Finite Group DatabasebetaFinite Group Databasebeta| order | 13503 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C7×C643 ⋊ C3 |
| Sylow | C643 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C643⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C643 | C3 |
| LCS | G ▷ C7×C643 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 650 |
| transitive degree | μt(G) = 4501 |
A minimal faithful permutation representation — 2 generators on 650 points:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,117)(76,97,114)(79,100,111)(82,103,108)(85,106,105)(88,109,102)(91,112,99)(93,115,96)(95,119,145)(98,122,142)(101,125,139)(104,128,136)(107,131,133)(110,134,130)(113,137,127)(116,140,124)(118,143,121)(120,147,176)(123,150,173)(126,153,170)(129,156,167)(132,159,164)(135,162,161)(138,165,158)(141,168,155)(144,171,152)(146,174,149)(148,178,210)(151,181,207)(154,184,204)(157,187,201)(160,190,198)(163,193,195)(166,196,192)(169,199,189)(172,202,186)(175,205,183)(177,208,180)(179,212,247)(182,215,244)(185,218,241)(188,221,238)(191,224,235)(194,227,232)(197,230,229)(200,233,226)(203,236,223)(206,239,220)(209,242,217)(211,245,214)(213,249,287)(216,252,284)(219,255,281)(222,258,278)(225,261,275)(228,264,272)(231,267,269)(234,270,266)(237,273,263)(240,276,260)(243,279,257)(246,282,254)(248,285,251)(250,289,330)(253,292,327)(256,295,324)(259,298,321)(262,301,318)(265,304,315)(268,307,312)(271,310,309)(274,313,306)(277,316,303)(280,319,300)(283,322,297)(286,325,294)(288,328,291)(290,332,375)(293,335,372)(296,333,369)(299,340,366)(302,343,363)(305,346,360)(308,349,357)(311,352,354)(314,355,351)(317,358,348)(320,361,345)(323,364,342)(326,367,339)(329,370,337)(331,373,334)(336,378,417)(338,379,414)(341,383,411)(344,385,408)(347,388,405)(350,391,402)(353,394,399)(356,397,396)(359,400,393)(362,403,390)(365,406,387)(368,409,384)(371,412,382)(374,415,380)(376,418,377)(381,421,454)(386,424,449)(389,426,446)(392,429,443)(395,432,440)(398,435,437)(401,438,434)(404,441,431)(407,444,428)(410,447,425)(413,450,423)(416,452,422)(419,455,420)(427,460,482)(430,462,479)(433,465,476)(436,468,473)(439,471,470)(442,474,467)(445,477,464)(448,480,461)(451,483,459)(453,485,458)(456,487,457)(463,492,511)(466,494,508)(469,497,505)(472,500,502)(475,503,499)(478,506,496)(481,509,493)(484,512,491)(486,514,490)(488,516,489)(495,521,537)(498,523,534)(501,526,531)(504,529,528)(507,532,525)(510,535,522)(513,538,520)(515,540,519)(517,542,518)(524,547,560)(527,549,557)(530,552,554)(533,555,551)(536,558,548)(539,561,546)(541,563,545)(543,565,544)(550,570,580)(553,572,577)(556,575,574)(559,578,571)(562,581,569)(564,583,568)(566,585,567)(573,590,597)(576,592,594)(579,595,591)(582,598,589)(584,600,588)(586,602,587)(593,607,611)(596,609,608)(599,612,606)(601,614,605)(603,616,604)(610,621,622)(613,623,620)(615,625,619)(617,627,618)(624,632,631)(626,633,630)(628,635,629)(634,639,638)(636,640,637)(641,643,642)(645,646,647)(648,650,649);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,290,333,252,178,211,246,283,323,365,407,445,478,507,533,556,576,593,602,563,512,447,364,279,205,143,92,114,139,167,198,232,269,309,351,393,431,464,493,520,545,567,570,523,465,391,304,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,302,344,386,418,330,372,414,332,376,419,456,488,517,543,566,586,603,617,628,636,641,640,625,598,558,506,441,358,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,297,339,382,422,457,460,385,298,218,150,94,118,144,172,203,237,274,314,356,398,436,469,498,524,542,485,412,325,245,176,207,241,278,318,360,402,440,473,502,528,551,571,589,605,618,621,592,552,500,435,352,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,379,292,212,248,286,326,368,410,448,481,510,536,559,579,596,610,616,583,538,480,406,319,239,171,115,70,89,111,136,164,195,229,266,306,348,390,428,461,491,519,544,547,494,429,346,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,347,389,427,455,373,287,327,369,411,449,482,511,537,560,580,597,611,622,631,638,642,643,633,612,578,532,474,400,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,337,380,420,424,340,255,181,119,146,175,206,240,277,317,359,401,439,472,501,527,550,565,514,450,367,282,208,145,173,204,238,275,315,357,399,437,470,499,525,548,569,588,604,607,572,526,468,394,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,338,381,421,335,249,288,329,371,413,451,484,513,539,562,582,599,613,624,627,600,561,509,444,361,276,202,140,90,51,67,86,108,133,161,192,226,263,303,345,387,425,459,490,518,521,462,388,301,221,153,97,53,71,91,113,138,166,197,231,268,308,350,392,430,463,487,415,328,247,284,324,366,408,446,479,508,534,557,577,594,608,620,630,637,639,623,595,555,503,438,355,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,291,334,377,383,295,215,147,177,209,243,280,320,362,404,442,475,504,530,553,573,585,540,483,409,322,242,174,117,142,170,201,235,272,312,354,396,434,467,496,522,546,568,587,590,549,497,432,349,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,341,375,417,454,378,289,331,374,416,453,486,515,541,564,584,601,615,626,634,635,614,581,535,477,403,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,342,384,423,458,489,492,426,343,258,184,122,72,93,116,141,169,200,234,271,311,353,395,433,466,495,516,452,370,285,210,244,281,321,363,405,443,476,505,531,554,574,591,606,619,629,632,609,575,529,471,397,310,230,162,106,62,30,10)(644,645,647,649,646,648,650)Polycyclic presentation:
3,7,643|b^a=b^4,c^a=c^1771503 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#27212596, #41008878, #41008882, … |
| maximal normal subgroup of | C2×#27212596, #41008878, #41008882, … |
| maximal quotient of | C2×#27212596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13503, 4);