D₁₈₄₇₁ #49149158

order	36942
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₈₄₇₁
unambiguous semidirect product	C₃⋊D₆₁₅₇
unambiguous semidirect product	C₄₇⋊D₃₉₃
unambiguous semidirect product	C₁₃₁⋊D₁₄₁

Splittings

Fitting, socle, derived, Hall	C₃×C₄₇×C₁₃₁ ⋊ C₂
Hall	C₄₇×C₁₃₁ ⋊ D₃
Hall	C₃×C₁₃₁ ⋊ D₄₇
Hall	C₃×C₄₇ ⋊ D₁₃₁
Sylow	C₁₃₁ ⋊ D₁₄₁
Sylow	C₄₇ ⋊ D₃₉₃
Sylow	C₃ ⋊ D₆₁₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₄₇×C₁₃₁	C₂
LCS	G ▷ C₃×C₄₇×C₁₃₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 181
transitive degree	μ_t(G) = 18471

A minimal faithful permutation representation — 2 generators on 181 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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181);(2,3)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(52,181)(53,180)(54,179)(55,178)(56,177)(57,176)(58,175)(59,174)(60,173)(61,172)(62,171)(63,170)(64,169)(65,168)(66,167)(67,166)(68,165)(69,164)(70,163)(71,162)(72,161)(73,160)(74,159)(75,158)(76,157)(77,156)(78,155)(79,154)(80,153)(81,152)(82,151)(83,150)(84,149)(85,148)(86,147)(87,146)(88,145)(89,144)(90,143)(91,142)(92,141)(93,140)(94,139)(95,138)(96,137)(97,136)(98,135)(99,134)(100,133)(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(111,122)(112,121)(113,120)(114,119)(115,118)(116,117)

Polycyclic presentation:

2,3,47,131|b^a=b^2,c^a=c^46,d^a=d^130

Conjugacy classes & character table

9237 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36942, 8);

D18471 #49149158