#72231808

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#139750 ⋊ C₂
Sylow	C₃ ⋊ B(5,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#72231807
commutator, cosocle	C₂³×C₃×D₄	C₂⁴
Frattini	C₂³×D₄	C₂³×D₃
Fitting	C₃×#139750	C₂
socle	C₂×C₃	#245225
hypercenter	#139750	D₃
hypocenter	C₃	B(5,𝔽₂)
LCS	G ▷ C₂³×C₃×D₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #139750

9576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^10,3|b^2=g,b^a=bf,c^a=cf,c^b=cgj,d^a=de,e^b=ei,e^c=ei,f^b=fh,f^d=fi,g^a=gh,g^e=gj,h^d=hj,i^b=ij,k^c=k^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×B(5,𝔽₂)
maximal normal subgroup of	D₃×B(5,𝔽₂)
maximal quotient of	#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, 2, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, 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);