#72220002

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄²×3₋¹⁺² ⋊ C₂²
Sylow	C₃×3₋¹⁺² ⋊ C₂×#104

Subgroups

	subgroup	G/H
center	C₄	#88646⋊C₃
commutator	C₃×C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂×C₃×D₃²
Fitting	C₃×C₄²×3₋¹⁺²	C₂²
socle	C₂×C₃²	C₃×D₃×C₂²⋊C₄
prime radical	#72219929	C₂
cosocle	C₂×C₃×C₄×C₉	C₂³×C₃
hypercenter	C₄²	D₃×AGL(1,ℤ/9)
hypocenter	C₃×C₉	C₂×C₃×#104
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

1376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^3,3^3|a^2=g,b^2=e,f^2=g,h^3=j,b^a=bf,e^a=eg,f^a=fg,h^a=h^2j^2,h^c=h^2j^2,h^d=hj^2,i^a=ij^2,i^c=i^2,j^a=j^2,j^c=j^2

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220005, #72220090
maximal normal subgroup of	#72220090
maximal quotient of	#72220005

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