of $$C_2^3$$ are still solvable.

What is the smallest finite group $$G$$ which is not solvable and requires more than two generators?

This question is the same as

The smallest group with 3 generators

but with "non abelian" replaced by "non solvable"

Kenta S points out that $$C_2^3 \times A_5$$ is order 480 and non solvable and requires at least 3 generators (since $$C_2^3$$ is a quotient). I have a feeling that we can do better and $$C_2^2 \times A_5$$ is also minimal 3 generated (EDIT: my feeling about $$2^2A_5$$ was wrong see the answer from ahulpke or for an explicit 2 generation of $$2^2A_5$$ see answer from Parcly Taxel)(Also note that ahulpke found another non-solvable group of size 480 which is minimal 3 generated, namely $$2^2S_5$$).

• So $A_5\times\mathbb F_2^3$ is an example, with order $480$. Sep 18, 2022 at 14:03
• If you can prove $C_2\times A_5$ is indeed not $2$-generated, you would be done. Indeed, the smallest non-abelian simple groups have order $60$ and $168$, so any minimal $3$-generated group must have $A_5$ as a component. But $A_5$ itself is $2$-generated. Sep 18, 2022 at 14:07
• @KentaS see my answer. Sep 18, 2022 at 14:28

An explicit search in GAP finds $$C_2\times C_2\times C_2\times A_5$$ and $$C_2\times C_2\times S_5$$ to be the two nonsolvable groups of smallest order that are not 2-generated:

gap> l:=AllSmallGroups(Size,[60..480],IsSolvableGroup,false);;
gap> f:=FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> sel:=Filtered([1..Length(l)],x->Length(GQuotients(f,l[x]:findall:=false))=0);
[ 50, 51 ]
gap> List(l{sel},StructureDescription);
[ "C2 x C2 x S5", "C2 x C2 x C2 x A5" ]


(both are not 2-generated because of the quotient $$C_2^3$$). A similar search finds the smallest perfect group (no abelian quotient) to be of order 15360, of the form $$C_2^4\rtimes(C_2^4\rtimes A_5)$$ (Note that this structure does not determine the group uniquely, and only one of such groups is not 2-generated.)

Peripherally related, it is well-known that $$A_5^{19}$$ is 2-generated, while $$A_5^{20}$$ is not.

• Is there an easy explanation or common reference for those facts about $A_5^{19}$ and $A_5^{20}$? I think that's very interesting. Sep 18, 2022 at 19:34
• The explanation for 19 is a count of 2-element generating sets (up to automorphisms). A reference is: P.Hall, THE EULERIAN FUNCTIONS OF A GROUP, Quart. J. Math, 1936, 134-151 Sep 18, 2022 at 23:54
• I was just rereading your answer and I was wondering if you would mind posting the code you used to find the perfect group of order 15360 that cannot be 2 generated? Oct 8, 2022 at 21:16
• @IanGershonTeixeira Will be in the next release of GAP. Currently: github.com/hulpke/gap/commit/… Oct 8, 2022 at 22:31

In fact $$2^2\rm A_5$$ can also be $$2$$-generated by pairing the extra transpositions with the generators of $$\rm A_5$$:

G := Group([(1,2,3)(6,7),(1,2,3,4,5)(8,9)]);
Print(StructureDescription(G));
C2 x C2 x A5


So the answer is $$2^3\rm A_5$$.