#72270070

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#111458 ⋊ C₂
Sylow	C₃ ⋊ #72219165

Subgroups

	subgroup	G/H
center	C₂³	#76753
commutator, cosocle	C₂⁶×C₃	C₂⁴
Frattini	C₂⁶	C₂³×D₃
Fitting	C₃×#111458	C₂
socle	C₂³×C₃	#2403
hypercenter	#111458	D₃
hypocenter	C₃	#72219165
LCS	G ▷ C₂⁶×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ #111458

44266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 384

Polycyclic presentation:

2^10,3|a^2=j,b^2=e,d^2=i,b^a=bg,c^a=ch,c^b=cf,d^a=dg,d^b=deij,e^a=ej,f^a=fj,g^b=gj,g^c=gi,h^b=hij,h^d=hi,k^d=k^2

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72219165
maximal normal subgroup of	D₃×#72219165
maximal quotient of	#72270432, #72271676

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