Finite Group DatabasebetaFinite Group Databasebeta| order | 14400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | Ω+(4,𝔽5) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, hypercenter | C2 | GO+(4,𝔽4) |
| commutator, cosocle, hypocenter | Ω+(4,𝔽5) | C2 |
| LCS | G ▷ Ω+(4,𝔽5) | |
| UCS | C1 ◁ C2 | |
181 conjugacy classes of subgroups; only the 4 normal subgroups are shown.
This group has 181 conjugacy classes of subgroups — too many to draw. Only its 4 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(5,20)(6,21)(7,22)(8,23)(9,24)(10,15)(11,16)(12,17)(13,18)(14,19)(25,100)(26,101)(27,102)(28,103)(29,104)(30,120)(31,121)(32,122)(33,123)(34,124)(35,115)(36,116)(37,117)(38,118)(39,119)(40,110)(41,111)(42,112)(43,113)(44,114)(45,105)(46,106)(47,107)(48,108)(49,109)(50,75)(51,76)(52,77)(53,78)(54,79)(55,95)(56,96)(57,97)(58,98)(59,99)(60,90)(61,91)(62,92)(63,93)(64,94)(65,85)(66,86)(67,87)(68,88)(69,89)(70,80)(71,81)(72,82)(73,83)(74,84)(130,145)(131,146)(132,147)(133,148)(134,149)(135,140)(136,141)(137,142)(138,143)(139,144)(150,225)(151,226)(152,227)(153,228)(154,229)(155,245)(156,246)(157,247)(158,248)(159,249)(160,240)(161,241)(162,242)(163,243)(164,244)(165,235)(166,236)(167,237)(168,238)(169,239)(170,230)(171,231)(172,232)(173,233)(174,234)(175,200)(176,201)(177,202)(178,203)(179,204)(180,220)(181,221)(182,222)(183,223)(184,224)(185,215)(186,216)(187,217)(188,218)(189,219)(190,210)(191,211)(192,212)(193,213)(194,214)(195,205)(196,206)(197,207)(198,208)(199,209)(255,270)(256,271)(257,272)(258,273)(259,274)(260,265)(261,266)(262,267)(263,268)(264,269)(275,350)(276,351)(277,352)(278,353)(279,354)(280,370)(281,371)(282,372)(283,373)(284,374)(285,365)(286,366)(287,367)(288,368)(289,369)(290,360)(291,361)(292,362)(293,363)(294,364)(295,355)(296,356)(297,357)(298,358)(299,359)(300,325)(301,326)(302,327)(303,328)(304,329)(305,345)(306,346)(307,347)(308,348)(309,349)(310,340)(311,341)(312,342)(313,343)(314,344)(315,335)(316,336)(317,337)(318,338)(319,339)(320,330)(321,331)(322,332)(323,333)(324,334)(380,395)(381,396)(382,397)(383,398)(384,399)(385,390)(386,391)(387,392)(388,393)(389,394)(400,475)(401,476)(402,477)(403,478)(404,479)(405,495)(406,496)(407,497)(408,498)(409,499)(410,490)(411,491)(412,492)(413,493)(414,494)(415,485)(416,486)(417,487)(418,488)(419,489)(420,480)(421,481)(422,482)(423,483)(424,484)(425,450)(426,451)(427,452)(428,453)(429,454)(430,470)(431,471)(432,472)(433,473)(434,474)(435,465)(436,466)(437,467)(438,468)(439,469)(440,460)(441,461)(442,462)(443,463)(444,464)(445,455)(446,456)(447,457)(448,458)(449,459)(505,520)(506,521)(507,522)(508,523)(509,524)(510,515)(511,516)(512,517)(513,518)(514,519)(525,600)(526,601)(527,602)(528,603)(529,604)(530,620)(531,621)(532,622)(533,623)(534,624)(535,615)(536,616)(537,617)(538,618)(539,619)(540,610)(541,611)(542,612)(543,613)(544,614)(545,605)(546,606)(547,607)(548,608)(549,609)(550,575