#72248602

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#60168 ⋊ C₂
Sylow	C₃ ⋊ #10576108

Subgroups

	subgroup	G/H
center	C₂²	C₂³×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁶
Frattini	C₂×C₄	C₂⁵×D₃
Fitting	C₃×#60168	C₂
socle	C₂²×C₃	C₂⁴×D₄
hypercenter	#60168	D₃
hypocenter	C₃	#10576108
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ #60168

22380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^9,3|a^2=h,c^2=g,f^2=i,h^2=i,b^a=bh,c^a=ch,d^a=dh,d^c=dg,e^a=ei,f^b=fi,f^d=fi,h^b=hi,h^c=hi,h^d=hi,j^d=j^2

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248632, D₃×#10576108
maximal normal subgroup of	#72248632, D₃×#10576108

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