D₁₅₈₆₁ #44540564

order	31722
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₈₆₁
unambiguous semidirect product	C₃⋊D₅₂₈₇
unambiguous semidirect product	C₁₇⋊D₉₃₃
unambiguous semidirect product	C₃₁₁⋊D₅₁

Splittings

Fitting, socle, derived, Hall	C₃×C₁₇×C₃₁₁ ⋊ C₂
Hall	C₁₇×C₃₁₁ ⋊ D₃
Hall	C₃×C₃₁₁ ⋊ D₁₇
Sylow	C₃₁₁ ⋊ D₅₁
Hall	C₃×C₁₇ ⋊ D₃₁₁
Sylow	C₁₇ ⋊ D₉₃₃
Sylow	C₃ ⋊ D₅₂₈₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₇×C₃₁₁	C₂
LCS	G ▷ C₃×C₁₇×C₃₁₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 331
transitive degree	μ_t(G) = 15861

A minimal faithful permutation representation — 2 generators on 331 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331);(2,3)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(22,331)(23,330)(24,329)(25,328)(26,327)(27,326)(28,325)(29,324)(30,323)(31,322)(32,321)(33,320)(34,319)(35,318)(36,317)(37,316)(38,315)(39,314)(40,313)(41,312)(42,311)(43,310)(44,309)(45,308)(46,307)(47,306)(48,305)(49,304)(50,303)(51,302)(52,301)(53,300)(54,299)(55,298)(56,297)(57,296)(58,295)(59,294)(60,293)(61,292)(62,291)(63,290)(64,289)(65,288)(66,287)(67,286)(68,285)(69,284)(70,283)(71,282)(72,281)(73,280)(74,279)(75,278)(76,277)(77,276)(78,275)(79,274)(80,273)(81,272)(82,271)(83,270)(84,269)(85,268)(86,267)(87,266)(88,265)(89,264)(90,263)(91,262)(92,261)(93,260)(94,259)(95,258)(96,257)(97,256)(98,255)(99,254)(100,253)(101,252)(102,251)(103,250)(104,249)(105,248)(106,247)(107,246)(108,245)(109,244)(110,243)(111,242)(112,241)(113,240)(114,239)(115,238)(116,237)(117,236)(118,235)(119,234)(120,233)(121,232)(122,231)(123,230)(124,229)(125,228)(126,227)(127,226)(128,225)(129,224)(130,223)(131,222)(132,221)(133,220)(134,219)(135,218)(136,217)(137,216)(138,215)(139,214)(140,213)(141,212)(142,211)(143,210)(144,209)(145,208)(146,207)(147,206)(148,205)(149,204)(150,203)(151,202)(152,201)(153,200)(154,199)(155,198)(156,197)(157,196)(158,195)(159,194)(160,193)(161,192)(162,191)(163,190)(164,189)(165,188)(166,187)(167,186)(168,185)(169,184)(170,183)(171,182)(172,181)(173,180)(174,179)(175,178)(176,177)

Polycyclic presentation:

2,3,17,311|b^a=b^2,c^a=c^16,d^a=d^310

Conjugacy classes & character table

7932 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₈₆₁
maximal normal subgroup of	C₂×D₁₅₈₆₁
maximal quotient of	Dic₁₅₈₆₁, C₂×D₁₅₈₆₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31722, 8);

D15861 #44540564