Dic₁₃₁₀₁ #72288694

order	52404
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₁₀₁

Splittings

derived, Hall	C₃×C₁₁×C₃₉₇ ⋊ C₄
Hall	C₁₁×C₃₉₇ ⋊ Dic₃
Hall	C₃×C₃₉₇ ⋊ Dic₁₁
Sylow	C₃₉₇ ⋊ Dic₃₃
Hall	C₃×C₁₁ ⋊ Dic₃₉₇
Sylow	C₁₁ ⋊ Dic₁₁₉₁
Sylow	C₃ ⋊ Dic₄₃₆₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₁₀₁
commutator, hypocenter	C₃×C₁₁×C₃₉₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₁×C₃₉₇	C₂
LCS	G ▷ C₃×C₁₁×C₃₉₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 415
transitive degree	μ_t(G) = 52404

A minimal faithful permutation representation — 2 generators on 415 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415);(1,3,2,4)(6,7)(9,18)(10,17)(11,16)(12,15)(13,14)(20,415)(21,414)(22,413)(23,412)(24,411)(25,410)(26,409)(27,408)(28,407)(29,406)(30,405)(31,404)(32,403)(33,402)(34,401)(35,400)(36,399)(37,398)(38,397)(39,396)(40,395)(41,394)(42,393)(43,392)(44,391)(45,390)(46,389)(47,388)(48,387)(49,386)(50,385)(51,384)(52,383)(53,382)(54,381)(55,380)(56,379)(57,378)(58,377)(59,376)(60,375)(61,374)(62,373)(63,372)(64,371)(65,370)(66,369)(67,368)(68,367)(69,366)(70,365)(71,364)(72,363)(73,362)(74,361)(75,360)(76,359)(77,358)(78,357)(79,356)(80,355)(81,354)(82,353)(83,352)(84,351)(85,350)(86,349)(87,348)(88,347)(89,346)(90,345)(91,344)(92,343)(93,342)(94,341)(95,340)(96,339)(97,338)(98,337)(99,336)(100,335)(101,334)(102,333)(103,332)(104,331)(105,330)(106,329)(107,328)(108,327)(109,326)(110,325)(111,324)(112,323)(113,322)(114,321)(115,320)(116,319)(117,318)(118,317)(119,316)(120,315)(121,314)(122,313)(123,312)(124,311)(125,310)(126,309)(127,308)(128,307)(129,306)(130,305)(131,304)(132,303)(133,302)(134,301)(135,300)(136,299)(137,298)(138,297)(139,296)(140,295)(141,294)(142,293)(143,292)(144,291)(145,290)(146,289)(147,288)(148,287)(149,286)(150,285)(151,284)(152,283)(153,282)(154,281)(155,280)(156,279)(157,278)(158,277)(159,276)(160,275)(161,274)(162,273)(163,272)(164,271)(165,270)(166,269)(167,268)(168,267)(169,266)(170,265)(171,264)(172,263)(173,262)(174,261)(175,260)(176,259)(177,258)(178,257)(179,256)(180,255)(181,254)(182,253)(183,252)(184,251)(185,250)(186,249)(187,248)(188,247)(189,246)(190,245)(191,244)(192,243)(193,242)(194,241)(195,240)(196,239)(197,238)(198,237)(199,236)(200,235)(201,234)(202,233)(203,232)(204,231)(205,230)(206,229)(207,228)(208,227)(209,226)(210,225)(211,224)(212,223)(213,222)(214,221)(215,220)(216,219)(217,218)

Polycyclic presentation:

2^2,3,11,397|a^2=b,c^a=c^2,d^a=d^10,e^a=e^396

Conjugacy classes & character table

13104 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", "e");;
gap> coll := SingleCollector(F, [2, 2, 3, 11, 397]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^396);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13101 #72288694