GroupTheory/ProjectiveSpecialOrthogonalGroup - Maple Help
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GroupTheory

  

ProjectiveSpecialOrthogonalGroup

  

construct a permutation group isomorphic to a projective special orthogonal group

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

ProjectiveSpecialOrthogonalGroup(d, n, q)

PSO(d, n, q)

Parameters

d

-

0, 1 or -1

n

-

a positive integer

q

-

power of a prime number

Description

• 

The projective special orthogonal group PSOd,n,q is the quotient of the special orthogonal group SOd,n,q by its center. The value of d must be 0 for odd n, or 1 or −1 for even n.

• 

The ProjectiveSpecialOrthogonalGroup( d, n, q ) command returns a permutation group isomorphic to the projective special orthogonal group PSOd,n,q .

• 

The PSO( d, n, q ) command is provided as an alias.

• 

If the argument q is not a prime power (and is non-numeric), then a symbolic group representing PSOd,n,q  is returned.

Examples

withGroupTheory:

GProjectiveSpecialOrthogonalGroup1,2,7

GC4

(1)

GroupOrderG

4

(2)

IsCyclicG

true

(3)

GProjectiveSpecialOrthogonalGroup1,2,8

GD7

(4)

AreIsomorphicG,DihedralGroup7

true

(5)

GPSO0,3,3

G2,36,117,128,9,1,2,45,6,127,11,138,9,10,1,35,76,138,10

(6)

AreIsomorphicG,Symm4

true

(7)

