Finite Group DatabasebetaFinite Group Databasebeta| order | 30930 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D1031⋊C5) |
| unambiguous semidirect product | (C1031⋊C5)⋊D3 |
| unambiguous semidirect product | D3093⋊C5 |
| Fitting, socle, derived, Hall | C3×C1031 ⋊ C2×C5 |
| Hall | C3×C1031⋊C5 ⋊ C2 |
| Hall | D3093 ⋊ C5 |
| Hall | C1031⋊C5 ⋊ D3 |
| Sylow | C1031 ⋊ C5×D3 |
| Sylow | C3 ⋊ D1031⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C1031 | C2×C5 |
| LCS | G ▷ C3×C1031 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1034 |
| transitive degree | μt(G) = 3093 |
A minimal faithful permutation representation — 2 generators on 1034 points:
(2,3)(5,9,10,28,33,6,7,15,17,56)(8,22,23,89,94,12,13,44,46,169)(11,38,39,40,67,19,20,77,79,81)(14,51,52,201,205,25,26,105,107,377)(16,61,62,237,219,30,31,124,126,114)(18,71,72,68,69,35,36,73,34,80)(21,83,84,248,178,41,42,157,159,334)(24,99,100,352,137,48,49,188,190,265)(27,112,60,231,232,54,55,121,122,417)(29,118,119,57,221,58,59,225,115,116)(32,131,132,442,447,64,65,252,254,320)(37,143,144,179,180,74,75,276,95,336)(43,164,165,519,383,86,87,310,312,604)(45,172,173,537,407,91,92,322,324,220)(47,182,183,138,139,96,97,341,70,267)(50,195,196,371,335,102,103,365,367,552)(53,210,211,615,266,109,110,391,392,469)(63,246,247,82,296,128,129,337,154,155)(66,259,260,704,708,134,135,458,460,596)(76,282,283,333,384,146,147,485,206,551)(78,289,101,360,361,150,151,192,193,222)(85,306,228,229,149,161,162,120,356,287)(88,317,318,774,419,167,168,530,532,648)(90,264,321,117,410,170,171,467,223,224)(93,328,329,613,781,175,176,390,546,535)(98,346,347,338,339,185,186,565,181,556)(104,372,373,830,605,198,199,586,588,851)(106,355,379,819,639,202,203,576,598,408)(108,385,386,268,269,207,208,607,140,472)(111,397,398,867,553,213,214,624,625,802)(113,249,250,671,470,217,130,436,437,435)(123,423,424,378,473,234,235,654,270,597)(125,278,343,344,294,239,240,184,242,153)(127,342,273,274,304,243,244,142,226,160)(133,452,453,572,502,256,257,691,693,751)(136,465,288,314,315,262,263,433,166,526)(141,279,280,357,358,271,145,238,191,227)(148,491,492,420,421,285,286,741,233,650)(152,496,212,620,403,291,292,394,395,216)(156,503,326,327,606,297,298,174,523,544)(158,507,454,784,640,301,302,695,696,409)(163,515,494,197,555,307,308,290,369,370)(177,549,349,350,811,331,332,187,567,568)(189,573,574,504,505,353,354,816,299,411)(194,561,482,483,557,362,363,281,434,368)(200,593,594,872,649,375,376,843,829,891)(204,584,600,703,968,380,381,769,850,782)(209,610,611,558,559,388,389,855,340,806)(215,630,631,999,852,400,401,878,879,990)(218,455,456,912,803,405,258,698,699,585)(230,493,506,438,719,414,415,300,674,675)(236,543,275,418,313,325,426,474,475,524)(241,661,662,684,685,427,428,788,448,554)(245,319,669,720,721,431,432,477,468,673)(251,677,678,498,499,439,440,847,293,749)(253,487,603,609,500,444,445,387,542,295)(255,687,688,643,644,449,450,560,412,754)(261,711,712,861,752,462,463,924,925,904)(272,488,489,617,618,416,284,443,393,536)(277,726,727,785,646