D₁₇₆₇₃ #47977796

order	35346
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₇₆₇₃
unambiguous semidirect product	C₃⋊D₅₈₉₁
unambiguous semidirect product	C₄₃⋊D₄₁₁
unambiguous semidirect product	C₁₃₇⋊D₁₂₉

Splittings

Fitting, socle, derived, Hall	C₃×C₄₃×C₁₃₇ ⋊ C₂
Hall	C₄₃×C₁₃₇ ⋊ D₃
Hall	C₃×C₁₃₇ ⋊ D₄₃
Sylow	C₁₃₇ ⋊ D₁₂₉
Hall	C₃×C₄₃ ⋊ D₁₃₇
Sylow	C₄₃ ⋊ D₄₁₁
Sylow	C₃ ⋊ D₅₈₉₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₄₃×C₁₃₇	C₂
LCS	G ▷ C₃×C₄₃×C₁₃₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 183
transitive degree	μ_t(G) = 17673

A minimal faithful permutation representation — 2 generators on 183 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)(47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183);(2,3)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(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)(48,183)(49,182)(50,181)(51,180)(52,179)(53,178)(54,177)(55,176)(56,175)(57,174)(58,173)(59,172)(60,171)(61,170)(62,169)(63,168)(64,167)(65,166)(66,165)(67,164)(68,163)(69,162)(70,161)(71,160)(72,159)(73,158)(74,157)(75,156)(76,155)(77,154)(78,153)(79,152)(80,151)(81,150)(82,149)(83,148)(84,147)(85,146)(86,145)(87,144)(88,143)(89,142)(90,141)(91,140)(92,139)(93,138)(94,137)(95,136)(96,135)(97,134)(98,133)(99,132)(100,131)(101,130)(102,129)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116)

Polycyclic presentation:

2,3,43,137|b^a=b^2,c^a=c^42,d^a=d^136

Conjugacy classes & character table

8838 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35346, 9);

D17673 #47977796