Dic₇₉₉₇ #44544573

order	31988
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₉₉₇
unambiguous semidirect product	C₁₁⋊Dic₇₂₇
unambiguous semidirect product	C₇₂₇⋊Dic₁₁

Splittings

derived, Hall	C₁₁×C₇₂₇ ⋊ C₄
Sylow	C₇₂₇ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₇₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₉₉₇
commutator, hypocenter	C₁₁×C₇₂₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₇₂₇	C₂
LCS	G ▷ C₁₁×C₇₂₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 742
transitive degree	μ_t(G) = 31988

A minimal faithful permutation representation — 2 generators on 742 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,742)(18,741)(19,740)(20,739)(21,738)(22,737)(23,736)(24,735)(25,734)(26,733)(27,732)(28,731)(29,730)(30,729)(31,728)(32,727)(33,726)(34,725)(35,724)(36,723)(37,722)(38,721)(39,720)(40,719)(41,718)(42,717)(43,716)(44,715)(45,714)(46,713)(47,712)(48,711)(49,710)(50,709)(51,708)(52,707)(53,706)(54,705)(55,704)(56,703)(57,702)(58,701)(59,700)(60,699)(61,698)(62,697)(63,696)(64,695)(65,694)(66,693)(67,692)(68,691)(69,690)(70,689)(71,688)(72,687)(73,686)(74,685)(75,684)(76,683)(77,682)(78,681)(79,680)(80,679)(81,678)(82,677)(83,676)(84,675)(85,674)(86,673)(87,672)(88,671)(89,670)(90,669)(91,668)(92,667)(93,666)(94,665)(95,664)(96,663)(97,662)(98,661)(99,660)(100,659)(101,658)(102,657)(103,656)(104,655)(105,654)(106,653)(107,652)(108,651)(109,650)(110,649)(111,648)(112,647)(113,646)(114,645)(115,644)(116,643)(117,642)(118,641)(119,640)(120,639)(121,638)(122,637)(123,636)(124,635)(125,634)(126,633)(127,632)(128,631)(129,630)(130,629)(131,628)(132,627)(133,626)(134,625)(135,624)(136,623)(137,622)(138,621)(139,620)(140,619)(141,618)(142,617)(143,616)(144,615)(145,614)(146,613)(147,612)(148,611)(149,610)(150,609)(151,608)(152,607)(153,606)(154,605)(155,604)(156,603)(157,602)(158,601)(159,600)(160,599)(161,598)(162,597)(163,596)(164,595)(165,594)(166,593)(167,592)(168,591)(169,590)(170,589)(171,588)(172,587)(173,586)(174,585)(175,584)(176,583)(177,582)(178,581)(179,580)(180,579)(181,578)(182,577)(183,576)(184,575)(185,574)(186,573)(187,572)(188,571)(189,570)(190,569)(191,568)(192,567)(193,566)(194,565)(195,564)(196,563)(197,562)(198,561)(199,560)(200,559)(201,558)(202,557)(203,556)(204,555)(205,554)(206,553)(207,552)(208,551)(209,550)(210,549)(211,548)(212,547)(213,546)(214,545)(215,544)(216,543)(217,542)(218,541)(219,540)(220,539)(221,538)(222,537)(223,536)(224,535)(225,534)(226,533)(227,532)(228,531)(229,530)(230,529)(231,528)(232,527)(233,526)(234,525)(235,524)(236,523)(237,522)(238,521)(239,520)(240,519)(241,518)(242,517)(243,516)(244,515)(245,514)(246,513)(247,512)(248,511)(249,510)(250,509)(251,508)(252,507)(253,506)(254,505)(255,504)(256,503)(257,502)(258,501)(259,500)(260,499)(261,498)(262,497)(263,496)(264,495)(265,494)(266,493)(267,492)(268,491)(269,490)(270,489)(271,488)(272,487)(273,486)(274,485)(275,484)(276,483)(277,482)(278,481)(279,480)(280,479)(281,478)(282,477)(283,476)(284,475)(285,474)(286,473)(287,472)(288,471)(289,470)(290,469)(291,468)(292,467)(293,466)(294,465)(295,464)(296,463)(297,462)(298,461)(299,460)(300,459)(301,458)(302,457)(303,456)(304,455)(305,454)(306,453)(307,452)(308,451)(309,450)(310,449)(311,448)(312,447)(313,446)(314,445)(315,444)(316,443)(317,442)(318,441)(319,440)(320,439)(321,438)(322,437)(323,436)(324,435)(325,434)(326,433)(327,432)(328,431)(329,430)(330,429)(331,428)(332,427)(333,426)(334,425)(335,424)(336,423)(337,422)(338,421)(339,420)(340,419)(341,418)(342,417)(343,416)(344,415)(345,414)(346,413)(347,412)(348,411)(349,410)(350,409)(351,408)(352,407)(353,406)(354,405)(355,404)(356,403)(357,402)(358,401)(359,400)(360,399)(361,398)(362,397)(363,396)(364,395)(365,394)(366,393)(367,392)(368,391)(369,390)(370,389)(371,388)(372,387)(373,386)(374,385)(375,384)(376,383)(377,382)(378,381)(379,380)

Polycyclic presentation:

2,11,727,2|a^2=d,b^a=b^10,c^a=c^726

Conjugacy classes & character table

8000 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₉₉₄
maximal normal subgroup of	Dic₁₅₉₉₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31988, 6);

Dic7997 #44544573