,478,479,538,694,413)(303,756,713,859,635,509,510,927,928,404)(305,481,566,562,563,512,513,348,345,359)(309,744,755,700,853,516,517,508,835,915)(311,764,765,983,887,520,521,823,961,641)(316,770,665,818,805,527,528,575,869,581)(323,786,787,666,667,539,540,689,430,471)(330,796,797,815,753,547,548,976,978,954)(351,592,814,651,652,570,571,841,422,893)(364,534,826,940,798,579,580,728,767,914)(366,827,629,971,888,582,583,875,876,642)(374,839,840,717,495,590,591,676,466,745)(382,834,759,760,973,601,602,514,900,812)(396,863,775,776,807,621,622,531,670,967)(399,873,874,947,497,627,628,742,762,747)(402,884,885,908,892,633,634,683,1004,1008)(406,714,715,1015,991,637,464,930,931,994)(425,848,501,459,743,657,658,616,705,706)(429,901,902,920,921,663,664,866,709,801)(441,905,564,966,525,679,680,808,809,768)(446,906,907,946,975,681,682,986,795,804)(451,710,854,645,938,461,690,810,722,838)(457,916,917,918,750,701,702,707,960,952)(476,655,533,599,656,608,725,778,779,849)(480,894,831,860,969,653,729,614,926,783)(484,672,589,697,660,731,732,836,837,898)(486,942,909,984,890,734,735,820,979,647)(490,692,626,817,763,738,739,870,871,518)(511,862,746,569,846,619,758,730,595,813)(522,910,789,790,858,686,766,541,972,636)(529,963,964,883,992,771,772,828,1001,1003)(545,864,865,844,939,793,623,923,792,723)(550,791,980,993,955,799,800,974,1030,922)(577,985,733,903,845,821,822,856,668,724)(578,956,737,857,777,824,825,929,612,911)(587,987,988,1032,1007,832,833,1013,1033,889)(632,1002,638,936,748,881,882,842,718,950)(659,1009,761,995,981,895,896,868,959,998)(716,1019,1020,1022,953,933,934,1034,937,1025)(736,1010,1000,935,1027,897,943,962,1021,970)(740,913,880,877,899,944,945,794,996,997)(757,1017,1018,948,949,957,932,1028,1011,1023)(773,941,977,1016,982,780,965,989,1029,1031)(886,919,1024,1026,951,1005,1006,1014,958,1012);(1,2,3)(4,5,8,14,27,46,95,181,340,475,122,233,422,653,478,75,145,281,484,733,389,612,858,635,403,55,113,218,406,638,571,590,372,587,834,785,691,873,890,646,358,225,306,492,743,837,695,452,692,764,559,807,583,828,800,789,539,92,174,110,212,396,623,866,591,437,673,758,958,934,931,1016,908,943,772,652,818,867,994,990,851,604,377,596,568,812,972,850,701,455,256,131,253,446,683,909,379,599,778,530,280,62,127,245,433,287,39,80,153,295,501,211,393,619,509,301,42,85,163,309,518,363,244,341,560,611,680,710,923,1007,888,505,410,503,753,955,920,704,677,903,666,344,237,360,504,297,154,79,38,78,152,293,312,523,767,939,667,139,151,290,495,746,513,240,276,477,424,656,822,986,1006,917,705,135,261,464,716,935,974,1005,884,921,384,606,853,992,1031,871,816,650,770,942,906,1013,1017,930,957,756,938,836,586,831,329,388,207,105,170,23,47,98,187,351,196,368,556,766,945,456,700,830,918,947,308,229,352,572,815,876,696,324,542,792,376,595,567,633,400,213,109,51,106,204,382,603,385,608,474,273,72,71,141,272,414,228,118,226,412,645,720,171,262,134,64,30,7,11,21,43,88,97,184,345,564,725,235,425,659,897,734,147,284,490,740,946,613,859,968,781,447,94,178,335,553,803,367,369,195,366,584,829,870,624,868