C₂×D₄₃₇₅ #72213652

order	17500
automorphism	Aut(G) order 26250000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₈₇₅₀
direct product	C₂×D₄₃₇₅

Splittings

Fitting	C₂×C₇×C₆₂₅ ⋊ C₂
derived, Hall	C₇×C₆₂₅ ⋊ C₂²
Sylow	C₆₂₅ ⋊ C₂×D₇
Sylow	C₇ ⋊ C₂×D₆₂₅

Subgroups

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

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 634
transitive degree	μ_t(G) = 8750

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₁₇₅₀₀, C₂×D₁₃₁₂₅
maximal normal subgroup of	D₁₇₅₀₀
maximal quotient of	Dic₈₇₅₀, D₁₇₅₀₀, C₂×D₁₃₁₂₅

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C2×D4375 #72213652