#72254152

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#62811 ⋊ C₂
Sylow	C₃ ⋊ #6437475

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#953
commutator, cosocle	C₂²×C₃×C₄	C₂⁵
Frattini	C₂²×C₄	C₂⁴×D₃
Fitting	C₃×#62811	C₂
socle	C₂²×C₃	C₂²×#120
hypercenter	#62811	D₃
hypocenter	C₃	#6437475
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ #62811

13420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^9,3|b^2=g,d^2=fghi,f^2=i,b^a=bg,c^a=ci,c^b=cg,d^a=di,d^b=df,d^c=dfgh,e^b=ei,f^b=fi,f^d=fi,h^b=hi,h^c=hi,h^d=hi,j^d=j^2

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254232, D₃×#6437475, #72255569
maximal normal subgroup of	#72254232, D₃×#6437475, #72255569

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