#72220350

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Sylow

C₃⁴ ⋊ C₂²×#41

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃⁴
commutator, socle, cosocle	C₂×C₃⁴	C₂⁵
Fitting	C₂²×C₃⁴	C₂⁴
hypercenter	C₂²	D₃⁴
hypocenter	C₃⁴	C₂²×#41
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

10073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^6,3^4|e^2=f,d^a=df,e^a=ef,g^a=g^2,g^c=g^2,g^e=g^2,h^a=g^2h,h^b=hj,h^c=h^2,h^e=g^2hi,i^b=i^2,i^c=i^2,i^e=i^2,j^b=j^2,j^c=j^2,j^e=ij

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 8, 5, F.7^2*F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> SetConjugate(coll, 10, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);