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#72218491

order6720
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC5×C7×SD16 ⋊ C22×C3
Hall#72218490 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C2×AGL(1,ℤ/8)
HallC5×C7 ⋊ C2×C3×AGL(1,ℤ/8)
HallΓL(1,𝔽8) ⋊ D5×AGL(1,ℤ/8)
SylowC7 ⋊ C3×D5×AGL(1,ℤ/8)
SylowC5 ⋊ AGL(1,ℤ/8)×AGL(1,𝔽7)

Subgroups

subgroupG/H
centerC2D4×D5×AGL(1,𝔽7)
commutator, cosocleC4×C5×C7C24×C3
FrattiniC4C22×D5×AGL(1,𝔽7)
FittingC5×C7×SD16C22×C3
socleC2×C5×C7C22×C3×D4
hypercenterSD16D5×AGL(1,𝔽7)
hypocenterC5×C7C2×C3×AGL(1,ℤ/8)
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C4 ◁ SD16

1192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^2,5,7|a^2=f,f^2=g,b^a=bf,c^a=cg,f^b=fg,h^b=h^4,h^c=h^4,h^d=h^4,i^d=i^6,i^e=i^2

Conjugacy classes & character table

157 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×AGL(1,ℤ/8)×AGL(1,𝔽7)
maximal normal subgroup ofD5×AGL(1,ℤ/8)×AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 5, 7]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^4);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> SetConjugate(coll, 9, 4, F.9^6);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);