#72248338

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#3484 ⋊ C₂
Sylow	C₃ ⋊ #7792761

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁵×D₃
commutator, socle, cosocle	C₂³×C₃	C₂⁶
Fitting	C₂×C₃×#3484	C₂
hypercenter	C₂×#3484	D₃
hypocenter	C₃	#7792761
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#3484

At least 6074 conjugacy classes of subgroups; only the 3913 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^9,3|a^2=h,f^2=i,c^b=cg,d^a=di,e^a=ehi,f^b=fi,f^c=fhi,f^e=fi,j^c=j^2

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248347, #72248495, D₃×#7792761
maximal normal subgroup of	#72248347, #72248495, D₃×#7792761
maximal quotient of	#72271201

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