C₃²×D₂₀₉ #16836788

order	3762
isoclinism	isoclinic class of D₂₀₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃²×D₂₀₉
unambiguous semidirect product	C₁₁⋊(C₃²×D₁₉)

Splittings

Fitting, socle, Hall	C₃²×C₁₁×C₁₉ ⋊ C₂
derived, Hall	C₁₁×C₁₉ ⋊ C₂×C₃²
Hall	C₃²×C₁₉ ⋊ D₁₁
Hall	C₃²×C₁₁ ⋊ D₁₉
Sylow	C₁₉ ⋊ C₃²×D₁₁
Sylow	C₁₁ ⋊ C₃²×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	D₂₀₉
commutator, cosocle, hypocenter	C₁₁×C₁₉	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₁₁×C₁₉	C₂
LCS	G ▷ C₁₁×C₁₉
UCS	C₁ ◁ C₃²

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 1881

A minimal faithful permutation representation — 2 generators on 36 points:

(1,2,3)(7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36);(4,5,6)(8,17)(9,16)(10,15)(11,14)(12,13)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)

Conjugacy classes & character table

954 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃²×C₁₉⋊AGL(1,𝔽₁₁), C₃²×D₁₀₄₅, C₃²×C₅×D₂₀₉, …
maximal normal subgroup of	C₃²×C₁₉⋊AGL(1,𝔽₁₁), C₃²×C₅×D₂₀₉, C₃²×C₇×D₂₀₉
maximal quotient of	C₃²×D₁₀₄₅, C₃²×C₅×D₂₀₉, C₃×C₁₁⋊(ΓL(1,𝔽₈)⋊D₁₉), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3762, 28);

C32×D209 #16836788