#72255176

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#359509 ⋊ C₂
Sylow	C₃ ⋊ #72213274

Subgroups

	subgroup	G/H
center	C₂²	#10701818
commutator	C₂³×C₃×D₄	C₂²×C₄
Frattini	C₂⁴×D₄	C₂²×D₃
Fitting	C₃×#359509	C₂
socle	C₂²×C₃	#10995
prime radical	#72254969	C₂
cosocle	C₂⁴×C₃×D₄	C₂³
hypercenter	#359509	D₃
hypocenter	C₃	#72213274
LCS	G ▷ C₂³×C₃×D₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×D₄ ◁ #359509

9146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^10,3|a^2=ij,b^2=efj,c^2=dij,e^2=j,f^2=j,b^a=be,c^a=cd,c^b=cfg,d^b=dh,e^c=ehi,f^a=fj,f^c=fij,g^a=gi,g^b=ghj,g^c=ghi,g^e=gj,g^f=gj,i^b=ij,k^c=k^2

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255255, D₃×#72213274, #72271470
maximal normal subgroup of	#72255255, D₃×#72213274, #72271470

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.9*F.10);;
gap> SetPower(coll, 2, F.5*F.6*F.10);;
gap> SetPower(coll, 3, F.4*F.9*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*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);