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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 $2^2S_5$).

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    $\begingroup$ So $A_5\times\mathbb F_2^3$ is an example, with order $480$. $\endgroup$
    – Kenta S
    Sep 18, 2022 at 14:03
  • $\begingroup$ 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. $\endgroup$
    – Kenta S
    Sep 18, 2022 at 14:07
  • $\begingroup$ @KentaS see my answer. $\endgroup$
    – Parcly Taxel
    Sep 18, 2022 at 14:28

2 Answers 2

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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.

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  • $\begingroup$ 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. $\endgroup$
    – Ian Gershon Teixeira
    Sep 18, 2022 at 19:34
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    $\begingroup$ 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 $\endgroup$
    – ahulpke
    Sep 18, 2022 at 23:54
  • $\begingroup$ 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? $\endgroup$
    – Ian Gershon Teixeira
    Oct 8, 2022 at 21:16
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    $\begingroup$ @IanGershonTeixeira Will be in the next release of GAP. Currently: github.com/hulpke/gap/commit/… $\endgroup$
    – ahulpke
    Oct 8, 2022 at 22:31
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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 $2^3\rm A_5$.

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