Dic₁₅₂₁₃ #72292764

order	60852
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₂₁₃

Splittings

derived, Hall	C₃×C₁₁×C₄₆₁ ⋊ C₄
Hall	C₁₁×C₄₆₁ ⋊ Dic₃
Hall	C₃×C₄₆₁ ⋊ Dic₁₁
Sylow	C₄₆₁ ⋊ Dic₃₃
Hall	C₃×C₁₁ ⋊ Dic₄₆₁
Sylow	C₁₁ ⋊ Dic₁₃₈₃
Sylow	C₃ ⋊ Dic₅₀₇₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₂₁₃
commutator, hypocenter	C₃×C₁₁×C₄₆₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₁×C₄₆₁	C₂
LCS	G ▷ C₃×C₁₁×C₄₆₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 479
transitive degree	μ_t(G) = 60852

A minimal faithful permutation representation — 2 generators on 479 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479);(1,3,2,4)(6,7)(9,18)(10,17)(11,16)(12,15)(13,14)(20,479)(21,478)(22,477)(23,476)(24,475)(25,474)(26,473)(27,472)(28,471)(29,470)(30,469)(31,468)(32,467)(33,466)(34,465)(35,464)(36,463)(37,462)(38,461)(39,460)(40,459)(41,458)(42,457)(43,456)(44,455)(45,454)(46,453)(47,452)(48,451)(49,450)(50,449)(51,448)(52,447)(53,446)(54,445)(55,444)(56,443)(57,442)(58,441)(59,440)(60,439)(61,438)(62,437)(63,436)(64,435)(65,434)(66,433)(67,432)(68,431)(69,430)(70,429)(71,428)(72,427)(73,426)(74,425)(75,424)(76,423)(77,422)(78,421)(79,420)(80,419)(81,418)(82,417)(83,416)(84,415)(85,414)(86,413)(87,412)(88,411)(89,410)(90,409)(91,408)(92,407)(93,406)(94,405)(95,404)(96,403)(97,402)(98,401)(99,400)(100,399)(101,398)(102,397)(103,396)(104,395)(105,394)(106,393)(107,392)(108,391)(109,390)(110,389)(111,388)(112,387)(113,386)(114,385)(115,384)(116,383)(117,382)(118,381)(119,380)(120,379)(121,378)(122,377)(123,376)(124,375)(125,374)(126,373)(127,372)(128,371)(129,370)(130,369)(131,368)(132,367)(133,366)(134,365)(135,364)(136,363)(137,362)(138,361)(139,360)(140,359)(141,358)(142,357)(143,356)(144,355)(145,354)(146,353)(147,352)(148,351)(149,350)(150,349)(151,348)(152,347)(153,346)(154,345)(155,344)(156,343)(157,342)(158,341)(159,340)(160,339)(161,338)(162,337)(163,336)(164,335)(165,334)(166,333)(167,332)(168,331)(169,330)(170,329)(171,328)(172,327)(173,326)(174,325)(175,324)(176,323)(177,322)(178,321)(179,320)(180,319)(181,318)(182,317)(183,316)(184,315)(185,314)(186,313)(187,312)(188,311)(189,310)(190,309)(191,308)(192,307)(193,306)(194,305)(195,304)(196,303)(197,302)(198,301)(199,300)(200,299)(201,298)(202,297)(203,296)(204,295)(205,294)(206,293)(207,292)(208,291)(209,290)(210,289)(211,288)(212,287)(213,286)(214,285)(215,284)(216,283)(217,282)(218,281)(219,280)(220,279)(221,278)(222,277)(223,276)(224,275)(225,274)(226,273)(227,272)(228,271)(229,270)(230,269)(231,268)(232,267)(233,266)(234,265)(235,264)(236,263)(237,262)(238,261)(239,260)(240,259)(241,258)(242,257)(243,256)(244,255)(245,254)(246,253)(247,252)(248,251)(249,250)

Polycyclic presentation:

2^2,3,11,461|a^2=b,c^a=c^2,d^a=d^10,e^a=e^460

Conjugacy classes & character table

15216 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, 461]);;
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.5^460);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15213 #72292764