C₁₀₉⋊C₃ #66489

order	327
isoclinism	isoclinic class of C₁₀₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₁₀₉⋊C₃

Splittings

Fitting, socle, derived, Sylow

C₁₀₉ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₀₉	C₃
LCS	G ▷ C₁₀₉
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 109

A minimal faithful permutation representation — 2 generators on 109 points:

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

Polycyclic presentation:

3,109|b^a=b^63

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₀₉⋊C₃, C₂×C₁₀₉⋊C₃, #11758281, …
maximal normal subgroup of	D₁₀₉⋊C₃, C₂×C₁₀₉⋊C₃, #11758281, …
maximal quotient of	C₂×C₁₀₉⋊C₃, #11758279, C₃×C₁₀₉⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(327, 1);

C109⋊C3 #66489