Dic₄₃₇₅ #72215821

order	17500
automorphism	Aut(G) order 26250000
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₇₅

Splittings

derived, Hall	C₇×C₆₂₅ ⋊ C₄
Sylow	C₆₂₅ ⋊ Dic₇
Sylow	C₇ ⋊ Dic₆₂₅

Subgroups

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

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 636
transitive degree	μ_t(G) = 17500

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

Polycyclic presentation:

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

Conjugacy classes & character table

4378 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₇₅₀, Dic₁₃₁₂₅
maximal normal subgroup of	Dic₈₇₅₀
maximal quotient of	Dic₁₃₁₂₅

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

Dic4375 #72215821