C₅×D₂₀₉ #14523453

order	2090
isoclinism	isoclinic class of D₂₀₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₂₀₉
unambiguous semidirect product	C₁₉⋊(C₅×D₁₁)

Splittings

Fitting, socle, Hall	C₅×C₁₁×C₁₉ ⋊ C₂
derived, Hall	C₁₁×C₁₉ ⋊ C₂×C₅
Hall	C₅×C₁₉ ⋊ D₁₁
Hall	C₅×C₁₁ ⋊ D₁₉
Sylow	C₁₉ ⋊ C₅×D₁₁
Sylow	C₁₁ ⋊ C₅×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₂₀₉
commutator, cosocle, hypocenter	C₁₁×C₁₉	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₁₁×C₁₉	C₂
LCS	G ▷ C₁₁×C₁₉
UCS	C₁ ◁ C₅

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 1045

A minimal faithful permutation representation — 2 generators on 35 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);(7,16)(8,15)(9,14)(10,13)(11,12)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×D₁₁×D₁₉, D₅×D₂₀₉, D₅₅⋊D₁₉, …
maximal normal subgroup of	C₅×D₁₁×D₁₉, D₅×D₂₀₉, D₅₅⋊D₁₉, …
maximal quotient of	C₅×Dic₂₀₉, C₂×C₅×D₂₀₉, C₅×C₇×D₂₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2090, 5);

C5×D209 #14523453