#72255204

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×#407 ⋊ C₂
Sylow	C₃ ⋊ #364416

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#937
commutator	C₂⁴×C₃	C₂³×C₄
Frattini	C₂³×C₄	C₂³×D₃
Fitting	C₂²×C₃×#407	C₂
socle	C₂²×C₃	C₂×#386
prime radical	#11001039	C₂
cosocle	C₂³×C₃×C₄	C₂⁴
hypercenter	C₂²×#407	D₃
hypocenter	C₃	#364416
LCS	G ▷ C₂⁴×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂²×#407

6096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^9,3|a^2=g,g^2=i,h^2=i,b^a=bghi,c^a=ci,c^b=cf,d^a=dghi,d^c=de,e^a=ei,f^a=fi,h^c=hi,j^c=j^2

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255214, #72255229, D₃×#364416, …
maximal normal subgroup of	#72255214, #72255229, D₃×#364416, …

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