Dic₁₃₀₂₁ #72288560

order	52084
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₀₂₁

Splittings

derived, Hall	C₂₉×C₄₄₉ ⋊ C₄
Sylow	C₄₄₉ ⋊ Dic₂₉
Sylow	C₂₉ ⋊ Dic₄₄₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₀₂₁
commutator, hypocenter	C₂₉×C₄₄₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₂₉×C₄₄₉	C₂
LCS	G ▷ C₂₉×C₄₄₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 482
transitive degree	μ_t(G) = 52084

A minimal faithful permutation representation — 2 generators on 482 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)(34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482);(1,3,2,4)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(35,482)(36,481)(37,480)(38,479)(39,478)(40,477)(41,476)(42,475)(43,474)(44,473)(45,472)(46,471)(47,470)(48,469)(49,468)(50,467)(51,466)(52,465)(53,464)(54,463)(55,462)(56,461)(57,460)(58,459)(59,458)(60,457)(61,456)(62,455)(63,454)(64,453)(65,452)(66,451)(67,450)(68,449)(69,448)(70,447)(71,446)(72,445)(73,444)(74,443)(75,442)(76,441)(77,440)(78,439)(79,438)(80,437)(81,436)(82,435)(83,434)(84,433)(85,432)(86,431)(87,430)(88,429)(89,428)(90,427)(91,426)(92,425)(93,424)(94,423)(95,422)(96,421)(97,420)(98,419)(99,418)(100,417)(101,416)(102,415)(103,414)(104,413)(105,412)(106,411)(107,410)(108,409)(109,408)(110,407)(111,406)(112,405)(113,404)(114,403)(115,402)(116,401)(117,400)(118,399)(119,398)(120,397)(121,396)(122,395)(123,394)(124,393)(125,392)(126,391)(127,390)(128,389)(129,388)(130,387)(131,386)(132,385)(133,384)(134,383)(135,382)(136,381)(137,380)(138,379)(139,378)(140,377)(141,376)(142,375)(143,374)(144,373)(145,372)(146,371)(147,370)(148,369)(149,368)(150,367)(151,366)(152,365)(153,364)(154,363)(155,362)(156,361)(157,360)(158,359)(159,358)(160,357)(161,356)(162,355)(163,354)(164,353)(165,352)(166,351)(167,350)(168,349)(169,348)(170,347)(171,346)(172,345)(173,344)(174,343)(175,342)(176,341)(177,340)(178,339)(179,338)(180,337)(181,336)(182,335)(183,334)(184,333)(185,332)(186,331)(187,330)(188,329)(189,328)(190,327)(191,326)(192,325)(193,324)(194,323)(195,322)(196,321)(197,320)(198,319)(199,318)(200,317)(201,316)(202,315)(203,314)(204,313)(205,312)(206,311)(207,310)(208,309)(209,308)(210,307)(211,306)(212,305)(213,304)(214,303)(215,302)(216,301)(217,300)(218,299)(219,298)(220,297)(221,296)(222,295)(223,294)(224,293)(225,292)(226,291)(227,290)(228,289)(229,288)(230,287)(231,286)(232,285)(233,284)(234,283)(235,282)(236,281)(237,280)(238,279)(239,278)(240,277)(241,276)(242,275)(243,274)(244,273)(245,272)(246,271)(247,270)(248,269)(249,268)(250,267)(251,266)(252,265)(253,264)(254,263)(255,262)(256,261)(257,260)(258,259)

Polycyclic presentation:

2^2,29,449|a^2=b,c^a=c^28,d^a=d^448

Conjugacy classes & character table

13024 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, 29, 449]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^28);;
gap> SetConjugate(coll, 4, 1, F.4^448);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13021 #72288560