#72219772

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#493 ⋊ C₂²
Sylow	C₃² ⋊ C₂×#1619

Subgroups

	subgroup	G/H
center	C₂²	#10595278
commutator, cosocle	C₃²×C₄²	C₂⁴
Frattini	C₄²	C₂²×D₃²
Fitting	C₃²×#493	C₂²
socle	C₂²×C₃²	C₂×#120
hypercenter	#493	D₃²
hypocenter	C₃²	C₂×#1619
LCS	G ▷ C₃²×C₄² ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

4056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|b^2=e,e^2=g,f^2=h,b^a=bf,c^b=ce,e^a=eh,e^c=eg,f^a=fh,f^c=fh,i^a=i^2,i^d=i^2,j^a=i^2j,j^d=j^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219801
maximal normal subgroup of	#72219801
maximal quotient of	#72219773, #72219791

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