D₁₇₀₁₉ #46801377

order	34038
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₇₀₁₉
unambiguous semidirect product	C₉⋊D₁₈₉₁
unambiguous semidirect product	C₃₁⋊D₅₄₉
unambiguous semidirect product	C₆₁⋊D₂₇₉

Splittings

Fitting, derived, Hall	C₉×C₃₁×C₆₁ ⋊ C₂
Hall	C₃₁×C₆₁ ⋊ D₉
Hall	C₉×C₆₁ ⋊ D₃₁
Hall	C₉×C₃₁ ⋊ D₆₁
Sylow	C₆₁ ⋊ D₂₇₉
Sylow	C₃₁ ⋊ D₅₄₉
Sylow	C₉ ⋊ D₁₈₉₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₃₁×C₆₁	C₂
Frattini	C₃	D₅₆₇₃
socle	C₃×C₃₁×C₆₁	D₃
LCS	G ▷ C₉×C₃₁×C₆₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 17019

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

Polycyclic presentation:

2,3,31,61,3|b^3=e^2,b^a=b^2e,c^a=c^30,d^a=d^60,e^a=e^2

Conjugacy classes & character table

8511 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34038, 12);

D17019 #46801377