Dic₁₅₀₄₇ #72292465

order	60188
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₀₄₇

Splittings

derived, Hall	C₄₁×C₃₆₇ ⋊ C₄
Sylow	C₃₆₇ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₃₆₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₀₄₇
commutator, hypocenter	C₄₁×C₃₆₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₁×C₃₆₇	C₂
LCS	G ▷ C₄₁×C₃₆₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 412
transitive degree	μ_t(G) = 60188

A minimal faithful permutation representation — 2 generators on 412 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)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412);(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)(47,412)(48,411)(49,410)(50,409)(51,408)(52,407)(53,406)(54,405)(55,404)(56,403)(57,402)(58,401)(59,400)(60,399)(61,398)(62,397)(63,396)(64,395)(65,394)(66,393)(67,392)(68,391)(69,390)(70,389)(71,388)(72,387)(73,386)(74,385)(75,384)(76,383)(77,382)(78,381)(79,380)(80,379)(81,378)(82,377)(83,376)(84,375)(85,374)(86,373)(87,372)(88,371)(89,370)(90,369)(91,368)(92,367)(93,366)(94,365)(95,364)(96,363)(97,362)(98,361)(99,360)(100,359)(101,358)(102,357)(103,356)(104,355)(105,354)(106,353)(107,352)(108,351)(109,350)(110,349)(111,348)(112,347)(113,346)(114,345)(115,344)(116,343)(117,342)(118,341)(119,340)(120,339)(121,338)(122,337)(123,336)(124,335)(125,334)(126,333)(127,332)(128,331)(129,330)(130,329)(131,328)(132,327)(133,326)(134,325)(135,324)(136,323)(137,322)(138,321)(139,320)(140,319)(141,318)(142,317)(143,316)(144,315)(145,314)(146,313)(147,312)(148,311)(149,310)(150,309)(151,308)(152,307)(153,306)(154,305)(155,304)(156,303)(157,302)(158,301)(159,300)(160,299)(161,298)(162,297)(163,296)(164,295)(165,294)(166,293)(167,292)(168,291)(169,290)(170,289)(171,288)(172,287)(173,286)(174,285)(175,284)(176,283)(177,282)(178,281)(179,280)(180,279)(181,278)(182,277)(183,276)(184,275)(185,274)(186,273)(187,272)(188,271)(189,270)(190,269)(191,268)(192,267)(193,266)(194,265)(195,264)(196,263)(197,262)(198,261)(199,260)(200,259)(201,258)(202,257)(203,256)(204,255)(205,254)(206,253)(207,252)(208,251)(209,250)(210,249)(211,248)(212,247)(213,246)(214,245)(215,244)(216,243)(217,242)(218,241)(219,240)(220,239)(221,238)(222,237)(223,236)(224,235)(225,234)(226,233)(227,232)(228,231)(229,230)

Polycyclic presentation:

2^2,41,367|a^2=b,c^a=c^40,d^a=d^366

Conjugacy classes & character table

15050 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, 367]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^40);;
gap> SetConjugate(coll, 4, 1, F.4^366);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15047 #72292465