Finite Group DatabasebetaFinite Group Databasebeta| order | 31530 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D1051⋊C5) |
| unambiguous semidirect product | (C1051⋊C5)⋊D3 |
| unambiguous semidirect product | D3153⋊C5 |
| Fitting, socle, derived, Hall | C3×C1051 ⋊ C2×C5 |
| Hall | C3×C1051⋊C5 ⋊ C2 |
| Hall | D3153 ⋊ C5 |
| Hall | C1051⋊C5 ⋊ D3 |
| Sylow | C1051 ⋊ C5×D3 |
| Sylow | C3 ⋊ D1051⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C1051 | C2×C5 |
| LCS | G ▷ C3×C1051 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1054 |
| transitive degree | μt(G) = 3153 |
A minimal faithful permutation representation — 2 generators on 1054 points:
(2,3)(5,9,10,28,33,6,7,15,17,57)(8,22,23,90,95,12,13,44,46,175)(11,38,39,40,68,19,20,78,80,82)(14,51,52,209,214,25,26,106,108,411)(16,62,63,257,238,30,31,129,131,119)(18,72,73,69,70,35,36,74,34,81)(21,84,85,268,186,41,42,162,164,367)(24,100,101,385,142,48,49,196,198,285)(27,113,114,442,444,54,55,225,227,532)(29,123,124,58,240,59,60,245,120,121)(32,136,137,343,495,65,66,272,274,344)(37,148,149,187,188,75,76,296,96,369)(43,169,170,563,421,87,88,331,333,683)(45,180,181,478,457,92,93,354,356,239)(47,190,191,143,144,97,98,373,71,287)(50,203,204,405,368,103,104,399,401,619)(53,219,220,698,286,110,111,431,433,514)(56,232,233,577,600,116,117,451,355,835)(61,251,252,176,585,126,127,470,345,346)(64,266,267,83,317,133,134,370,159,160)(67,279,280,775,646,139,140,503,505,667)(77,302,303,422,423,151,152,526,215,685)(79,310,102,394,395,155,156,200,201,241)(86,327,248,249,154,166,167,125,390,308)(89,338,339,832,531,172,173,575,576,307)(91,348,349,122,460,177,178,590,242,243)(94,361,362,863,743,183,184,607,609,584)(99,379,380,371,372,193,194,629,189,622)(105,406,407,912,684,206,207,656,657,518)(107,416,417,511,512,211,212,677,284,458)(109,425,426,288,289,216,217,690,145,516)(112,438,439,961,620,222,223,708,709,875)(115,447,448,915,515,229,230,658,719,784)(118,290,453,228,496,235,236,465,446,275)(128,474,475,412,927,254,255,608,668,669)(130,298,376,377,315,259,260,192,262,158)(132,375,293,294,325,263,264,147,246,165)(135,487,488,486,839,269,270,757,754,755)(138,498,499,638,543,276,277,769,770,803)(141,509,332,428,429,282,283,565,218,694)(146,299,300,391,392,291,150,258,199,247)(153,208,530,662,717,305,306,409,445,920)(157,537,221,583,705,312,313,435,436,586)(161,544,358,359,686,318,319,182,567,603)(163,550,500,741,510,322,323,772,773,459)(168,559,535,205,621,328,329,311,403,404)(171,570,571,593,309,335,336,350,703,533)(174,581,480,481,857,341,342,261,744,745)(179,595,596,347,841,351,352,851,587,588)(185,613,614,842,389,364,365,838,811,645)(195,632,633,687,688,382,383,884,424,948)(197,642,643,647,508,387,388,897,898,461)(202,627,231,525,623,396,397,301,450,402)(210,671,672,244,502,413,414,932,462,463)(213,681,624,625,479,419,420,374,877,742)(224,529,666,704,517,304,441,434,592,292)(226,555,556,381,378,393,326,524,631,628)(234,519,568,569,601,295,452,334,357,691)(237,722,723,566,706,454,455,721,825,437)(250,730,548,489,846,467,468,321,759,760)(253,734,639,640,558,471,472,386,870,817)(256