Finite Group DatabasebetaFinite Group Databasebeta| order | 28800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| general orthogonal | GO+(4,𝔽5) |
| derived | Ω+(4,𝔽5) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, hypercenter | C2 | #72219552 |
| commutator, cosocle, hypocenter | Ω+(4,𝔽5) | C22 |
| LCS | G ▷ Ω+(4,𝔽5) | |
| UCS | C1 ◁ C2 | |
536 conjugacy classes of subgroups; only the 7 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 7 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(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)(50,52,54,51,53)(55,57,59,56,58)(60,62,64,61,63)(65,67,69,66,68)(70,72,74,71,73)(75,78,76,79,77)(80,83,81,84,82)(85,88,86,89,87)(90,93,91,94,92)(95,98,96,99,97)(100,104,103,102,101)(105,109,108,107,106)(110,114,113,112,111)(115,119,118,117,116)(120,124,123,122,121)(125,145,140,135,130)(126,146,141,136,131)(127,147,142,137,132)(128,148,143,138,133)(129,149,144,139,134)(150,171,167,163,159)(151,172,168,164,155)(152,173,169,160,156)(153,174,165,161,157)(154,170,166,162,158)(175,197,194,186,183)(176,198,190,187,184)(177,199,191,188,180)(178,195,192,189,181)(179,196,193,185,182)(200,223,216,214,207)(201,224,217,210,208)(202,220,218,211,209)(203,221,219,212,205)(204,222,215,213,206)(225,249,243,237,231)(226,245,244,238,232)(227,246,240,239,233)(228,247,241,235,234)(229,248,242,236,230)(250,265,255,270,260)(251,266,256,271,261)(252,267,257,272,262)(253,268,258,273,263)(254,269,259,274,264)(275,291,282,298,289)(276,292,283,299,285)(277,293,284,295,286)(278,294,280,296,287)(279,290,281,297,288)(300,317,309,321,313)(301,318,305,322,314)(302,319,306,323,310)(303,315,307,324,311)(304,316,308,320,312)(325,343,331,349,337)(326,344,332,345,338)(327,340,333,346,339)(328,341,334,347,335)(329,342,330,348,336)(350,369,358,372,361)(351,365,359,373,362)(352,366,355,374,363)(353,367,356,370,364)(354,368,357,371,360)(375,385,395,380,390)(376,386,396,381,391)(377,387,397,382,392)(378,388,398,383,393)(379,389,399,384,394)(400,411,422,408,419)(401,412,423,409,415)(402,413,424,405,416)(403,414,420,406,417)(404,410,421,407,418)(425,437,449,431,443)(426,438,445,432,444)(427,439,446,433,440)(428,435,447,434,441)(429,436,448,430,442)(450,463,471,459,467)(451,464,472,455,468)(452,460,473,456,469)(453,461,474,457,465)(454,462,470,458,466)(475,489,498,482,491)(476,485,499,483,492)(477,486,495,484,493)(478,487,496,480,494)(479,488,497,481,490)(500,505,510,515,520)(501,506,511,516,521)(502,507,512,517,522)(503,508,513,518,523)(504,509,514,519,524)(525,531,537,543,549)(526,532,538,544,545)(527,533,539,540,546)(528,534,535,541,547)(529,530,536,542,548)(550,557,564,566,573)(551,558,560,567,574)(552,559,561,568,570)(553,555,562,569,571)(554,556,563,565,572)(575,583,586,594,597)(576,584,587,590,598)(577,580,588,591,599)(578,581,589,592,595)(579,582,585,593,596)(600,609,613,617,621)(601,605,614,618,622)(602,606,610,619,623)(603,607,611,615,624)(604,608,612,616,620);(5,6,7,8,9)(10,12,14,11,13)(15,18,16,19,17)(20,24,23,22,21)(30,31,32,33,34)(35,37,39,36,38)(40,43