Dic₁₂₅₅₂ #72287696

order	50208
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₅₅₂

Splittings

Hall	C₃×C₅₂₃ ⋊ Q₃₂
Sylow	C₅₂₃ ⋊ Dic₂₄
Sylow	C₃ ⋊ Dic₄₁₈₄

Subgroups

	subgroup	G/H
center	C₂	D₁₂₅₅₂
commutator, cosocle	C₃×C₈×C₅₂₃	C₂²
Frattini	C₈	C₂×D₁₅₆₉
Fitting	C₃×C₁₆×C₅₂₃	C₂
socle, prime radical	C₂×C₃×C₅₂₃	D₈
hypercenter	C₁₆	D₁₅₆₉
hypocenter	C₃×C₅₂₃	Q₃₂
LCS	G ▷ C₃×C₈×C₅₂₃ ▷ C₃×C₄×C₅₂₃ ▷ C₂×C₃×C₅₂₃ ▷ C₃×C₅₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 558
transitive degree	μ_t(G) = 50208

A minimal faithful permutation representation — 2 generators on 558 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(33,34,35)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,35)(37,558)(38,557)(39,556)(40,555)(41,554)(42,553)(43,552)(44,551)(45,550)(46,549)(47,548)(48,547)(49,546)(50,545)(51,544)(52,543)(53,542)(54,541)(55,540)(56,539)(57,538)(58,537)(59,536)(60,535)(61,534)(62,533)(63,532)(64,531)(65,530)(66,529)(67,528)(68,527)(69,526)(70,525)(71,524)(72,523)(73,522)(74,521)(75,520)(76,519)(77,518)(78,517)(79,516)(80,515)(81,514)(82,513)(83,512)(84,511)(85,510)(86,509)(87,508)(88,507)(89,506)(90,505)(91,504)(92,503)(93,502)(94,501)(95,500)(96,499)(97,498)(98,497)(99,496)(100,495)(101,494)(102,493)(103,492)(104,491)(105,490)(106,489)(107,488)(108,487)(109,486)(110,485)(111,484)(112,483)(113,482)(114,481)(115,480)(116,479)(117,478)(118,477)(119,476)(120,475)(121,474)(122,473)(123,472)(124,471)(125,470)(126,469)(127,468)(128,467)(129,466)(130,465)(131,464)(132,463)(133,462)(134,461)(135,460)(136,459)(137,458)(138,457)(139,456)(140,455)(141,454)(142,453)(143,452)(144,451)(145,450)(146,449)(147,448)(148,447)(149,446)(150,445)(151,444)(152,443)(153,442)(154,441)(155,440)(156,439)(157,438)(158,437)(159,436)(160,435)(161,434)(162,433)(163,432)(164,431)(165,430)(166,429)(167,428)(168,427)(169,426)(170,425)(171,424)(172,423)(173,422)(174,421)(175,420)(176,419)(177,418)(178,417)(179,416)(180,415)(181,414)(182,413)(183,412)(184,411)(185,410)(186,409)(187,408)(188,407)(189,406)(190,405)(191,404)(192,403)(193,402)(194,401)(195,400)(196,399)(197,398)(198,397)(199,396)(200,395)(201,394)(202,393)(203,392)(204,391)(205,390)(206,389)(207,388)(208,387)(209,386)(210,385)(211,384)(212,383)(213,382)(214,381)(215,380)(216,379)(217,378)(218,377)(219,376)(220,375)(221,374)(222,373)(223,372)(224,371)(225,370)(226,369)(227,368)(228,367)(229,366)(230,365)(231,364)(232,363)(233,362)(234,361)(235,360)(236,359)(237,358)(238,357)(239,356)(240,355)(241,354)(242,353)(243,352)(244,351)(245,350)(246,349)(247,348)(248,347)(249,346)(250,345)(251,344)(252,343)(253,342)(254,341)(255,340)(256,339)(257,338)(258,337)(259,336)(260,335)(261,334)(262,333)(263,332)(264,331)(265,330)(266,329)(267,328)(268,327)(269,326)(270,325)(271,324)(272,323)(273,322)(274,321)(275,320)(276,319)(277,318)(278,317)(279,316)(280,315)(281,314)(282,313)(283,312)(284,311)(285,310)(286,309)(287,308)(288,307)(289,306)(290,305)(291,304)(292,303)(293,302)(294,301)(295,300)(296,299)(297,298)

Polycyclic presentation:

2^5,3,523|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^522

Conjugacy classes & character table

12555 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", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 523]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^522);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12552 #72287696