D₁₀₄₃₁ #35209953

order	20862
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₄₃₁
unambiguous semidirect product	C₉⋊D₁₁₅₉
unambiguous semidirect product	C₁₉⋊D₅₄₉
unambiguous semidirect product	C₆₁⋊D₁₇₁

Splittings

Fitting, derived, Hall	C₉×C₁₉×C₆₁ ⋊ C₂
Hall	C₁₉×C₆₁ ⋊ D₉
Hall	C₉×C₆₁ ⋊ D₁₉
Hall	C₉×C₁₉ ⋊ D₆₁
Sylow	C₆₁ ⋊ D₁₇₁
Sylow	C₁₉ ⋊ D₅₄₉
Sylow	C₉ ⋊ D₁₁₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₉×C₆₁	C₂
Frattini	C₃	D₃₄₇₇
socle	C₃×C₁₉×C₆₁	D₃
LCS	G ▷ C₉×C₁₉×C₆₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 10431

A minimal faithful permutation representation — 2 generators on 89 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);(2,9)(3,8)(4,7)(5,6)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2,3,19,61,3|b^3=e^2,b^a=b^2e,c^a=c^18,d^a=d^60,e^a=e^2

Conjugacy classes & character table

5217 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁₂₉₃
maximal quotient of	D₃₁₂₉₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20862, 12);

D10431 #35209953