Mathematics Asked by YONG YANG on January 3, 2022

I am having trouble construct the following group in GAP. It is a solvable primitive linear group acting on V where |V|=5^8. We know the Fitting subgroup is of order 2^6*4 (central product of extra special group E of order 2^7 with a cyclic group of order 4). On top of E/Z(E) we have a group A of order 6^4 acts on E/Z(E). Here A itself has a normal extra special group D of order 27 and A/D acts on D/Z(D) and A/D cong GL(2,3). In some sense, G would be a maximal solvable primitive group on V=5^8.

If it is possible, I need similar construction in |V|=7^8.

I cannot easily tell you how I did this calculation, but in case it is helpful anyway, here is the group that you are looking for.

```
F1 := Identity(GF(5));;
G := Subgroup (GL(8,5), [
F1*[
[ 1, 1, 4, 4, 2, 3, 2, 3 ],
[ 3, 2, 2, 3, 4, 4, 4, 4 ],
[ 3, 3, 3, 3, 1, 4, 4, 1 ],
[ 1, 4, 1, 4, 3, 3, 2, 2 ],
[ 3, 2, 2, 3, 1, 1, 1, 1 ],
[ 1, 1, 4, 4, 3, 2, 3, 2 ],
[ 1, 4, 1, 4, 2, 2, 3, 3 ],
[ 3, 3, 3, 3, 4, 1, 1, 4 ]
],
F1*[
[ 2, 0, 4, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 2, 0, 4, 0 ],
[ 0, 0, 0, 0, 3, 0, 4, 0 ],
[ 2, 0, 1, 0, 0, 0, 0, 0 ],
[ 0, 3, 0, 1, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 2, 0, 4 ],
[ 0, 0, 0, 0, 0, 2, 0, 1 ],
[ 0, 2, 0, 1, 0, 0, 0, 0 ]
]
]);;
gap> Size(G);
331776
gap> StructureDescription(G);
"((((((C2 x ((C4 x C2) : C2)) : C2) : C2) : ((C3 x C3) : C3)) : Q8) : C3) : C4"
```

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