,41,44,42)(45,49,48,47,46)(55,56,57,58,59)(60,62,64,61,63)(65,68,66,69,67)(70,74,73,72,71)(80,81,82,83,84)(85,87,89,86,88)(90,93,91,94,92)(95,99,98,97,96)(105,106,107,108,109)(110,112,114,111,113)(115,118,116,119,117)(120,124,123,122,121)(125,225,200,175,150)(126,226,201,176,151)(127,227,202,177,152)(128,228,203,178,153)(129,229,204,179,154)(130,231,207,183,159)(131,232,208,184,155)(132,233,209,180,156)(133,234,205,181,157)(134,230,206,182,158)(135,237,214,186,163)(136,238,210,187,164)(137,239,211,188,160)(138,235,212,189,161)(139,236,213,185,162)(140,243,216,194,167)(141,244,217,190,168)(142,240,218,191,169)(143,241,219,192,165)(144,242,215,193,166)(145,249,223,197,171)(146,245,224,198,172)(147,246,220,199,173)(148,247,221,195,174)(149,248,222,196,170)(250,325,275,350,300)(251,326,276,351,301)(252,327,277,352,302)(253,328,278,353,303)(254,329,279,354,304)(255,331,282,358,309)(256,332,283,359,305)(257,333,284,355,306)(258,334,280,356,307)(259,330,281,357,308)(260,337,289,361,313)(261,338,285,362,314)(262,339,286,363,310)(263,335,287,364,311)(264,336,288,360,312)(265,343,291,369,317)(266,344,292,365,318)(267,340,293,366,319)(268,341,294,367,315)(269,342,290,368,316)(270,349,298,372,321)(271,345,299,373,322)(272,346,295,374,323)(273,347,296,370,324)(274,348,297,371,320)(375,425,475,400,450)(376,426,476,401,451)(377,427,477,402,452)(378,428,478,403,453)(379,429,479,404,454)(380,431,482,408,459)(381,432,483,409,455)(382,433,484,405,456)(383,434,480,406,457)(384,430,481,407,458)(385,437,489,411,463)(386,438,485,412,464)(387,439,486,413,460)(388,435,487,414,461)(389,436,488,410,462)(390,443,491,419,467)(391,444,492,415,468)(392,440,493,416,469)(393,441,494,417,465)(394,442,490,418,466)(395,449,498,422,471)(396,445,499,423,472)(397,446,495,424,473)(398,447,496,420,474)(399,448,497,421,470)(500,525,550,575,600)(501,526,551,576,601)(502,527,552,577,602)(503,528,553,578,603)(504,529,554,579,604)(505,531,557,583,609)(506,532,558,584,605)(507,533,559,580,606)(508,534,555,581,607)(509,530,556,582,608)(510,537,564,586,613)(511,538,560,587,614)(512,539,561,588,610)(513,535,562,589,611)(514,536,563,585,612)(515,543,566,594,617)(516,544,567,590,618)(517,540,568,591,619)(518,541,569,592,615)(519,542,565,593,616)(520,549,573,597,621)(521,545,574,598,622)(522,546,570,599,623)(523,547,571,595,624)(524,548,572,596,620);(25,29,28,27,26)(30,34,33,32,31)(35,39,38,37,36)(40,44,43,42,41)(45,49,48,47,46)(50,53,51,54,52)(55,58,56,59,57)(60,63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179,229,204)(155,190,245,210)(156,191,246,211)(157,192,247,212)(158,193,248,213)(159,194,249,214)(160,180,240,220)(161,181,241,221)(162,182,242,222)(163,183,243,223)(164,184,244,224)(165,195,235,205)(166,196,236,206)(167,197,237,207)(168,198,238,208)(169,199,239,209)(170,185,230,215)(171,186,231,216)(172,187,232,217)(173,188,233,218)(174,189,234,219)(255,265,270,260)(256,266,271,261)(257,267,272,262)(258,268,273,263)(259,269,274,264)(275,300,350,325)(276,301,351,326)(277,302,352,327)(278,303,353,328)(279,304,354,329