Dic₄₉₀₇ #34087241

order	19628
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₉₀₇
unambiguous semidirect product	C₇⋊Dic₇₀₁

Splittings

derived, Hall	C₇×C₇₀₁ ⋊ C₄
Sylow	C₇₀₁ ⋊ Dic₇
Sylow	C₇ ⋊ Dic₇₀₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₉₀₇
commutator, hypocenter	C₇×C₇₀₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₇₀₁	C₂
LCS	G ▷ C₇×C₇₀₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 712
transitive degree	μ_t(G) = 19628

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

Polycyclic presentation:

2,7,701,2|a^2=d,b^a=b^6,c^a=c^700

Conjugacy classes & character table

4910 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₈₁₄, Dic₁₄₇₂₁
maximal normal subgroup of	Dic₉₈₁₄
maximal quotient of	Dic₁₄₇₂₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19628, 6);

Dic4907 #34087241