#72243697

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#14552224 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃²	C₃²×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₃⁴	C₂×C₃³
Frattini, socle	C₃³	C₃³×D₃
Fitting	#14552224	C₂
prime radical	C₃²×(C₃⋊D₃)⋊C₃	C₃²
hypercenter	C₉⋊C₉	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₂×#6894
LCS	G ▷ C₃⁴ ▷ C₃²
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 486

Polycyclic presentation:

2,3^7|c^3=f,d^3=e^2f^2,c^b=ce,d^b=de,d^c=de^2f^2,g^a=g^2,g^b=gh,h^a=h^2

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243705
maximal normal subgroup of	#72243705

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