Dic₈₇₅₀ #72273075

order	35000
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₇₅₀

Splittings

Hall	C₇×C₆₂₅ ⋊ Q₈
Sylow	C₆₂₅ ⋊ Dic₁₄
Sylow	C₇ ⋊ Dic₁₂₅₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄₃₇₅
commutator, cosocle	C₂×C₇×C₆₂₅	C₂²
Frattini	C₂×C₁₂₅	C₂×D₃₅
Fitting	C₄×C₇×C₆₂₅	C₂
socle, prime radical	C₂×C₅×C₇	C₂×D₁₂₅
hypercenter	C₄	D₄₃₇₅
hypocenter	C₇×C₆₂₅	Q₈
LCS	G ▷ C₂×C₇×C₆₂₅ ▷ C₇×C₆₂₅
UCS	C₁ ◁ C₂ ◁ C₄

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 640
transitive degree	μ_t(G) = 35000

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

Polycyclic presentation:

2^3,5,7,5^3|a^2=c,b^2=c,d^5=f,f^5=g,g^5=h,b^a=bc,d^a=d^4f^4g^4h^4,e^a=e^6,f^a=f^4g^4h^4,g^a=g^4h^4,h^a=h^4

Conjugacy classes & character table

8753 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 7, 5, 5, 5]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.6^4*F.7^4*F.8^4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^4*F.7^4*F.8^4);;
gap> SetConjugate(coll, 7, 1, F.7^4*F.8^4);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic8750 #72273075