#72271019

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#24404 ⋊ C₂
Sylow	C₃ ⋊ #6533135

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×#120
commutator, cosocle	C₂⁴×C₃	C₂⁵
Frattini	C₂⁴	C₂⁴×D₃
Fitting	C₃×#24404	C₂
socle	C₂²×C₃	C₂²×#120
hypercenter	#24404	D₃
hypocenter	C₃	#6533135
LCS	G ▷ C₂⁴×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #24404

At least 7666 conjugacy classes of subgroups; only the 709 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^9,3|a^2=h,c^2=fhi,b^a=bi,c^a=cf,c^b=cfg,d^a=dghi,d^c=dfhi,e^a=efh,e^b=efg,e^c=efhi,f^a=fi,f^c=fi,f^d=fi,f^e=fi,g^a=gi,g^c=gi,g^d=gi,g^e=gi,h^c=hi,h^d=hi,h^e=hi,j^e=j^2

Conjugacy classes & character table

117 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271077, #72271939, D₃×#6533135
maximal normal subgroup of	#72271077, #72271939, D₃×#6533135

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