#72270302

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#359501 ⋊ C₂
Sylow	C₃ ⋊ #72219162

Subgroups

	subgroup	G/H
center	C₂³	#71671
commutator	C₂³×C₃×D₄	C₂²×C₄
Frattini	C₂⁴×D₄	C₂²×D₃
Fitting	C₃×#359501	C₂
socle	C₂³×C₃	#2021
prime radical	C₂×#10742700	C₂
cosocle	C₂⁴×C₃×D₄	C₂³
hypercenter	#359501	D₃
hypocenter	C₃	#72219162
LCS	G ▷ C₂³×C₃×D₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁴×D₄ ◁ #359501

30790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^10,3|b^2=df,b^a=bd,c^a=ce,c^b=ceg,d^c=dh,e^b=ehi,f^c=fh,g^a=gij,g^b=ghi,g^d=gj,g^f=gj,h^b=hj,i^b=ij,k^c=k^2

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270313, #72271676
maximal normal subgroup of	#72270313, #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, 2, F.4*F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 11, 3, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);