#72243555

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#14552199 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃²	C₃²×AGL(1,ℤ/9)
commutator, cosocle	C₃²×C₉	C₂×C₃³
Frattini, socle	C₃³	C₃³×D₃
Fitting	#14552199	C₂
prime radical	C₃³×D₉	C₃²
hypercenter	C₃²⋊C₉	AGL(1,ℤ/9)
hypocenter	C₉	C₂×#6894
LCS	G ▷ C₃²×C₉ ▷ C₉
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 162

Polycyclic presentation:

2,3^7|b^3=e,d^3=f^2,g^3=h,b^a=bh^2,c^b=cef^2,d^a=dh,d^b=df,g^a=g^2h^2,g^b=gh^2,g^d=gh^2,h^a=h^2

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243574
maximal normal subgroup of	#72243574

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