)(551,576)(552,577)(553,578)(554,579)(555,595)(556,596)(557,597)(558,598)(559,599)(560,590)(561,591)(562,592)(563,593)(564,594)(565,585)(566,586)(567,587)(568,588)(569,589)(570,580)(571,581)(572,582)(573,583)(574,584);(1,76)(2,27)(3,103)(4,54)(5,395)(6,471)(7,422)(8,498)(9,449)(10,140)(11,216)(12,167)(13,243)(14,194)(15,510)(16,586)(17,537)(18,613)(19,564)(20,255)(21,331)(22,282)(23,358)(24,309)(25,100)(26,51)(28,78)(30,495)(31,446)(32,397)(33,473)(34,424)(35,240)(36,191)(37,142)(38,218)(39,169)(40,610)(41,561)(42,512)(43,588)(44,539)(45,355)(46,306)(47,257)(48,333)(49,284)(50,75)(52,102)(55,470)(56,421)(57,497)(58,448)(59,399)(60,215)(61,166)(62,242)(63,193)(64,144)(65,585)(66,536)(67,612)(68,563)(69,514)(70,330)(71,281)(72,357)(73,308)(74,259)(79,104)(80,445)(81,396)(82,472)(83,423)(84,499)(85,190)(86,141)(87,217)(88,168)(89,244)(90,560)(91,511)(92,587)(93,538)(94,614)(95,305)(96,256)(97,332)(98,283)(99,359)(105,420)(106,496)(107,447)(108,398)(109,474)(110,165)(111,241)(112,192)(113,143)(114,219)(115,535)(116,611)(117,562)(118,513)(119,589)(120,280)(121,356)(122,307)(123,258)(124,334)(126,201)(127,152)(128,228)(129,179)(130,520)(131,596)(132,547)(133,623)(134,574)(135,265)(136,341)(137,292)(138,368)(139,319)(145,380)(146,456)(147,407)(148,483)(149,434)(150,225)(151,176)(153,203)(155,620)(156,571)(157,522)(158,598)(159,549)(160,365)(161,316)(162,267)(163,343)(164,294)(170,480)(171,431)(172,382)(173,458)(174,409)(175,200)(177,227)(180,595)(181,546)(182,622)(183,573)(184,524)(185,340)(186,291)(187,367)(188,318)(189,269)(195,455)(196,406)(197,482)(198,433)(199,384)(204,229)(205,570)(206,521)(207,597)(208,548)(209,624)(210,315)(211,266)(212,342)(213,293)(214,369)(220,430)(221,381)(222,457)(223,408)(224,484)(230,545)(231,621)(232,572)(233,523)(234,599)(235,290)(236,366)(237,317)(238,268)(239,344)(245,405)(246,481)(247,432)(248,383)(249,459)(251,326)(252,277)(253,353)(254,304)(260,390)(261,466)(262,417)(263,493)(264,444)(270,505)(271,581)(272,532)(273,608)(274,559)(275,350)(276,301)(278,328)(285,490)(286,441)(287,392)(288,468)(289,419)(295,605)(296,556)(297,507)(298,583)(299,534)(300,325)(302,352)(310,465)(311,416)(312,492)(313,443)(314,394)(320,580)(321,531)(322,607)(323,558)(324,509)(329,354)(335,440)(336,391)(337,467)(338,418)(339,494)(345,555)(346,506)(347,582)(348,533)(349,609)(360,415)(361,491)(362,442)(363,393)(364,469)(370,530)(371,606)(372,557)(373,508)(374,584)(376,451)(377,402)(378,478)(379,429)(385,515)(386,591)(387,542)(388,618)(389,569)(400,475)(401,426)(403,453)(410,615)(411,566)(412,517)(413,593)(414,544)(425,450)(427,477)(435,590)(436,541)(437,617)(438,568)(4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conjugacy classes — view the full character table.
| maximal subgroup of | GO+(4,𝔽5) |
| maximal normal subgroup of | GO+(4,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := 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