D₁₀₂₀₉ #35205281

order	20418
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₂₀₉
unambiguous semidirect product	C₃⋊D₃₄₀₃
unambiguous semidirect product	C₄₁⋊D₂₄₉
unambiguous semidirect product	C₈₃⋊D₁₂₃

Splittings

Fitting, socle, derived, Hall	C₃×C₄₁×C₈₃ ⋊ C₂
Hall	C₄₁×C₈₃ ⋊ D₃
Hall	C₃×C₈₃ ⋊ D₄₁
Hall	C₃×C₄₁ ⋊ D₈₃
Sylow	C₈₃ ⋊ D₁₂₃
Sylow	C₄₁ ⋊ D₂₄₉
Sylow	C₃ ⋊ D₃₄₀₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₄₁×C₈₃	C₂
LCS	G ▷ C₃×C₄₁×C₈₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 10209

A minimal faithful permutation representation — 2 generators on 127 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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(2,3)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,121)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)

Polycyclic presentation:

2,3,41,83|b^a=b^2,c^a=c^40,d^a=d^82

Conjugacy classes & character table

5106 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₀₆₂₇
maximal quotient of	D₃₀₆₂₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20418, 9);

D10209 #35205281