#72244875

order	9747
automorphism	?
homology	Abelianization (H₁) = C₃×C₉ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow

C₁₉² ⋊ C₃×C₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₁₉²	C₃×C₉
prime radical	C₁₉⋊C₃²	C₃
cosocle	#11762246	C₃²
LCS	G ▷ C₁₉²
UCS	C₁

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 171

Polycyclic presentation:

3^3,19^2|a^3=c,d^a=d^9,d^b=d^11,d^c=d^7,e^a=d^4e^16,e^b=e^11,e^c=d^5e^11

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#4296², #72244900
maximal normal subgroup of	#4296², #72244900

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [3, 3, 3, 19, 19]);;
gap> SetPower(coll, 1, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^9);;
gap> SetConjugate(coll, 4, 2, F.4^11);;
gap> SetConjugate(coll, 4, 3, F.4^7);;
gap> SetConjugate(coll, 5, 1, F.4^4*F.5^16);;
gap> SetConjugate(coll, 5, 2, F.5^11);;
gap> SetConjugate(coll, 5, 3, F.4^5*F.5^11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);