Dic₅₀₅₂ #72216632

order	20208
automorphism	Aut(G) order 33949440
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₀₅₂

Splittings

Hall	C₃×C₄₂₁ ⋊ Q₁₆
Sylow	C₄₂₁ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₁₆₈₄

Subgroups

	subgroup	G/H
center	C₂	D₅₀₅₂
commutator, cosocle	C₃×C₄×C₄₂₁	C₂²
Frattini	C₄	C₂×D₁₂₆₃
Fitting	C₃×C₈×C₄₂₁	C₂
socle, prime radical	C₂×C₃×C₄₂₁	D₄
hypercenter	C₈	D₁₂₆₃
hypocenter	C₃×C₄₂₁	Q₁₆
LCS	G ▷ C₃×C₄×C₄₂₁ ▷ C₂×C₃×C₄₂₁ ▷ C₃×C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 440
transitive degree	μ_t(G) = 20208

A minimal faithful permutation representation — 2 generators on 440 points:

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

Polycyclic presentation:

2^4,3,421|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^420

Conjugacy classes & character table

5055 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₁₀₄, Dic₁₅₁₅₆
maximal normal subgroup of	Dic₁₀₁₀₄
maximal quotient of	Dic₁₅₁₅₆

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 421]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^420);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5052 #72216632