Dic₇₆₀₁ #44477303

order	30404
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₆₀₁
unambiguous semidirect product	C₁₁⋊Dic₆₉₁
unambiguous semidirect product	C₆₉₁⋊Dic₁₁

Splittings

derived, Hall	C₁₁×C₆₉₁ ⋊ C₄
Sylow	C₆₉₁ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₆₉₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₆₀₁
commutator, hypocenter	C₁₁×C₆₉₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₆₉₁	C₂
LCS	G ▷ C₁₁×C₆₉₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 706
transitive degree	μ_t(G) = 30404

A minimal faithful permutation representation — 2 generators on 706 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,706)(18,705)(19,704)(20,703)(21,702)(22,701)(23,700)(24,699)(25,698)(26,697)(27,696)(28,695)(29,694)(30,693)(31,692)(32,691)(33,690)(34,689)(35,688)(36,687)(37,686)(38,685)(39,684)(40,683)(41,682)(42,681)(43,680)(44,679)(45,678)(46,677)(47,676)(48,675)(49,674)(50,673)(51,672)(52,671)(53,670)(54,669)(55,668)(56,667)(57,666)(58,665)(59,664)(60,663)(61,662)(62,661)(63,660)(64,659)(65,658)(66,657)(67,656)(68,655)(69,654)(70,653)(71,652)(72,651)(73,650)(74,649)(75,648)(76,647)(77,646)(78,645)(79,644)(80,643)(81,642)(82,641)(83,640)(84,639)(85,638)(86,637)(87,636)(88,635)(89,634)(90,633)(91,632)(92,631)(93,630)(94,629)(95,628)(96,627)(97,626)(98,625)(99,624)(100,623)(101,622)(102,621)(103,620)(104,619)(105,618)(106,617)(107,616)(108,615)(109,614)(110,613)(111,612)(112,611)(113,610)(114,609)(115,608)(116,607)(117,606)(118,605)(119,604)(120,603)(121,602)(122,601)(123,600)(124,599)(125,598)(126,597)(127,596)(128,595)(129,594)(130,593)(131,592)(132,591)(133,590)(134,589)(135,588)(136,587)(137,586)(138,585)(139,584)(140,583)(141,582)(142,581)(143,580)(144,579)(145,578)(146,577)(147,576)(148,575)(149,574)(150,573)(151,572)(152,571)(153,570)(154,569)(155,568)(156,567)(157,566)(158,565)(159,564)(160,563)(161,562)(162,561)(163,560)(164,559)(165,558)(166,557)(167,556)(168,555)(169,554)(170,553)(171,552)(172,551)(173,550)(174,549)(175,548)(176,547)(177,546)(178,545)(179,544)(180,543)(181,542)(182,541)(183,540)(184,539)(185,538)(186,537)(187,536)(188,535)(189,534)(190,533)(191,532)(192,531)(193,530)(194,529)(195,528)(196,527)(197,526)(198,525)(199,524)(200,523)(201,522)(202,521)(203,520)(204,519)(205,518)(206,517)(207,516)(208,515)(209,514)(210,513)(211,512)(212,511)(213,510)(214,509)(215,508)(216,507)(217,506)(218,505)(219,504)(220,503)(221,502)(222,501)(223,500)(224,499)(225,498)(226,497)(227,496)(228,495)(229,494)(230,493)(231,492)(232,491)(233,490)(234,489)(235,488)(236,487)(237,486)(238,485)(239,484)(240,483)(241,482)(242,481)(243,480)(244,479)(245,478)(246,477)(247,476)(248,475)(249,474)(250,473)(251,472)(252,471)(253,470)(254,469)(255,468)(256,467)(257,466)(258,465)(259,464)(260,463)(261,462)(262,461)(263,460)(264,459)(265,458)(266,457)(267,456)(268,455)(269,454)(270,453)(271,452)(272,451)(273,450)(274,449)(275,448)(276,447)(277,446)(278,445)(279,444)(280,443)(281,442)(282,441)(283,440)(284,439)(285,438)(286,437)(287,436)(288,435)(289,434)(290,433)(291,432)(292,431)(293,430)(294,429)(295,428)(296,427)(297,426)(298,425)(299,424)(300,423)(301,422)(302,421)(303,420)(304,419)(305,418)(306,417)(307,416)(308,415)(309,414)(310,413)(311,412)(312,411)(313,410)(314,409)(315,408)(316,407)(317,406)(318,405)(319,404)(320,403)(321,402)(322,401)(323,400)(324,399)(325,398)(326,397)(327,396)(328,395)(329,394)(330,393)(331,392)(332,391)(333,390)(334,389)(335,388)(336,387)(337,386)(338,385)(339,384)(340,383)(341,382)(342,381)(343,380)(344,379)(345,378)(346,377)(347,376)(348,375)(349,374)(350,373)(351,372)(352,371)(353,370)(354,369)(355,368)(356,367)(357,366)(358,365)(359,364)(360,363)(361,362)

Polycyclic presentation:

2,11,691,2|a^2=d,b^a=b^10,c^a=c^690

Conjugacy classes & character table

7604 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₂₀₂
maximal normal subgroup of	Dic₁₅₂₀₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30404, 4);

Dic7601 #44477303