D₉×C₁₀₉ #14520051

order	1962
isoclinism	isoclinic class of D₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₀₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₉×C₁₀₉
unambiguous semidirect product	C₉⋊(C₂×C₁₀₉)

Splittings

Fitting, Hall	C₉×C₁₀₉ ⋊ C₂
derived, Sylow	C₉ ⋊ C₂×C₁₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₀₉	D₉
commutator, cosocle, hypocenter	C₉	C₂×C₁₀₉
Frattini	C₃	D₃×C₁₀₉
Fitting	C₉×C₁₀₉	C₂
socle	C₃×C₁₀₉	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₁₀₉

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 118
transitive degree	μ_t(G) = 981

A minimal faithful permutation representation — 2 generators on 118 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118);(2,9)(3,8)(4,7)(5,6)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₉×D₁₀₉, C₂×D₉×C₁₀₉, C₇×D₉×C₁₀₉, …
maximal normal subgroup of	D₉×D₁₀₉, C₂×D₉×C₁₀₉, C₇×D₉×C₁₀₉, …
maximal quotient of	Dic₉×C₁₀₉, C₂×D₉×C₁₀₉, C₇×D₉×C₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1962, 3);

D9×C109 #14520051