#72247653

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×B(4,𝔽₂) ⋊ C₂
Sylow	C₃ ⋊ #394066

Subgroups

	subgroup	G/H
center	C₂³	D₃×#120
commutator, cosocle	C₂⁵×C₃	C₂⁴
Frattini	C₂⁵	C₂³×D₃
Fitting	C₂²×C₃×B(4,𝔽₂)	C₂
socle	C₂³×C₃	C₂×#120
hypercenter	C₂²×B(4,𝔽₂)	D₃
hypocenter	C₃	#394066
LCS	G ▷ C₂⁵×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂²×B(4,𝔽₂)

12412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^9,3|a^2=g,b^2=egi,c^2=efhi,b^a=be,c^a=cef,c^b=ch,d^a=degi,d^b=degi,d^c=defh,e^a=ei,e^b=ei,e^d=ei,g^b=gi,g^c=gi,g^d=gi,h^a=hi,h^b=hi,h^d=hi,j^c=j^2

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#394066, #72248603, #72248627
maximal normal subgroup of	D₃×#394066, #72248603, #72248627
maximal quotient of	#72248603, #72248808, #72269887

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