#72220142

order	15552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×#6900 ⋊ C₂²
Sylow	#6900 ⋊ B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#72220044
commutator, cosocle	C₂³×C₉²	C₂³×C₃
Frattini	C₂³×C₃²	C₂×C₃×D₃²
Fitting	C₂⁴×#6900	C₂²
socle	C₂×C₃²	C₃×#63983
hypercenter	C₂⁴	#92265⋊C₂
hypocenter	C₉²	C₃×B(4,𝔽₂)
LCS	G ▷ C₂³×C₉² ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁴

3283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^3,3^4|a^2=g,h^3=j,i^3=k,c^a=cf,c^b=ce,e^a=eg,f^b=fg,h^a=h^2j^2,h^b=h^2j^2,h^d=hj^2,i^a=h^2i,i^b=i^2k^2,i^d=ik^2,j^a=j^2,j^b=j^2,k^a=j^2k,k^b=k^2

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220184
maximal normal subgroup of	#72220184

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.10^2);;
gap> SetConjugate(coll, 8, 2, F.8^2*F.10^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.10^2);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^2*F.11^2);;
gap> SetConjugate(coll, 9, 4, F.9*F.11^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.10^2*F.11);;
gap> SetConjugate(coll, 11, 2, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);