#72247167

order	2904
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₁₁² ⋊ D₄
Sylow	C₁₁² ⋊ #67
Sylow	C₃ ⋊ #11757246

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₁×D₃₃
commutator, Fitting, socle, cosocle	C₂×C₃×C₁₁²	C₂²
hypocenter	C₃×C₁₁²	D₄
LCS	G ▷ C₂×C₃×C₁₁² ▷ C₃×C₁₁²
UCS	C₁ ◁ C₂

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 132

Polycyclic presentation:

2^3,3,11^2|b^a=bc,d^b=d^2,e^a=e^10,e^b=e^10,f^a=e^2f,f^b=f^10

Conjugacy classes & character table

249 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72247171, #72247204, D₃×#11757246, …
maximal normal subgroup of	#72247171, #72247204, D₃×#11757246, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 11, 11]);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 5, 2, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);