)(280,315,370,335)(281,316,371,336)(282,317,372,337)(283,318,373,338)(284,319,374,339)(285,305,365,345)(286,306,366,346)(287,307,367,347)(288,308,368,348)(289,309,369,349)(290,320,360,330)(291,321,361,331)(292,322,362,332)(293,323,363,333)(294,324,364,334)(295,310,355,340)(296,311,356,341)(297,312,357,342)(298,313,358,343)(299,314,359,344)(380,390,395,385)(381,391,396,386)(382,392,397,387)(383,393,398,388)(384,394,399,389)(400,425,475,450)(401,426,476,451)(402,427,477,452)(403,428,478,453)(404,429,479,454)(405,440,495,460)(406,441,496,461)(407,442,497,462)(408,443,498,463)(409,444,499,464)(410,430,490,470)(411,431,491,471)(412,432,492,472)(413,433,493,473)(414,434,494,474)(415,445,485,455)(416,446,486,456)(417,447,487,457)(418,448,488,458)(419,449,489,459)(420,435,480,465)(421,436,481,466)(422,437,482,467)(423,438,483,468)(424,439,484,469)(505,515,520,510)(506,516,521,511)(507,517,522,512)(508,518,523,513)(509,519,524,514)(525,550,600,575)(526,551,601,576)(527,552,602,577)(528,553,603,578)(529,554,604,579)(530,565,620,585)(531,566,621,586)(532,567,622,587)(533,568,623,588)(534,569,624,589)(535,555,615,595)(536,556,616,596)(537,557,617,597)(538,558,618,598)(539,559,619,599)(540,570,610,580)(541,571,611,581)(542,572,612,582)(543,573,613,583)(544,574,614,584)(545,560,605,590)(546,561,606,591)(547,562,607,592)(548,563,608,593)(549,564,609,594);(5,100)(6,101)(7,102)(8,103)(9,104)(10,75)(11,76)(12,77)(13,78)(14,79)(15,50)(16,51)(17,52)(18,53)(19,54)(20,25)(21,26)(22,27)(23,28)(24,29)(30,120)(31,121)(32,122)(33,123)(34,124)(35,95)(36,96)(37,97)(38,98)(39,99)(40,70)(41,71)(42,72)(43,73)(44,74)(55,115)(56,116)(57,117)(58,118)(59,119)(60,90)(61,91)(62,92)(63,93)(64,94)(80,110)(81,111)(82,112)(83,113)(84,114)(130,225)(131,226)(132,227)(133,228)(134,229)(135,200)(136,201)(137,202)(138,203)(139,204)(140,175)(141,176)(142,177)(143,178)(144,179)(145,150)(146,151)(147,152)(148,153)(149,154)(155,245)(156,246)(157,247)(158,248)(159,249)(160,220)(161,221)(162,222)(163,223)(164,224)(165,195)(166,196)(167,197)(168,198)(169,199)(180,240)(181,241)(182,242)(183,243)(184,244)(185,215)(186,216)(187,217)(188,218)(189,219)(205,235)(206,236)(207,237)(208,238)(209,239)(255,350)(256,351)(257,352)(258,353)(259,354)(260,325)(261,326)(262,327)(263,328)(264,329)(265,300)(266,301)(267,302)(268,303)(269,304)(270,275)(271,276)(272,277)(273,278)(274,279)(280,370)(281,371)(282,372)(283,373)(284,374)(285,345)(286,346)(287,347)(288,348)(289,349)(290,320)(291,321)(292,322)(293,323)(294,324)(305,365)(306,366)(307,367)(308,368)(309,369)(310,340)(311,341)(312,342)(313,343)(314,344)(330,360)(331,361)(332,362)(333,363)(334,364)(380,475)(381,476)(382,477)(383,478)(384,479)(385,450)(386,451)(387,452)(388,453)(389,454)(390,425)(391,426)(392,427)(393,428)(394,429)(395,400)(396,401)(397,402)(398,403)(399,404)(405,495)(406,496)(407,497)(408,498)(409,499)(410,470)(411,471)(412,472)(413,473)(414,474)(415,445)(416,446)(417,447)(418,448)(419,449)(430,490)(431,491)(432,492)(433,493)(434,494)(435,465)(436,466)(437,467)(438,468)(439,469