#72268693

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Sylow

#6900 ⋊ #106

Subgroups

	subgroup	G/H
center	C₂²	C₄⋊#92314
commutator	C₄×C₉²	C₂×C₃×C₄
Frattini	C₂×C₃²×C₄	C₂×C₃×C₃⋊D₃
Fitting	C₂×C₈×#6900	C₂
socle	C₂²×C₃²	C₃×C₄⋊(C₃⋊D₃)
prime radical	C₂²×C₃³	C₄⋊(C₃⋊D₃)
cosocle	C₂×C₄×C₉²	C₂²×C₃
hypercenter	C₂×C₈	#92314
hypocenter	C₉²	C₃×#106
LCS	G ▷ C₄×C₉² ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 2592

Polycyclic presentation:

2^2,3,2^3,3^4|a^2=d,b^2=e,e^2=f,g^3=i,h^3=j,b^a=be,e^a=ef,g^a=g^2i^2,g^c=gi,h^a=h^2j^2,h^c=hj,i^a=i^2,j^a=j^2

Conjugacy classes & character table

426 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#106×#92314
maximal normal subgroup of	#106×#92314

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