C₉×D₂₀₉ #16836768

order	3762
isoclinism	isoclinic class of D₂₀₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₉×D₂₀₉
unambiguous semidirect product	C₁₁⋊(C₉×D₁₉)

Splittings

Fitting, Hall	C₉×C₁₁×C₁₉ ⋊ C₂
derived, Hall	C₁₁×C₁₉ ⋊ C₂×C₉
Hall	C₉×C₁₉ ⋊ D₁₁
Hall	C₉×C₁₁ ⋊ D₁₉
Sylow	C₁₉ ⋊ C₉×D₁₁
Sylow	C₁₁ ⋊ C₉×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₉	D₂₀₉
commutator, hypocenter	C₁₁×C₁₉	C₂×C₉
Frattini	C₃	C₃×D₂₀₉
Fitting	C₉×C₁₁×C₁₉	C₂
socle, cosocle	C₃×C₁₁×C₁₉	C₂×C₃
prime radical	C₃×D₂₀₉	C₃
LCS	G ▷ C₁₁×C₁₉
UCS	C₁ ◁ C₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 1881

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39);(11,20)(12,19)(13,18)(14,17)(15,16)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×C₉×D₂₀₉, C₉×D₁₀₄₅, C₉×C₁₉⋊AGL(1,𝔽₁₁), …
maximal normal subgroup of	C₅×C₉×D₂₀₉, C₉×C₁₉⋊AGL(1,𝔽₁₁), C₇×C₉×D₂₀₉
maximal quotient of	C₅×C₉×D₂₀₉, C₉×D₁₀₄₅, C₇×C₉×D₂₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3762, 8);

C9×D209 #16836768