)(455,485)(456,486)(457,487)(458,488)(459,489)(505,600)(506,601)(507,602)(508,603)(509,604)(510,575)(511,576)(512,577)(513,578)(514,579)(515,550)(516,551)(517,552)(518,553)(519,554)(520,525)(521,526)(522,527)(523,528)(524,529)(530,620)(531,621)(532,622)(533,623)(534,624)(535,595)(536,596)(537,597)(538,598)(539,599)(540,570)(541,571)(542,572)(543,573)(544,574)(555,615)(556,616)(557,617)(558,618)(559,619)(560,590)(561,591)(562,592)(563,593)(564,594)(580,610)(581,611)(582,612)(583,613)(584,614)41 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((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)(50,52,54,51,53)(55,57,59,56,58)(60,62,64,61,63)(65,67,69,66,68)(70,72,74,71,73)(75,78,76,79,77)(80,83,81,84,82)(85,88,86,89,87)(90,93,91,94,92)(95,98,96,99,97)(100,104,103,102,101)(105,109,108,107,106)(110,114,113,112,111)(115,119,118,117,116)(120,124,123,122,121)(125,145,140,135,130)(126,146,141,136,131)(127,147,142,137,132)(128,148,143,138,133)(129,149,144,139,134)(150,171,167,163,159)(151,172,168,164,155)(152,173,169,160,156)(153,174,165,161,157)(154,170,166,162,158)(175,197,194,186,183)(176,198,190,187,184)(177,199,191,188,180)(178,195,192,189,181)(179,196,193,185,182)(200,223,216,214,207)(201,224,217,210,208)(202,220,218,211,209)(203,221,219,212,205)(204,222,215,213,206)(225,249,243,237,231)(226,245,244,238,232)(227,246,240,239,233)(228,247,241,235,234)(229,248,242,236,230)(250,265,255,270,260)(251,266,256,271,261)(252,267,257,272,262)(253,268,258,273,263)(254,269,259,274,264)(275,291,282,298,289)(276,292,283,299,285)(277,293,284,295,286)(278,294,280,296,287)(279,290,281,297,288)(300,317,309,321,313)(301,318,305,322,314)(302,319,306,323,310)(303,315,307,324,311)(304,316,308,320,312)(325,343,331,349,337)(326,344,332,345,338)(327,340,333,346,339)(328,341,334,347,335)(329,342,330,348,336)(350,369,358,372,361)(351,365,359,373,362)(352,366,355,374,363)(353,367,356,370,364)(354,368,357,371,360)(375,385,395,380,390)(376,386,396,381,391)(377,387,397,382,392)(378,388,398,383,393)(379,389,399,384,394)(400,411,422,408,419)(401,412,423,409,415)(402,413,424,405,416)(403,414,420,406,417)(404,410,421,407,418)(425,437,449,431,443)(426,438,445,432,444)(427,439,446,433,440)(428,435,447,434,441)(429,436,448,430,442)(450,463,471,459,467)(451,464,472,455,468)(452,460,473,456,469)(453,461,474,457,465)(454,462,470,458,466)(475,489,498,482,491)(476,485,499,483,492)(477,486,495,484,493)(478,487,496,480,494)(479,488,497,481,490)(500,505,510,515,520)(501,506,511,516,521)(502,507,512,517,522)(503,508,513,518,523)(504,509,514,519,524)(525,531,537,543,549)(526,532,538,544,545)(527,533,539,540,546)(528,534,535,541,547)(529,530,536,542,548)(550,557,564,566,573)(551,558,560,567,574)(552,559,561,568,570)(553,555,562,569,571)(554,556,563,565,572)(575,583,586,594,597)(576,584,587,590,598)(577,580,588,591,599)(578,581,589,592,595)(579,582,585,593,596)(600,609,613,617,621)(601,605,614,618,622)(602,606,610,619,623)(603,607,611,615,624)(604,608,612,616,620), 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