#72270089

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#115133 ⋊ C₂
Sylow	C₃ ⋊ #72219165

Subgroups

	subgroup	G/H
center	C₂³	#74884
commutator, cosocle	C₂⁶×C₃	C₂⁴
Frattini	C₂⁶	C₂³×D₃
Fitting	C₃×#115133	C₂
socle	C₂³×C₃	#2403
hypercenter	#115133	D₃
hypocenter	C₃	#72219165
LCS	G ▷ C₂⁶×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ #115133

At least 6266 conjugacy classes of subgroups; only the 447 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10,3|a^2=j,b^2=f,c^2=i,d^2=i,b^a=bh,c^a=cg,c^b=ce,d^a=dghi,d^b=defj,f^a=fj,g^b=gi,g^d=gi,h^b=hj,h^c=hi,h^d=hi,k^d=k^2

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72219165
maximal normal subgroup of	D₃×#72219165
maximal quotient of	#72270642, #72271584

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