#72259782

order	11664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×#6900 ⋊ C₂²
Sylow	C₃×#6900 ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂×#72214299
commutator, cosocle	C₂×C₃×C₉²	C₂³×C₃
Frattini	C₂×C₃²	C₂×C₃×#1113
Fitting	C₂²×C₃×#6900	C₂²
socle	C₂×C₃³	C₂×C₃×D₃²
hypercenter	C₂²	#72214299
hypocenter	C₃×C₉²	C₂×C₃×D₄
LCS	G ▷ C₂×C₃×C₉² ▷ C₃×C₉²
UCS	C₁ ◁ C₂ ◁ C₂²

2530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Polycyclic presentation:

2^3,3,2,3^5|f^3=i,g^3=j,b^a=be,f^a=f^2i^2,f^c=f^2i^2,f^d=fi^2,g^a=f^2g,g^c=g^2j^2,g^d=gj^2,h^a=h^2,i^a=i^2,i^c=i^2,j^a=i^2j,j^c=j^2

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72213202
maximal normal subgroup of	D₃×#72213202

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