#72220179

order	10368
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₃×3₋¹⁺²×#120 ⋊ C₂²
Sylow	C₃×3₋¹⁺² ⋊ C₂×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#72220089
commutator, cosocle	C₂³×C₃×C₉	C₂⁴×C₃
Frattini	C₂³×C₃	C₂²×C₃×D₃²
Fitting	C₃×3₋¹⁺²×#120	C₂²
socle	C₂×C₃²	C₃×D₃×#120
hypercenter	#120	D₃×AGL(1,ℤ/9)
hypocenter	C₃×C₉	C₂×C₃×B(4,𝔽₂)
LCS	G ▷ C₂³×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #120

8057 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^3,3^3|a^2=h,i^3=k,c^a=cg,c^b=cf,f^a=fh,g^b=gh,i^a=i^2k^2,i^d=i^2k^2,i^e=ik^2,j^a=jk^2,j^d=j^2,k^a=k^2,k^d=k^2

Conjugacy classes & character table

264 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220184
maximal quotient 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, 2, 3, 2, 2, 2, 3, 3, 3]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2*F.11^2);;
gap> SetConjugate(coll, 9, 4, F.9^2*F.11^2);;
gap> SetConjugate(coll, 9, 5, F.9*F.11^2);;
gap> SetConjugate(coll, 10, 1, F.10*F.11^2);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.11^2);;
gap> SetConjugate(coll, 11, 4, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);