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H(3,ℤ/32) #72234422

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monolithic
• monomial
• stem

Aliases

Subgroups

2099 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 1024 points:

(1,2,4,13,7,16,34,77)(3,21,22,111,25,114,122,391,27,117,124,393,146,411,435,830,29,119,125,395,144,413,436,831,30,118,123,396,147,412,437,832)(5,39,40,191,42,193,198,572,44,194,199,520,207,580,556,758)(6,15,33,76,48,93,172,305)(8,18,36,79,56,101,180,315)(9,17,38,81,57,103,182,317)(10,19,35,83,58,102,183,319)(11,20,37,82,55,104,181,318)(12,65,66,250,68,252,259,473,71,255,261,651,276,660,679,475,73,256,263,650,278,661,680,476,72,253,264,652,277,662,681,137)(14,84,85,208,87,331,336,209,90,333,298,600,346,772,735,43)(23,127,128,443,130,445,449,873,132,446,427,499,143,483,484,906,134,447,450,864,457,876,630,910,153,496,497,874,485,908,879,1024)(24,113,121,390,136,404,429,717,138,405,431,723,471,703,870,757,140,406,432,290,470,721,871,722,141,279,430,292,472,369,807,312)(26,116,31,157,145,410,159,500,148,418,161,502,488,854,512,912,150,420,162,504,487,856,513,913,151,419,160,505,489,855,514,914)(28,115,126,394,53,229,230,625,154,415,441,834,234,628,631,829,155,417,442,835,232,409,632,852,156,416,438,392,235,629,633,833)(32,164,142,345,166,519,212,349,169,521,482,579,531,921,45,89)(41,192,197,571,201,575,584,950,203,576,585,816,591,954,933,1001)(46,195,170,299,54,236,237,640,152,493,494,573,238,642,586,952)(47,92,171,304,215,352,535,737)(49,95,174,307,220,356,539,741)(50,94,176,309,221,358,541,743)(51,96,173,311,222,357,542,745)(52,97,175,310,219,359,540,744)(59,105,188,326,244,383,560,760)(60,108,184,328,245,382,561,762)(61,109,187,327,241,384,559,761)(62,106,186,78,246,320,562,321)(63,107,189,324,242,100,378,379)(64,110,185,80,243,329,179,316)(67,265,266,590,267,686,689,888,270,687,675,606,273,697,698,889,271,688,672,960,692,992,809,904,287,709,690,967,700,991,990,455)(69,257,262,608,98,366,367,464,280,669,684,892,370,806,678,893,282,670,685,969,274,701,702,894,281,664,260,682,371,808,683,474)(70,254,74,217,275,605,289,479,283,665,291,609,704,964,718,480,285,666,293,607,706,965,719,481,284,663,294,610,705,966,720,139)(75,295,272,433,297,601,347,533,300,724,344,786,733,210,88,168)(86,330,335,592,338,767,775,593,341,769,726,961,780,1011,998,202)(91,332,302,239,99,372,373,491,286,439,337,643,376,771,764,211)(112,196,397,458,398,581,836,459,401,510,824,486,408,529,849,461,402,582,821,890,839,550,945,492,422,862,837,909,851,928,1022,131)(120,423,206,598,424,863,462,842,425,865,231,634,434,872,463,843,426,867,599,875,869,943,498,860,440,866,635,907,868,1023,133,400)(129,444,448,674,452,699,877,988,454,268,656,258,469,343,785,671,456,693,878,673,883,368,810,677,478,694,903,676,896,691,993,989)(135,403,428,826,465,844,551,911,466,845,518,917,637,853,552,944,467,846,515,501,638,915,553,916,149,414,163,503,233,627,177,549)(158,389,506,777,507,820,882,840,508,822,636,626,511,828,891,841,509,825,918,838,920,1020,451,858,517,823,881,850,919,1019,460,399)(165,388,468,779,524,815,596,783,527,817,895,953,925,1017,204,340)(167,523,240,375,178,554,495,707,516,522,490,641,530,922,213,350)(190,566,567,884,568,946,948,885,569,658,288,649,710,923,711,887,570,947,713,980,811,805,716,899,847,985,715,979,955,1016,994,453)(200,574,583,301,588,374,813,734,589,348,532,303,644,377,555,313)(205,577,528,727,218,611,612,827,477,900,901,951,613,971,957,1021)(214,351,534,736,602,787,929,999)(216,353,536,322,604,763,770,765)(223,360,547,752,619,800,937,1003)(224,363,543,754,620,799,938,1005)(225,364,546,753,616,801,936,1004)(226,361,545,306,621,746,939,747)(227,362,548,750,617,355,795,796)(228,365,544,308,618,755,538,742)(247,385,565,738,645,380,812,814)(248,387,563,325,646,766,557,759)(249,386,564,323,647,381,558,314)(251,334,653,578,654,773,981,695,655,714,968,639,659,732,986,696,657,712,982,949,984,756,880,708,668,774,987,977,983,995,886,269)(296,648,407,848,725,962,781,927,728,978,778,1012,996,594,339,526)(342,768,730,614,354,789,790,898,421,861,776,972,793,1010,1006,595)(525,819,615,792,537,931,902,859,667,818,897,970,924,1018,597,784)(587,857,956,729,958,791,1014,997,959,782,926,731,973,794,932,739)(603,788,930,748,963,905