Dic₁₄₇₁₉ #72291669

order	58876
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₇₁₉

Splittings

derived, Hall	C₄₁×C₃₅₉ ⋊ C₄
Sylow	C₃₅₉ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₃₅₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₇₁₉
commutator, hypocenter	C₄₁×C₃₅₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₁×C₃₅₉	C₂
LCS	G ▷ C₄₁×C₃₅₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 404
transitive degree	μ_t(G) = 58876

A minimal faithful permutation representation — 2 generators on 404 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)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404);(1,3,2,4)(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)(47,404)(48,403)(49,402)(50,401)(51,400)(52,399)(53,398)(54,397)(55,396)(56,395)(57,394)(58,393)(59,392)(60,391)(61,390)(62,389)(63,388)(64,387)(65,386)(66,385)(67,384)(68,383)(69,382)(70,381)(71,380)(72,379)(73,378)(74,377)(75,376)(76,375)(77,374)(78,373)(79,372)(80,371)(81,370)(82,369)(83,368)(84,367)(85,366)(86,365)(87,364)(88,363)(89,362)(90,361)(91,360)(92,359)(93,358)(94,357)(95,356)(96,355)(97,354)(98,353)(99,352)(100,351)(101,350)(102,349)(103,348)(104,347)(105,346)(106,345)(107,344)(108,343)(109,342)(110,341)(111,340)(112,339)(113,338)(114,337)(115,336)(116,335)(117,334)(118,333)(119,332)(120,331)(121,330)(122,329)(123,328)(124,327)(125,326)(126,325)(127,324)(128,323)(129,322)(130,321)(131,320)(132,319)(133,318)(134,317)(135,316)(136,315)(137,314)(138,313)(139,312)(140,311)(141,310)(142,309)(143,308)(144,307)(145,306)(146,305)(147,304)(148,303)(149,302)(150,301)(151,300)(152,299)(153,298)(154,297)(155,296)(156,295)(157,294)(158,293)(159,292)(160,291)(161,290)(162,289)(163,288)(164,287)(165,286)(166,285)(167,284)(168,283)(169,282)(170,281)(171,280)(172,279)(173,278)(174,277)(175,276)(176,275)(177,274)(178,273)(179,272)(180,271)(181,270)(182,269)(183,268)(184,267)(185,266)(186,265)(187,264)(188,263)(189,262)(190,261)(191,260)(192,259)(193,258)(194,257)(195,256)(196,255)(197,254)(198,253)(199,252)(200,251)(201,250)(202,249)(203,248)(204,247)(205,246)(206,245)(207,244)(208,243)(209,242)(210,241)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(217,234)(218,233)(219,232)(220,231)(221,230)(222,229)(223,228)(224,227)(225,226)

Polycyclic presentation:

2^2,41,359|a^2=b,c^a=c^40,d^a=d^358

Conjugacy classes & character table

14722 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d");;
gap> coll := SingleCollector(F, [2, 2, 41, 359]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^40);;
gap> SetConjugate(coll, 4, 1, F.4^358);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14719 #72291669