[GAP Forum] Regarding split extension of the unit group of a group algebra

[Surinder Kaur]

Dear all

Suppose I have a group algebra FG, where F is the field of order p and G is
dihedral group of order 2p. If V(FG) denotes the set of all invertible
elements of FG with augmentation 1, then it is known that |V(FG)| =
|F|^(2(p-1)) (|F|-1).

I want to find a normal subgroup N of V(FG) such that V(FG) is the split
extension of G by N. In case of p=3, I followed the following method:

I found all normal subgroups of V(F3D6) of order |V|/|D_6| and then for
each N, I found its automorphism group (Aut(N)). Finally a non-trivial
homomorphisms in "AllHomomorphismClasses(G, Aut(N))" helped me getting
V(F3D6) as split extension of D6. The following is the code I used:

p:=3;;
g:=DihedralGroup(6);;
f:=GF(3);;
fg:=GroupRing(f,g);;
e:=Identity(fg);;
m:=MinimalGeneratingSet(g);;
l:=List(m,x-> x^Embedding(g,fg));;
g1:=List(g,x-> x^Embedding(g,fg));;
u:=Units(fg);;
s:=Filtered(u, x-> Augmentation(x) = (Z(p)^(0)) );;
v:=AsGroup(s);;

h:=Subgroups(v);;
h1:=Filtered(h, x-> Size(x) = 27);;

h2:=[];;
for i in h1 do
if IsNormal(u, i)
then
#Print(i, "\n");
Append(h2, [i]);
fi;
od;
#Print(h2, "\n");

for i in h2 do
#Print(l[2] in i, "\n");
od;

t:=h2[5];;
#gen:=GeneratorsOfGroup(t);;
#Print(gen, "\n");
au:=AutomorphismGroup(t);;
hom:=AllHomomorphismClasses(g, au);;
#Print(Size(au) , "\n");

for i in hom do
sdp:=SemidirectProduct(g,i,t);;
Print(IsomorphismGroups(v, sdp), "\n");
od;

But when I tried the same procedure for p=7 (in order to check whether my
conjecture is right or not in general), GAP is not responding at all.

Is there some alternate way to check my conjecture?
Many thanks in advance.