,541,492,493,427,477,316,273,528,692)(265,749,750,778,847,483,484,520,591,758)(271,714,715,821,727,490,443,967,894,464)(278,700,701,655,1016,501,432,957,910,901)(281,652,653,954,804,507,400,907,712,977)(297,786,787,763,728,521,522,599,771,466)(314,724,449,780,781,539,540,720,506,928)(320,774,678,679,971,545,546,418,834,942)(324,810,654,812,813,552,553,909,554,840)(330,819,808,702,947,560,561,551,914,791)(337,828,829,899,876,572,573,644,963,1026)(340,836,837,934,534,578,579,673,975,798)(353,746,747,670,779,598,482,831,929,930)(360,860,861,756,1017,604,605,1024,984,896)(363,865,366,871,805,610,611,616,617,1000)(384,889,890,972,973,635,636,1029,718,1021)(398,902,580,991,949,648,649,788,826,958)(408,916,917,880,536,659,660,761,959,800)(410,924,925,634,1023,664,665,792,887,888)(415,936,937,589,809,674,675,986,843,844)(430,951,748,562,950,695,696,979,689,820)(440,964,615,456,538,710,711,869,725,726)(469,867,822,823,1042,731,732,564,1004,1005)(473,976,906,661,1018,735,736,651,918,883)(476,868,955,956,818,738,739,699,707,783)(485,980,981,993,940,751,752,793,933,1025)(491,814,815,1012,693,762,557,978,886,547)(494,987,858,799,785,764,765,602,864,582)(497,513,542,504,797,766,767,782,776,777)(523,574,992,1010,966,789,790,830,908,848)(527,995,996,872,873,794,795,941,618,729)(549,968,969,594,960,806,716,1019,849,845)(597,911,895,866,827,853,854,641,612,1009)(606,1013,1014,892,893,862,833,1053,637,733)(626,802,965,796,919,878,879,713,998,852)(630,740,1022,990,945,881,882,856,962,682)(650,1036,923,1050,1020,903,904,997,1015,855)(663,1039,1040,753,801,921,922,1006,982,983)(676,1043,891,931,913,938,939,737,1032,1030)(680,1045,1046,985,1001,943,944,1051,989,1035)(697,1007,1008,1048,953,952,824,1052,974,850)(768,874,816,1011,1031,988,807,1002,1003,935)(859,1037,926,1049,1047,905,999,994,1041,1054)(885,970,1038,1033,1034,1027,1028,1044,900,946);(1,2,3)(4,5,8,14,27,56,118,237,456,329,249,385,638,871,1022,787,181,357,602,859,903,648,396,200,100,197,389,646,214,421,684,517,291,36,74,147,295,232,377,257,394,647,901,755,346,586,840,723,525,134,268,486,756,985,988,766,496,272,441,713,966,704,590,846,1018,881,194,381,629,782,541,315,70,78,154,309,534,799,863,812,552,322,42,86,168,330,562,687,422,209,412,670,931,1034,945,681,470,570,827,696,717,971,979,746,480,259,31,64,135,271,491,763,769,964,1000,319,243,461,507,279,504,778,178,350,594,850,952,695,428,216,106,210,415,676,940,1024,936,1041,815,481,522,788,578,338,372,623,399,651,904,752,879,726,537,516,519,127,253,473,737,889,1030,407,658,777,409,663,923,995,987,967,1039,944,1014,1054,1028,968,714,114,228,323,551,809,1001,1033,829,642,419,211,26,53,112,224,302,527,796,701,958,948,862,604,358,180,355,575,833,1008,1046,1040,1004,860,579,475,184,363,612,867,950,955,220,434,664,306,206,103,48,22,45,94,185,366,618,874,806,548,85,165,326,557,816,633,886,576,834,1015,969,596,852,918,659,406,280,506,505,779,893,731,467,248,123,246,465,540,722,877,749,902,1026,404,267,369,342,582,750,149,298,524,792,939,1005,672,657,914,700,111,221,437,707,960,843,587,345,46,96,189,283,510,395,240,317,543,804,773,356,477,254