#72276965

order	4104
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×C₁₉ ⋊ C₂×C₉
Hall	C₃×#4296 ⋊ D₄
Sylow	C₁₉ ⋊ C₉×D₁₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊AGL(1,𝔽₁₉)
commutator, socle	C₂×C₃×C₁₉	C₂²×C₉
Fitting	C₃×C₄×C₁₉	C₂×C₉
prime radical	C₄⋊(C₃⋊(D₁₉⋊C₃))	C₃
cosocle	C₂×C₃×C₁₉⋊C₃	C₂²×C₃
hypercenter	C₄	C₃⋊AGL(1,𝔽₁₉)
hypocenter	C₃×C₁₉	D₄×C₉
LCS	G ▷ C₂×C₃×C₁₉ ▷ C₃×C₁₉
UCS	C₁ ◁ C₂ ◁ C₄

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 228

Polycyclic presentation:

2^2,3,2,3^2,19|a^2=d,c^3=e,b^a=bd,f^b=f^2,g^b=g^18,g^c=g^6,g^e=g^7

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×C₃⋊AGL(1,𝔽₁₉), D₃×C₄⋊AGL(1,𝔽₁₉), D₁₂×AGL(1,𝔽₁₉)
maximal normal subgroup of	D₄×C₃⋊AGL(1,𝔽₁₉), D₃×C₄⋊AGL(1,𝔽₁₉), D₁₂×AGL(1,𝔽₁₉)

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");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 3, 3, 19]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7^18);;
gap> SetConjugate(coll, 7, 3, F.7^6);;
gap> SetConjugate(coll, 7, 5, F.7^7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);