GPSO1,4,9

G2,15,41,19,51,3,17,465,20,52,130,113,16,43,1147,23,60,146,98,235,452,1589,26,69,167,323,545,572,37810,29,78,190,80,194,31,8411,12,32,85,203,408,497,3913,35,94,227,287,252,284,50814,38,103,247,479,709,259,49118,48,125,275,57,56,141,28021,55,139,127,119,44,116,2224,63,154,299,520,517,315,50025,66,162,316,537,458,532,45027,72,175,345,562,743,782,39228,75,182,363,616,787,620,59730,81,196,393,553,676,355,40733,88,212,264,499,511,724,43134,91,219,441,699,288,40,10836,97,65,159,263,472,477,54137,100,239,298,296,294,516,46442,112,270,279,276,129,53,13345,118,123,272,281,49,50,12847,122,137,134,277,120,126,11554,136,273,282,131,271,59,14558,143,117,111,268,121,274,14461,149,241,322,544,319,422,15262,151,292,233,459,229,68,16564,157,306,236,462,166,320,6771,172,337,573,649,586,77,18873,177,351,601,555,581,763,34174,179,349,598,778,329,558,74776,185,370,626,653,748,780,81879,191,383,648,170,331,181,36082,199,399,384,650,643,797,8386,206,208,417,687,418,101,24287,209,419,234,460,253,93,22489,214,430,16090,216,435,95,230,109,262,49592,221,446,701,246,107,257,42596,232,164,153,297,148,102,24599,213,416,470,522,494,521,218104,249,475,715,482,291,434,694105,251,414,442,409,478,110,265106,254,484,718,258,490,698,321124,142,138,140,132,278,135,269147,285,461,207,415,501,440,696150,289,309,530,313,506,722,713155,301,496,719,293,436,318,453156,304,525,690,445,686,161,314163,317,539,474,714,483,240,411168,326,201,404,198,397,186,373169,328,556,613,642,369,174,342171,334,567,625,552,739,336,571173,340,579,745,815,609,359,607176,348,596,389,570,338,574,760178,354,580,757,402,674,765,791180,358,382,647,379,560,750,557183,365,332,563,636,795,582,387184,367,551,627,547,734,189,380187,376,635,735,197,395,564,754192,386,377,638,372,630,764,606193,388,347,594,755,675,587,610195,391,659,753,790,362,614,357200,202,406,678,742,775,405,634204,410,524,711,688,256,486,444205,413,492,502,489,423,211,424210,421,682,720,681,468,226,448215,432,510,488,691,451,536,693217,437,439,523,438,695,712,808220,443,454,526,540,721,238,467222,447,702,283,505,307,528,704223,420,476,717,267,504,261,426225,449,703,807,710,716,533,311228,266,463,515,255,412,684,428231,455,456,480,457,706,727,705237,303,302,514,487,427,485,250243,473,503,260,433,692,493,290248,481,512,527,726,685,542,683286,429,689,295,305,498,300,469308,529,465,707,531,519,310,518312,534,535,466,538,471,543,507324,374,619,789,350,600,335,568325,550,737,361,612,578,330,559327,554,593,663,673,672,668,646333,565,621,599,656,583,766,670339,569,736,812,731,777,344,588343,585,768,368,624,792,615,786346,575,665,798,730,667,788,814352,603,677,781,577,679,651,799353,605,738,631,640,590,356,366364,584,655,618,548,660,608,589371,628,576,666,657,611,767,785375,633,381,644,752,756,803,662385,632,793,773,813,802,390,658394,661,744,561,751,403,398,401396,546,592,664,776,800,774,816400,669,732,794,811,671,779,566513,708,723,700,697,725,806,729549,617,759,761591,772,749,652,770,604,595,645602,637,783,817,623,629,622,639641,796,820,746,740,680,805,819741,762,758,784,769,771,801,810,1,2,16,443,18,49,1124,5,21,566,7,24,648,9,27,7310,30,82,7811,13,36,9812,33,89,9614,39,106,25515,42,113,27117,47,123,1920,53,134,11422,58,144,27523,61,150,29025,67,163,30326,70,170,33228,76,186,37429,79,192,36731,71,173,17932,86,207,22934,92,222,15335,95,231,22137,101,243,47438,104,250,32240,109,263,49741,111,117,4345,119,116,14246,121,141,14348,126,124,11850,57,59,5551,129,277,13552,131,136,12854,137,274,14060,147,238,46862,152,295,8863,155,302,6865,160,312,45566,159,311,26469,168,325,35672,176,349,59974,180,359,61175,183,80,19577,182,364,61981,197,396,66384,20285,204,411,41387,210,422,41090,217,419,53691,220,444,28593,225,448,14894,228,304,48197,234,461,71099,237,465,712100,240,471,689102,246,477,445103,248,453,420105,252,449,162107,258,298,310108,260,434,423110,266,503,690115,120,272,139122,268,138,282125,276,132,280127,133,273,145130,279,269,278146,283,506,706149,288,512,504151,293,486,529154,256,305,527156,161,315,526157,307,257,488158,309,531,432164,226,450,447165,211,425,296166,321,212,427167,324,548,628169,329,405,344171,335,569,754172,338,575,739174,343,586,766175,346,592,773177,352,604,650178,355,608,574181,361,399,667184,368,620,614185,371,629,549187,348,597,603188,378,640,783189,381,645,600190,353,388,655191,384,589,771193,389,370,627194,354,342,583196,372,584,767198,395,662,330199,400,670,201200,402,675,334203,308,416,498206,209,236,463208,245,476,235213,428,412,458214,431,683,806215,433,219,442216,436,227,451218,439,697