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ubuntu2204
Kernel: SageMath 9.6
# SmallGroup(32, 6) is realizable but not fully realizable.
#
# In this group, the action of C_4 on C_2^3 is faithful.

%run -i FPENAG.py

G = gap.SmallGroup(32,6)
[tg, ag, bg, cg, jjj] = G.Generators()
gens = [tg, ag, bg, cg]
kG = gap.GroupRing(GF(2), G)
f = gap.Embedding(G, kG)
id = gap.Image(f, cg^0)
[t, a, b, c] = [ gap.Image(f, x) for x in gens ]
Igens = [id + a + t + a*t,  id + c + t + c*t]
I = gap.Ideal(kG, Igens)
kGmodI = kG/I

analyzekGmodI(G, f, id, I, kGmodI, Igens, skipFRCheck=True)
G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes G. I didn't check whether it fully realizes G. 
# Here we see that 1 + a + t and 1 + c + t are all units in the group algebra.

(id + a + t).IsUnit(), (id + c + t).IsUnit()
(true, true)
# Consider all possible ideals containing 1 + a + t + u and
# 1 + c + t + v where u, v are elements of G.

for ug in G.Elements():
    for vg in G.Elements():
        Xgens = [id + a + t + ug, id + c + t + vg]
        X = gap.Ideal(kG, Xgens)
        if GEmbeds(G, f, id, X):
            print(ug, vg)
            analyzekGmodI(G, f, id, X, kG/X, Xgens, skipFRCheck=False)
            print()
f1*f2 f1*f4 G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes but does not fully realize G. f1*f2 f1*f4*f5 G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes but does not fully realize G. f1*f2*f3 f1*f4 G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes but does not fully realize G. f1*f2*f3 f1*f4*f5 G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes but does not fully realize G. f1*f2*f4 f1*f4 G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes but does not fully realize G. f1*f2*f5 f1*f4 G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes but does not fully realize G. f1*f2*f5 f1*f4*f5 G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes but does not fully realize G. f1*f2*f3*f4 f1*f4 G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes but does not fully realize G. f1*f2*f3*f5 f1*f4 G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes but does not fully realize G. f1*f2*f3*f5 f1*f4*f5 G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes but does not fully realize G. f1*f2*f4*f5 f1*f4 G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes but does not fully realize G. f1*f2*f3*f4*f5 f1*f4 G = (C2 x C2 x C2) : C4 of order 32. G embeds in kG/I. kG/I has 64 elements and 32 units. kG/I realizes but does not fully realize G.