,126,59,10,18,37,77,153,307,532,694,865,573,472,643,709,963,976,992,362,217,427,693,545,318,159,80,38,79,157,314,426,691,784,514,285,82,57,119,239,458,628,376,190,374,626,880,593,698,954,710,438,277,500,719,975,982,751,483,263,129,258,479,743,495,95,186,368,620,397,264,373,236,288,143,69,28,58,122,244,464,669,928,909,652,104,205,405,655,911,1038,996,417,600,857,1023,873,728,392,245,327,558,818,767,511,282,139,65,30,7,11,21,43,89,174,343,583,838,561,468,640,869,613,868,991,359,563,821,892,635,382,193,97,44,91,179,353,423,686,947,789,521,76,150,301,229,113,226,391,196,386,641,896,588,842,810,720,447,270,489,761,986,1048,972,662,442,493,526,793,994,924,847,757,916,1027,383,634,882,738,474,233,144,156,311,536,801,999,1010,907,825,565,88,171,337,574,831,1011,993,414,673,935,1029,943,678,416,568,331,564,824,973,1020,853,595,764,492,66,138,278,502,774,990,439,708,962,546,463,501,276,136,273,494,352,597,855,1021,1007,822,170,334,212,418,680,890,1031,837,671,933,1003,747,790,572,335,169,332,566,653,908,1037,983,800,535,310,287,451,255,476,740,887,632,885,660,488,758,665,926,981,303,528,227,445,718,974,937,917,759,487,230,449,553,811,841,1017,895,639,101,199,393,55,115,231,299,148,297,523,791,603,685,598,351,177,23,47,99,195,384,637,894,333,567,826,739,365,615,870,730,819,614,436,274,341,172,87,41,19,9,16,32,67,141,284,513,783,817,167,124,247,466,729,888,745,835,677,941,1044,854,910,401,403,203,400,654,781,927,727,462,242,120,17,34,71,145,290,191,375,627,875,619,367,175,344,584,742,300,63,132,265,485,753,934,915,772,498,223,440,712,433,703,959,878,624,98,192,378,512,457,238,33,68,142,286,515,601,690,539,312,155,20,40,83,161,320,547,692,425,607,830,736,899,961,897,734,828,611,429,444,531,305,151,75,35,15,29,61,128,256,478,741,977,803,160,121,241,459,509,371,187,90,176,347,589,845,956,706,435,219,432,702,912,932,1042,1043,925,556,260,296,520,785,581,188,370,621,876,649,484,625,455,724,453,294,60,125,250,469,733,930,775,780,503,207,408,661,919,851,1019,1050,679,832,1012,884,1002,814,555,325,162,321,549,807,872,616,364,183,92,13,24,50,105,208,410,666,431,699,820,732,866,610,361,608,836,605,823,1006,1051,1052,1013,339,577,93,182,360,606,688,949,957,998,794,152,304,222,110,51,107,213,388,644,900,1035,844,808,550,446,225,443,716,970,1047,1053,1045,922,715,765,795,997,921,530,797,448,656,913,636,891,735,471,251,452,289,313,538,802,980,1036,906,204,402,622,173,340,580,786,744,978,1049,675,861,1025,938,674,413,52,109,218,430,697,953,849,571,348,591,776,140,281,508,460,544,805,711,499,771,762,490,269,133,62,130,261,482,748,942,920,951,1009,336,252,420,682,946,1032,929,668,108,215,424,689,560,328,166,84,163,324,554,609,864,798,533,308,39,81,158,316,542,380,631,379,630,883,760,349,592,848,965,529,137,275,497,768,989,984,754,164,266,450,721,813,705,585,839,1016,898,201,131,262,117,234,293,73,72,146,292,518,683,411,667,645,387,49,102,202,398,650,905,858,569,354,599,856,617,770,198,390,559,725,454,235,116,54,25,12,6)Polycyclic presentation:
2,3,5,1051|b^a=b^2,d^a=d^1050,d^c=d^413330 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31530, 14);