,299223,424,414,224230,435,294,267232,457,289,314233,262,496,721239,470,691,244241,472,534,544242,456,511,525247,480,487,508249,441,700,520251,462,542,722253,300254,473,528,718259,438,696,723261,494,499,489265,501284,509,479,539286,510,702,408287,297,478,681291,513,490,717292,514,725,484301,523,493,409306,318,541,729313,535,475,716316,538,728,711317,540,320,543319,524,491,452323,546,732,787326,552,740,814327,555,390,347328,360,331,561333,566,756,793336,572,379,641337,554,743,357339,577,665,653340,580,582,764341,581,671,394345,590,387,654350,550,656,644351,602,781,404358,556,587,362363,617,788,605365,621,558,748366,568,382,403369,625,676,804373,632,747,815375,634,794,626376,636,661,678380,643,790,559383,649,677,737385,595386,652,547,735391,545,730,642392,660,760,775393,647,799,674397,664,795,606398,666,571,631401,672,750,622406,679,648,792407,680,615,601415,522,492,709417,688,715,500418,521,727,714421,459,692,517426,485,516,430429,495,530,724437,502,460,708440,698,454,515443,699,533,701464,507,720,695466,686,482,518467,713,808,704469,537,519,726483,693,685,703505,719,809,687532,682,707,705551,738,813,820553,741,736,564557,746,673,758560,570,759,578562,752,745,593563,749,780,819565,637,567,757573,623,791,812576,762,633,651579,618,761,796585,769,658,802588,770,744,782591,765,596,613594,776,798,797598,779,635,772607,609,753,816610,742,646,777612,785,638,751616,786,639,763624,755,817,774630,659,731,784657,800,734,811669,733,810,789801,803,818,805,1,3,19,46,51,17,15,415,22,59,125,126,143,276,497,25,68,166,322,508,465,5168,10,31,78,84,29,80,1949,28,77,189,382,550,548,56911,14,40,110,267,413,221,42112,34,93,226,246,478,479,47613,37,102,61,64,158,310,53216,45,120,273,144,21,57,14218,50,55,140,112,268,271,11320,54,138,123,133,48,127,27723,62,153,298,519,462,537,14924,65,161,33,90,218,440,54026,71,174,344,589,734,616,18027,74,181,362,615,772,670,63230,83,201,405,352,381,646,79832,87,211,426,491,468,699,44635,96,233,316,529,152,296,32036,99,238,469,109,264,500,46138,105,253,408,681,717,219,42439,107,259,492,720,209,420,25142,44,117,128,278,134,56,11443,115,53,135,280,136,281,12147,124,119,130,58,141,270,14552,132,274,122,116,118,131,12960,148,287,422,450,239,459,70763,156,305,443,285,431,477,26266,67,164,235,100,241,292,25269,169,330,560,597,777,649,66070,171,336,567,571,334,552,73972,79,193,390,621,790,778,62473,178,356,609,783,818,332,56475,184,369,545,731,358,337,55976,187,377,639,173,341,582,76581,198,398,346,593,775,384,65182,200,403,676,788,799,186,37585,205,414,425,108,261,460,70986,208,418,101,244,206,242,41788,213,429,690,472,501,230,45489,215,434,510,473,506,150,29191,210,423,497,504,701,419,44292,223,448,265,502,234,441,24794,229,98,236,464,162,297,51895,97,220,445,696,154,300,522103,224,257,489,288,203,409,682104,243,475,530,432,691,290,214106,256,487,702,481,471,684,240111,269,282,139,279,137,272,275146,284,157,308,165,319,232,458147,286,511,304,526,470,315,216151,294,227,452,544,531,245,306155,303,204,412,685,714,283,507159,212,295,517,467,495,525,494160,313,536,722,694,260,493,451163,318,542,307,321,463,711,543167,325,551,655,188,379,642,787168,327,196,394,662,781,797,814170,333,175,347,595,585,747,802172,339,578,572,647,608,556,627176,350,568,758,759,611,784,785177,353,606,735,795,815,402,602179,357,610,743,813,792,383,645182,342,584,612,586,323,547,736183,366,623,630,674,601,780,340185,372,631,757,365,622,197,359191,385,604,782,786,565,755,659192,387,370,590,555,745,754,580195,392,599,558,749,773,594,368199,401,673,756,326,553,742,730202,407,663,643,644,667,661,679207,416,686,520,724,721,263,498217,438,437,523,712,808,728,439222,411,444,293,515,312,514,258225,311,533,716,710,807,703,449228,453,704,250,483,534,698,512231,456,706,727,809,455,705,480237,466,486,505,496,490,317,248249,482,289,430,503,309,433,693254,485,719,474,266,447,410,683255,427,527,301,524,718,538,528299,521,499,415,314,435,689,541302,428,484,436,535,726,688,539324,549,348,374,389,657,801,762328,557,367,378,364,620,737,812329,345,591,391,583,360,343,587331,562,753,793,349,388,656,770335,570,741,371338,576,617,789,666,600,769,596351,386,653,629,791,635,640,563354,607,764,581,376,637,738,748355,603,554,744,678,397,665,803361,613,573,363,618,750,588,380373,399,668,577,575,634,752,561393,633,406,672,404,677,751,650395,626,763,817,636,638,579,605396,546,733,776,592,774,664,800400,671,566,732,794,654,779,811488,713,692,715509,723,513,697,708,700,806,729574,761,771,760,810,619,767,628598,766,768,658,614,652,675,648641,746,740,804,819,680,796,805

