Let $G$ be the group $[729,57]$, using GAP's notation. I have so far two descriptions of the group:


* a presentation
* an embedding (not surjective!) of the group into a Sylow $3$-subgroup of the unit group of a finite ring


Is $G$ isomorphic to a Sylow $3$-subgroup of some well-known group?



Is $[729,57]$ a Sylow $3$-subgroup of some well-known group?
Yes (to my surprise) it appears to be isomorphic to the Sylow $3$-subgroup of $3.J\_3$, the $3$-fold cover of the Janko sporadic simple group $J\_3$. Here is some Magma code:



```
> C:=MatrixGroup("3J3",1);
> P:=Sylow(C,3);   
> IdentifyGroup(P);  
<729, 57>

```

And here is GAP code for the same computation (thanks to Stefan Kohl and Frieder Ladisch)



```
LoadPackage("AtlasRep");;
G:=AtlasGroup("3.J3");;
v:= Z(4)^0*[1,0,1,0,0,0,0,0,0];;
Gperm:= Action( G, Orbit( G, v, OnRight), OnRight );;
P := SylowSubgroup(Gperm,3);;
IdGroup(P);

```

[ 729, 57 ]


