C₄₁×C₁₀₉⋊C₃ #27211571

order	13407
isoclinism	isoclinic class of C₁₀₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₄₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄₁×C₁₀₉⋊C₃
unambiguous semidirect product	C₁₀₉⋊(C₃×C₄₁)
unambiguous semidirect product	(C₄₁×C₁₀₉)⋊C₃

Splittings

Fitting, socle, Hall	C₄₁×C₁₀₉ ⋊ C₃
derived, Sylow	C₁₀₉ ⋊ C₃×C₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₄₁	C₁₀₉⋊C₃
commutator, cosocle, hypocenter	C₁₀₉	C₃×C₄₁
Frattini	C₁	G
Fitting, socle	C₄₁×C₁₀₉	C₃
LCS	G ▷ C₁₀₉
UCS	C₁ ◁ C₄₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 150
transitive degree	μ_t(G) = 4469

A minimal faithful permutation representation — 2 generators on 150 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)(110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150);(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)

Conjugacy classes & character table

1599 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₄₁×C₁₀₉⋊C₃, C₄₁×D₁₀₉⋊C₃, D₄₁×C₁₀₉⋊C₃, …
maximal normal subgroup of	C₂×C₄₁×C₁₀₉⋊C₃, C₄₁×D₁₀₉⋊C₃, D₄₁×C₁₀₉⋊C₃, …
maximal quotient of	C₂×C₄₁×C₁₀₉⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13407, 1);

C41×C109⋊C3 #27211571