(8)

GroupOrderG

265680

(9)

IsSimpleG

true

(10)

GPSO1,4,9

G2,15,39,19,46,3,17,445,20,48,108,187,14,16,417,23,57,119,207,328,371,3689,26,66,145,257,383,375,17210,29,75,169,77,173,31,8111,12,32,83,186,198,33,8613,35,89,92,192,91,116,1821,51,40,98,38,95,34,4222,54,99,191,94,188,199,20524,60,128,228,149,266,379,13225,63,136,241,337,329,335,44427,69,153,272,342,255,378,13528,72,67,148,264,154,275,40630,78,175,310,430,431,318,37737,93,106,203,55,115,53,11343,101,200,52,112,195,97,19745,105,201,104,88,100,85,11047,107,103,202,327,90,193,19649,109,204,114,118,102,96,8450,111,194,87,189,206,56,11758,122,79,178,315,140,120,20959,125,65,14261,130,232,242,365,240,239,18462,133,237,361,363,289,360,27064,139,248,370,312,314,82,18368,151,269,299,225,295,74,16670,156,279,369,251,316,435,43371,157,282,210,333,283,286,41273,163,294,263,392,259,357,40776,170,305,385,276,168,249,16280,180,319,437,349,364,323,426121,212,244,131,235,247,127,137123,217,177,176,311,372,332,346124,220,185,134,238,356,443,354126,224,271,398,171,227,253,344129,230,340,436,320,141,252,211138,245,218,341,330,214,144,208143,256,381,352,280,300,421,452146,260,231,355,181,322,179,317147,262,390,165,298,293,152,150155,277,229,334,441,359,160,288158,174,307,427,384,246,258,350159,285,274,403,362,450,395,397161,290,415,460,455,428,292,223164,296,380,373,287,236,358,234167,302,284,388,281,409,446,442182,324,439,449,389,419,429,410213,336,402,399,267,268,306,265215,338,445,347,221,345,447,339216,321,254,326,250,374,222,219226,243,273,301,393,308,391,394233,325,440,459,454,303,425,366261,387,451,278,408,438,367,386291,417,422,456,404,413,424,416297,348,400,458,304,420,457,331309,313,432,434,376,453,411,418343,448,405,353,401,351,414,396,1,2,16,42,1013,18,45,106,344,5,21,52,1106,7,24,61,1318,9,27,70,15710,30,79,75,13211,13,17,19,4712,33,15,40,3614,37,94,92,18820,49,48,46,9322,55,116,98,10023,58,123,218,34225,64,140,251,29326,67,149,267,39628,73,164,297,21229,76,171,62,13431,82,184,263,30632,84,113,50,5435,90,117,108,4138,96,196,191,3943,103,104,197,19244,102,201,51,10753,114,200,97,19856,87,190,95,8557,120,210,334,25659,126,225,245,36760,129,208,331,12863,137,243,133,16565,143,153,273,15266,146,261,388,45068,72,161,177,31369,154,276,282,34571,159,286,413,36574,167,231,260,24477,174,308,417,17378,176,312,264,34080,181,323,122,21581,182,180,320,28183,86,187,193,20488,105,118,189,20389,115,195,206,11191,194,99,186,327109,205,202,112,199119,163,175,250,375121,213,309,428,385124,221,346,415,439125,223,336,322,394127,227,311,406,425130,233,166,300,332135,239,349,437,372136,242,366,419,414138,246,271,252,147139,249,373,301,310141,253,228,325,411142,255,379,374,455144,145,258,291,262148,220,344,433,335150,169,155,278,285151,270,389,289,330156,280,294,420,324158,283,319,216,259160,279,397,431,230162,292,403,222,348168,304,381,298,247172,224,268,307,409178,316,226,299,424179,266,338,360,377183,326,441,269,380185,219,343,378,209207,329,442,347,376214,274,404,460,234217,314,434,370,427229,352,361,272,400232,295,421,284,408235,357,390,328,410236,359,368,386,443237,362,351,401,418238,363,451,305,426240,321,438,454,436241,364,452,432,398248,371,393,459,423254,317,358,440,405257,384,277,353,449265,356,315,391,448275,341,444,399,333287,412,456,339,446288,392,337,395,453290,416,430,354,429296,422,387,458,383303,402,447,407,369350,355,435,382,457,1,3,19,44,46,17,15,395,22,56,118,197,106,105,517,25,65,144,63,138,247,3288,10,31,75,81,29,77,1739,28,74,168,243,357,213,22711,14,38,97,194,37,54,9612,34,88,114,94,195,20,5013,36,92,35,91,89,192,1816,43,104,198,109,111,21,5323,59,127,57,121,214,337,44424,62,135,240,286,414,296,40326,68,152,271,399,276,264,39427,71,160,289,228,351,130,23430,80,123,219,129,231,356,17832,85,115,206,187,204,40,9933,87,191,45,52,42,102,5541,100,199,203,98,186,112,8447,107,190,327,103,193,202,9048,83,113,200,205,95,189,20149,110,117,101,86,188,108,9358,124,222,349,252,78,177,31460,69,155,159,287,365,363,33361,132,236,360,396,450,334,21064,141,254,377,220,323,355,14066,147,263,393,406,224,225,33667,150,268,163,295,257,385,25370,158,284,411,455,367,454,42272,162,293,383,126,226,269,39273,165,299,148,265,308,398,24976,172,267,275,407,344,262,39179,179,318,230,354,139,250,17682,185,322,346,430,374,315,319119,208,218,131,142,241,235,329120,211,311,146,180,321,312,431122,216,181,248,372,437,238,320125,207,330,371,136,137,244,341128,229,353,157,270,397,255,380133,184,288,266,395,283,272,401134,217,340,370,364,310,260,326143,256,382,280,381,421,352,300145,259,170,306,166,301,298,154149,242,362,237,277,358,448,378151,171,294,375,273,402,390,305153,274,405,379,412,373,441,232156,281,410,416,366,453,384,415161,291,418,261,389,442,427,459164,282,285,239,343,359,342,361167,303,279,324,278,223,350,424174,309,429,316,413,460,446,408175,183,317,436,426,209,332,443182,325,387,302,251,376,456,307212,335,245,368215,339,447,345,221,347,445,338233,258,386,404,439,290,313,433246,369,438,432,440,388,292,419297,423,458,348,420,400,304,331409,417,451,435,428,425,449,434

(11)

GroupOrderG

259200

(12)

IsSimpleG

false

(13)

See Also

GroupTheory[ProjectiveGeneralOrthogonalGroup]

GroupTheory[SpecialOrthogonalGroup]