#72248202

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#1795 ⋊ C₂
Sylow	C₃ ⋊ #394080

Subgroups

	subgroup	G/H
center	C₂²	#73852
commutator, cosocle	C₂³×C₃×C₄	C₂⁴
Frattini	C₂³×C₄	C₂³×D₃
Fitting	C₂×C₃×#1795	C₂
socle	C₂²×C₃	#2403
hypercenter	C₂×#1795	D₃
hypocenter	C₃	#394080
LCS	G ▷ C₂³×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#1795

8274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^9,3|a^2=i,b^2=g,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cefi,c^b=cefg,d^b=dh,d^c=dhi,e^a=ei,e^b=ei,e^c=ei,f^a=fi,f^c=fi,g^d=gi,h^b=hi,j^c=j^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#394080, #72254214, #72269995
maximal normal subgroup of	D₃×#394080, #72254214, #72269995

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