D₁₀₄₁₃ #35209698

order	20826
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₄₁₃
unambiguous semidirect product	C₉⋊D₁₁₅₇
unambiguous semidirect product	C₁₃⋊D₈₀₁
unambiguous semidirect product	C₈₉⋊D₁₁₇

Splittings

Fitting, derived, Hall	C₉×C₁₃×C₈₉ ⋊ C₂
Hall	C₁₃×C₈₉ ⋊ D₉
Hall	C₉×C₈₉ ⋊ D₁₃
Hall	C₉×C₁₃ ⋊ D₈₉
Sylow	C₈₉ ⋊ D₁₁₇
Sylow	C₁₃ ⋊ D₈₀₁
Sylow	C₉ ⋊ D₁₁₅₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₃×C₈₉	C₂
Frattini	C₃	D₃₄₇₁
socle	C₃×C₁₃×C₈₉	D₃
LCS	G ▷ C₉×C₁₃×C₈₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 10413

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

Polycyclic presentation:

2,3,13,89,3|b^3=e^2,b^a=b^2e,c^a=c^12,d^a=d^88,e^a=e^2

Conjugacy classes & character table

5208 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁₂₃₉
maximal quotient of	D₃₁₂₃₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20826, 9);

D10413 #35209698