#72254307

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#3537 ⋊ C₂
Sylow	C₃ ⋊ #7837029

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁵×D₃
commutator, socle, cosocle	C₂³×C₃	C₂⁶
Fitting	C₂×C₃×#3537	C₂
hypercenter	C₂×#3537	D₃
hypocenter	C₃	#7837029
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#3537

20116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^9,3|a^2=h,b^2=g,e^2=i,f^2=i,b^a=bh,c^b=cg,d^a=dh,d^b=dh,e^b=ehi,e^c=ehi,e^d=ei,f^a=fhi,f^c=fh,f^d=fi,f^e=fi,j^b=j^2

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254320, D₃×#7837029, #72269887
maximal normal subgroup of	#72254320, D₃×#7837029, #72269887
maximal quotient of	#72270150

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