,759,512,304,157,300,260,461,449,252,443,165,313,525,769,962,631,880,793,176,330,214,399,629,877,998,841,675,321,536,783,970,1018,797,979,1009,965,894,901,995,1034,1025,952,749,597,847,813,662,786,679,439,249,128,61,125,241,429,665,573,774,966,776,483,129,248,435,469,265,81,56,114,220,408,395,254,448,686,520,87,166,316,529,773,580,432,317,531,777,787,687,863,887,640,361,221,296,502,752,840,674,436,672,824,563,357,188,285,146,74,35,15,29,60,123,236,322,538,594,845,269,292,497,748,951,760,428,479,728,658,819,983,1032,1030,925,392,263,466,718,937,879,1001,881,630,799,549,798,912,1002,1008,891,648,417,526,283,487,737,927,711,755,507,754,956,833,989,548,765,821,203,353,49,101,194,364,581,370,247,336,554,804,702,712,926,1014,1023,517,415,491,742,948,997,898,543,607,854,347,566,346,375,198,102,48,22,45,93,177,333,201,378,268,138,68,28,57,117,223,115,17,34,70,140,270,107,206,331,175,91,13,24,50,104,200,186,348,565,810,386,426,660,899,1011,874,286,493,744,949,1024,860,924,916,975,685,180,337,555,805,579,362,192,99,189,355,577,823,796,977,987,825,644,302,508,463,713,929,445,485,315,232,419,649,892,882,699,914,332,550,401,632,883,999,1022,936,717,465,615,861,993,1033,961,598,848,826,726,661,900,1012,950,747,496,472,724,843,727,173,325,234,121,58,10,18,37,76,148,100,191,359,578,732,250,438,676,904,751,155,116,222,409,641,622,450,689,911,775,168,319,534,780,771,527,314,164,311,522,684,442,620,639,407,219,33,67,137,266,470,159,246,434,670,809,532,354,575,663,427,239,31,63,130,251,441,540,788,846,678,473,499,750,953,1020,959,664,729,941,896,984,976,1028,1027,969,618,467,719,814,981,996,875,627,397,547,328,545,794,878,1000,600,768,524,310,489,132,255,451,458,506,84,160,305,514,761,398,626,872,381,582,103,197,371,585,802,552,334,169,320,535,782,928,546,795,944,738,488,282,486,736,895,657,423,655,808,562,343,182,167,86,41,19,9,16,32,66,136,264,468,722,643,274,59,120,230,416,271,36,73,142,275,476,208,387,601,380,202,26,53,111,215,402,350,569,593,844,609,537,784,971,978,693,190,356,515,762,960,588,835,698,913,910,339,557,365,570,349,185,96,44,90,52,108,209,390,614,373,589,838,510,757,715,932,988,682,735,528,421,574,817,982,1003,915,544,551,801,634,886,980,1021,933,714,462,259,459,707,919,907,985,849,654,669,144,278,481,730,745,494,289,267,471,723,940,327,519,498,291,150,20,40,82,156,299,193,126,242,430,668,440,460,709,922,954,298,224,411,642,889,865,690,905,855,688,183,342,561,763,516,307,161,83,158,303,511,617,391,616,500,294,69,77,149,288,431,243,124,238,318,533,779,576,820,885,681,444,65,133,258,457,703,790,973,811,708,205,383,605,852,991,625,869,893,963,1010,1004,1029,1015,1019,1026,862,721,671,839,902,864,621,394,210,326,172,323,541,791,964,827,967,806,521,739,257,454,697,706,741,162,119,227,413,647,628,453,694,602,832,199,374,592,842,637,405,217,112,216,404,636,857,610,856,731,482,279,143,277,480,651,420,231,418,558,338,179,89,54,25,12,6)Polycyclic presentation:
2,3,5,1031|b^a=b^2,d^a=d^1030,d^c=d^264324 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30930, 12);