,1009,1007)(622,802,942,1000,974,797,1013,1015)(623,804,940,751,975,1008,934,1002)(624,803,941,749,976,798,935,740);(1,3,5,23)(2,12,14,67,17,69,88,268,19,72,91,270,106,281,348,693,20,73,90,271,107,282,347,694,110,71,286,287,386,280,574,444)(4,31,32,158,35,160,167,508,37,162,169,509,185,161,516,517)(6,24,41,129)(7,53,54,231,55,232,238,635)(8,26,43,131)(9,28,45,133)(10,30,46,132)(11,29,44,134)(13,74,75,288,78,290,298,712,80,291,301,567,81,292,302,714,82,293,300,715,323,390,85,334,83,294,303,713,324,723,337,774)(15,21,86,112,94,115,339,399,96,118,342,401,361,416,782,840,97,119,341,402,362,417,781,841,365,117,421,422,803,415,857,858)(16,98,99,368,100,276,374,267,102,371,346,691,381,68,272,273,104,274,376,699,103,277,377,692,329,370,87,343,320,278,344,700)(18,70,89,269,105,279,195,569,108,284,350,695,353,406,194,570,109,285,349,696,385,405,493,847,387,283,707,708,92,113,39,190)(22,84,120,188,123,332,425,536,125,333,426,565,124,439,440,171)(25,142,143,246,144,482,485,647)(27,152,153,64)(33,65,165,251,173,253,525,655,175,256,527,657,544,255,667,668)(34,177,178,550,179,551,166,529,181,552,530,581,183,553,531,928)(36,126,168,427,184,438,532,450,187,442,533,497,563,441,583,128)(38,163,170,510,186,515,199,582,189,518,494,862,564,428,40,196)(42,206,58,235,207,599,243,234)(47,135,200,451)(48,217,218,606,219,607,613,967)(49,137,202,453)(50,139,204,455)(51,141,205,454)(52,140,203,456)(56,233,239,636,241,637,643,881)(57,147,240,457,242,146,490,130)(59,149,210,460)(60,151,211,459)(61,150,209,461)(62,156,213,462)(63,155,212,463)(66,164,258,328,260,520,672,738,261,522,674,79,262,299,675,322,263,521,676,325,684,191,266,326,264,523,673,327,685,573,690,304)(76,157,296,499,306,501,726,867,308,502,729,443,309,503,730,865,310,504,728,874,749,826,335,423,311,505,731,864,750,917,776,866)(77,312,313,756,314,289,373,716,316,717,297,732,379,718,336,711,318,757,734,773,321,719,764,948,319,722,733,995,317,720,735,994)(93,229,354,626,355,411,791,398,357,629,780,838,798,114,407,408,359,409,793,850,358,412,794,839,755,628,338,777,746,413,778,851)(95,116,340,400,360,414,577,446,363,419,784,842,788,846,576,447,364,420,783,843,802,845,900,496,804,418,859,860,351,403,192,127)(101,369,375,811,380,704,193,578,382,721,579,955,352,275,236,639,384,703,641,568,383,705,580,949,766,404,345,710,763,706,642,977)(111,388,389,754,392,816,821,1000,393,818,823,307,394,727,824,748,395,817,825,751,834,571,397,752,396,819,822,753,835,951,837,736)(121,330,265,547,430,768,687,930,432,769,688,942,431,861,709,534)(122,433,434,559,435,813,424,561,436,786,868,180,437,555,869,557)(136,468,469,621,470,895,896,976)(138,477,478,228)(145,295,486,604,487,724,909,215)(148,491,492,248)(154,495,498,249)(159,237,511,378,512,198,507,562,513,586,919,182,514,556,920,558)(172,366,537,805,538,806,524,923,540,701,924,946,542,808,925,1016)(174,254,526,656,543,663,926,878,546,666,927,903,940,665,956,448)(176,257,528,658,545,664,585,947,548,670,901,985,941,669,197,566)(201,590,222,610,591,960,618,609)(208,506,245,638,600,918,646,465)(214,464,587,880)(216,467,589,882)(220,608,614,968,616,969,972,987)(221,472,615,883,617,471,897,452)(223,474,594,886)(224,476,595,885)(225,475,593,887)(226,481,597,888)(227,480,596,889)(230,372,630,812,631,331,484,770,632,771,449,560,633,772,879,535)(244,489,644,890,645,488,588,458)(247,466,601,891)(250,648,649,963,650,978,979,602)(252,612,659,795,660,584,654,939,661,957,983,541,662,933,984,935)(259,640,677,814,678,519,698,761,679,572,671,765,702,922,689,762,680,952,988,760,683,921,990,315,681,758,989,737,367,554,809,759)(305,549,739,943,740,500,790,910,742,911,725,872,796,912,775,906,744,944,997,863,747,913,1006,873,745,916,996,1023,743,914,998,1024)(356,627,792,876,797,854,575,598,799,915,953,908,787,410,611,634,801,853,970,445,800,855,954,875,1008,844,779,483,905,856,971,907)(391,827,828,1015,829,815,849,1004,830,950,820,1007,852,1018,836,1005,831,1021,1019,1003,833,1017,1022,741,832,1001,1020,999,625,931,945,1002)(429,848,697,936,870,1014,686,938,871,1012,991,539,807,932,992,934)(473,898,899,623)(479,902,904,624)(592,653,620,682,961,982,975,892)(603,894,959,981)(605,789,810,1013,964,767,785,1009,965,1010,877,937,966,1011,993,929)(619,652,973,980,974,651,958,884)(622,893,962,986)

