D₁₈₅₃₁ #49150429

order	37062
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₈₅₃₁
unambiguous semidirect product	C₉⋊D₂₀₅₉
unambiguous semidirect product	C₂₉⋊D₆₃₉
unambiguous semidirect product	C₇₁⋊D₂₆₁

Splittings

Fitting, derived, Hall	C₉×C₂₉×C₇₁ ⋊ C₂
Hall	C₂₉×C₇₁ ⋊ D₉
Hall	C₉×C₇₁ ⋊ D₂₉
Hall	C₉×C₂₉ ⋊ D₇₁
Sylow	C₇₁ ⋊ D₂₆₁
Sylow	C₂₉ ⋊ D₆₃₉
Sylow	C₉ ⋊ D₂₀₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₂₉×C₇₁	C₂
Frattini	C₃	D₆₁₇₇
socle	C₃×C₂₉×C₇₁	D₃
LCS	G ▷ C₉×C₂₉×C₇₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 18531

A minimal faithful permutation representation — 2 generators on 109 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);(2,9)(3,8)(4,7)(5,6)(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)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)

Polycyclic presentation:

2,3,29,71,3|b^3=e^2,b^a=b^2e,c^a=c^28,d^a=d^70,e^a=e^2

Conjugacy classes & character table

9267 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37062, 8);

D18531 #49150429