D₁₆₅₈₅ #45684265

order	33170
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₅₈₅
unambiguous semidirect product	C₅⋊D₃₃₁₇
unambiguous semidirect product	C₃₁⋊D₅₃₅
unambiguous semidirect product	C₁₀₇⋊D₁₅₅

Splittings

Fitting, socle, derived, Hall	C₅×C₃₁×C₁₀₇ ⋊ C₂
Hall	C₃₁×C₁₀₇ ⋊ D₅
Hall	C₅×C₁₀₇ ⋊ D₃₁
Hall	C₅×C₃₁ ⋊ D₁₀₇
Sylow	C₁₀₇ ⋊ D₁₅₅
Sylow	C₃₁ ⋊ D₅₃₅
Sylow	C₅ ⋊ D₃₃₁₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₃₁×C₁₀₇	C₂
LCS	G ▷ C₅×C₃₁×C₁₀₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 143
transitive degree	μ_t(G) = 16585

A minimal faithful permutation representation — 2 generators on 143 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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(2,5)(3,4)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(38,143)(39,142)(40,141)(41,140)(42,139)(43,138)(44,137)(45,136)(46,135)(47,134)(48,133)(49,132)(50,131)(51,130)(52,129)(53,128)(54,127)(55,126)(56,125)(57,124)(58,123)(59,122)(60,121)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)

Polycyclic presentation:

2,5,31,107|b^a=b^4,c^a=c^30,d^a=d^106

Conjugacy classes & character table

8294 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33170, 9);

D16585 #45684265