#72219962

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄×#6900 ⋊ C₂
Sylow	#6900 ⋊ C₂²⋊C₄

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×C₃×AGL(1,ℤ/9)
commutator	C₂×C₃×C₉	C₂×C₃²×C₄
Frattini, socle	C₂²×C₃²	C₂×C₃²×D₃
Fitting	C₂×C₄×#6900	C₂
prime radical	C₂²×C₃×AGL(1,ℤ/9)	C₂×C₃
cosocle	C₂²×C₃×C₉	C₂²×C₃²
hypercenter	C₂×C₄×C₉	AGL(1,ℤ/9)
hypocenter	C₉	C₂²⋊C₄×3₋¹⁺²
LCS	G ▷ C₂×C₃×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₄×C₉

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3^2,2^2,3^3|b^2=e,d^3=g^2,h^3=i,b^a=bf,d^a=di^2,d^c=dg,h^a=h^2i^2,h^c=hi^2,i^a=i^2

Conjugacy classes & character table

270 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220003, #72220021, #72220027
maximal normal subgroup of	#72220003, #72220021, #72220027
maximal quotient of	#72219963, #72220074

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