Polycyclic presentation:

2^15|a^2=c,b^2=d,c^2=f,d^2=g,e^2=hkno,f^2=i,g^2=j,h^2=k,i^2=l,j^2=m,k^2=n,n^2=o,b^a=be,c^b=ch,d^a=dhkno,d^c=dkno,f^b=fk,f^d=fn,g^a=gkno,g^c=gno,g^f=go,i^b=in,i^d=io,j^a=jno,j^c=jo,l^b=lo,m^a=mo

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m", "n", "o");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.11*F.14*F.15);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.12);;
gap> SetPower(coll, 10, F.13);;
gap> SetPower(coll, 11, F.14);;
gap> SetPower(coll, 14, F.15);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.11*F.14*F.15);;
gap> SetConjugate(coll, 4, 3, F.4*F.11*F.14*F.15);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.14);;
gap> SetConjugate(coll, 7, 1, F.7*F.11*F.14*F.15);;
gap> SetConjugate(coll, 7, 3, F.7*F.14*F.15);;
gap> SetConjugate(coll, 7, 6, F.7*F.15);;
gap> SetConjugate(coll, 9, 2, F.9*F.14);;
gap> SetConjugate(coll, 9, 4, F.9*F.15);;
gap> SetConjugate(coll, 10, 1, F.10*F.14*F.15);;
gap> SetConjugate(coll, 10, 3, F.10*F.15);;
gap> SetConjugate(coll, 12, 2, F.12*F.15);;
gap> SetConjugate(coll, 13, 1, F.13*F.15);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);