#72246730

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#14679 ⋊ C₂
Sylow	C₃ ⋊ #394058

Subgroups

	subgroup	G/H
center	C₂³	C₂×#937
commutator, cosocle	C₂⁵×C₃	C₂⁴
Frattini	C₂⁵	C₂³×D₃
Fitting	C₃×#14679	C₂
socle	C₂³×C₃	C₂×#120
hypercenter	#14679	D₃
hypocenter	C₃	#394058
LCS	G ▷ C₂⁵×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ #14679

25258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^9,3|b^2=ef,c^2=efgi,b^a=be,c^a=cefg,c^b=ch,d^b=defh,d^c=defg,e^c=ei,f^c=fi,g^b=gi,h^c=hi,j^c=j^2

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#394058, #72253433, #72253440, …
maximal normal subgroup of	D₃×#394058, #72253433, #72253440, …
maximal quotient of	#72253440, #72271203

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