Dic₆₉₁₀ #72236331

order	27640
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₉₁₀

Splittings

Hall	C₅×C₆₉₁ ⋊ Q₈
Sylow	C₆₉₁ ⋊ Dic₁₀
Sylow	C₅ ⋊ Dic₁₃₈₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₄₅₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₆₉₁	C₂²
Fitting	C₄×C₅×C₆₉₁	C₂
hypercenter	C₄	D₃₄₅₅
hypocenter	C₅×C₆₉₁	Q₈
LCS	G ▷ C₂×C₅×C₆₉₁ ▷ C₅×C₆₉₁
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 704
transitive degree	μ_t(G) = 27640

A minimal faithful permutation representation — 2 generators on 704 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,704)(16,703)(17,702)(18,701)(19,700)(20,699)(21,698)(22,697)(23,696)(24,695)(25,694)(26,693)(27,692)(28,691)(29,690)(30,689)(31,688)(32,687)(33,686)(34,685)(35,684)(36,683)(37,682)(38,681)(39,680)(40,679)(41,678)(42,677)(43,676)(44,675)(45,674)(46,673)(47,672)(48,671)(49,670)(50,669)(51,668)(52,667)(53,666)(54,665)(55,664)(56,663)(57,662)(58,661)(59,660)(60,659)(61,658)(62,657)(63,656)(64,655)(65,654)(66,653)(67,652)(68,651)(69,650)(70,649)(71,648)(72,647)(73,646)(74,645)(75,644)(76,643)(77,642)(78,641)(79,640)(80,639)(81,638)(82,637)(83,636)(84,635)(85,634)(86,633)(87,632)(88,631)(89,630)(90,629)(91,628)(92,627)(93,626)(94,625)(95,624)(96,623)(97,622)(98,621)(99,620)(100,619)(101,618)(102,617)(103,616)(104,615)(105,614)(106,613)(107,612)(108,611)(109,610)(110,609)(111,608)(112,607)(113,606)(114,605)(115,604)(116,603)(117,602)(118,601)(119,600)(120,599)(121,598)(122,597)(123,596)(124,595)(125,594)(126,593)(127,592)(128,591)(129,590)(130,589)(131,588)(132,587)(133,586)(134,585)(135,584)(136,583)(137,582)(138,581)(139,580)(140,579)(141,578)(142,577)(143,576)(144,575)(145,574)(146,573)(147,572)(148,571)(149,570)(150,569)(151,568)(152,567)(153,566)(154,565)(155,564)(156,563)(157,562)(158,561)(159,560)(160,559)(161,558)(162,557)(163,556)(164,555)(165,554)(166,553)(167,552)(168,551)(169,550)(170,549)(171,548)(172,547)(173,546)(174,545)(175,544)(176,543)(177,542)(178,541)(179,540)(180,539)(181,538)(182,537)(183,536)(184,535)(185,534)(186,533)(187,532)(188,531)(189,530)(190,529)(191,528)(192,527)(193,526)(194,525)(195,524)(196,523)(197,522)(198,521)(199,520)(200,519)(201,518)(202,517)(203,516)(204,515)(205,514)(206,513)(207,512)(208,511)(209,510)(210,509)(211,508)(212,507)(213,506)(214,505)(215,504)(216,503)(217,502)(218,501)(219,500)(220,499)(221,498)(222,497)(223,496)(224,495)(225,494)(226,493)(227,492)(228,491)(229,490)(230,489)(231,488)(232,487)(233,486)(234,485)(235,484)(236,483)(237,482)(238,481)(239,480)(240,479)(241,478)(242,477)(243,476)(244,475)(245,474)(246,473)(247,472)(248,471)(249,470)(250,469)(251,468)(252,467)(253,466)(254,465)(255,464)(256,463)(257,462)(258,461)(259,460)(260,459)(261,458)(262,457)(263,456)(264,455)(265,454)(266,453)(267,452)(268,451)(269,450)(270,449)(271,448)(272,447)(273,446)(274,445)(275,444)(276,443)(277,442)(278,441)(279,440)(280,439)(281,438)(282,437)(283,436)(284,435)(285,434)(286,433)(287,432)(288,431)(289,430)(290,429)(291,428)(292,427)(293,426)(294,425)(295,424)(296,423)(297,422)(298,421)(299,420)(300,419)(301,418)(302,417)(303,416)(304,415)(305,414)(306,413)(307,412)(308,411)(309,410)(310,409)(311,408)(312,407)(313,406)(314,405)(315,404)(316,403)(317,402)(318,401)(319,400)(320,399)(321,398)(322,397)(323,396)(324,395)(325,394)(326,393)(327,392)(328,391)(329,390)(330,389)(331,388)(332,387)(333,386)(334,385)(335,384)(336,383)(337,382)(338,381)(339,380)(340,379)(341,378)(342,377)(343,376)(344,375)(345,374)(346,373)(347,372)(348,371)(349,370)(350,369)(351,368)(352,367)(353,366)(354,365)(355,364)(356,363)(357,362)(358,361)(359,360)

Polycyclic presentation:

2^3,5,691|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^690

Conjugacy classes & character table

6913 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₈₂₀
maximal normal subgroup of	Dic₁₃₈₂₀

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, 2, 5, 691]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^690);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6910 #72236331