Finite Group Databasebeta

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C₅⋊(D₁₂₁₃⋊C₃) #49138605

• metacyclic
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

Splittings

Subgroups

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 1218 points:

(2,3)(4,5)(7,11,12,8,9,13)(10,19,20,14,15,29)(16,32,33,22,23,46)(17,31,27,25,21,18)(24,51,52,35,36,69)(26,49,50,38,34,43)(28,30,48,40,41,44)(37,74,75,54,55,98)(39,72,73,57,53,66)(42,47,71,59,60,67)(45,87,85,63,64,62)(56,103,104,77,78,133)(58,101,102,80,76,95)(61,70,100,82,83,96)(65,116,117,88,89,91)(68,122,114,92,93,86)(79,138,139,106,107,174)(81,136,137,109,105,130)(84,99,135,111,112,131)(90,151,152,118,119,126)(94,158,156,123,124,121)(97,163,149,127,128,115)(108,179,180,141,142,221)(110,177,178,144,140,171)(113,134,176,146,147,172)(120,192,193,153,154,167)(125,199,200,159,160,162)(129,205,197,164,165,157)(132,210,190,168,169,150)(143,226,227,182,183,274)(145,224,225,185,181,218)(148,175,223,187,188,219)(155,239,240,194,195,214)(161,246,247,201,202,209)(166,253,251,206,207,204)(170,258,244,211,212,198)(173,263,237,215,216,191)(184,279,280,229,230,333)(186,277,278,232,228,271)(189,222,276,234,235,272)(196,292,293,241,242,267)(203,299,300,248,249,262)(208,306,307,254,255,257)(213,312,304,259,260,252)(217,317,297,264,265,245)(220,322,290,268,269,238)(231,338,339,282,283,398)(233,336,337,285,281,330)(236,275,335,287,288,331)(243,351,352,294,295,326)(250,358,359,301,302,321)(256,365,366,308,309,316)(261,372,370,313,314,311)(266,377,363,318,319,305)(270,382,356,323,324,298)(273,387,349,327,328,291)(284,403,404,341,342,469)(286,401,402,344,340,395)(289,334,400,346,347,396)(296,416,417,353,354,391)(303,423,424,360,361,386)(310,430,431,367,368,381)(315,437,438,373,374,376)(320,443,435,378,379,371)(325,448,428,383,384,364)(329,453,421,388,389,357)(332,458,414,392,393,350)(343,474,475,406,407,546)(345,472,473,409,405,466)(348,399,471,411,412,467)(355,487,488,418,419,462)(362,494,495,425,426,457)(369,501,502,432,433,452)(375,508,509,439,440,447)(380,515,513,444,445,442)(385,520,506,449,450,436)(390,525,499,454,455,429)(394,530,492,459,460,422)(397,535,485,463,464,415)(408,551,552,477,478,629)(410,549,550,480,476,543)(413,470,548,482,483,544)(420,564,565,489,490,539)(427,571,572,496,497,534)(434,578,579,503,504,529)(441,585,586,510,511,524)(446,592,593,516,517,519)(451,598,590,521,522,514)(456,603,583,526,527,507)(461,608,576,531,532,500)(465,613,569,536,537,493)(468,618,562,540,541,486)(479,634,635,554,555,718)(481,632,633,557,553,626)(484,547,631,559,560,627)(491,647,648,566,567,622)(498,654,655,573,574,617)(505,661,662,580,581,612)(512,668,669,587,588,607)(518,675,676,594,595,602)(523,682,680,599,600,597)(528,687,673,604,605,591)(533,692,666,609,610,584)(538,697,659,614,615,577)(542,702,652,619,620,570)(545,707,645,623,624,563)(556,723,724,637,638,813)(558,721,722,640,636,715)(561,630,720,642,643,716)(568,736,737,649,650,711)(575,743,744,656,657,706)(582,750,751,663,664,701)(589,757,758,670,671,696)(596,764,765,677,678,691)(601,771,772,683,684,686)(606,777,769,688,689,681)(611,782,762,693,694,674)(616,787,755,698,699,667)(621,792,748,703,704,660)(625,797,741,708,709,653)(628,802,734,712,713,646)(639,818,819,726,727,914)(641,816,817,729,725,810)(644,719,815,731,732,811)(651,831,832,738,739,806)(658,838,839,745,746,801)(665,845,846,752,753,796)(672,852,853,759,760,791)(679,859,860,766,767,786)(685,866,867,773,774,781)(690,873,871,778,779,776)(695,878,864,783,784,770)(700,883,857,788,789,763)(705,888,850,793,794,756)(710,893,843,798,799,749)(714,898,836,803,804,742)(717,903,829,807,808,735)(728,919,920,821,822,1021)(730,917,918,824,820,911)(733,814,916,826,827,912)(740,932,933,833,834,907)(747,939,940,840,841,902)(754,946,947,847,848,897)(761,953,954,854,855,892)(768,960,961,861,862,887)(775,967,968,868,869,882)(780,974,975,874,875,877)(785,980,972,879,880,872)(790,985,965,884,885,865)(795,990,958,889,890,858)(800,995,951,894,895,851)(805,1000,944,899,900,844)(809,1005,937,904,905,837)(812,1010,930,908,909,830)(823,1026,1027,922,923,1121)(825,1024,1025,925,921,1018)(828,915,1023,927,928,1019)(835,1038,1039,934,935,1014)(842,1045,1046,941,942,1009)(849,1052,1053,948,949,1004)(856,1059,1060,955,956,999)(863,1066,1067,962,963,994)(870,1073,1030,969,970,989)(876,1079,1080,976,977,984)(881,1086,1084,981,982,979)(886,1090,1077,986,987,973)(891,1093,1071,991,992,966)(896,1098,1064,996,997,959)(901,1103,1057,1001,1002,952)(906,1108,1050,1006,1007,945)(910,1112,1043,1011,1012,938)(913,1115,1036,1015,1016,931)(924,1081,1082,1029,978,1164)(926,1123,1124,1031,1028,1119)(929,1022,1122,1033,1034,1120)(936,1129,1130,1040,1041,1117)(943,1136,1137,1047,1048,1114)(950,1143,1144,1054,1055,1111)(957,1150,1151,1061,1062,1107)(964,1155,1156,1068,1069,1102)(971,1160,1126,1074,1075,1097)(983,1167,1035,1087,1088,1089)(988,1168,1128,1091,1092,1085)(993,1169,1135,1094,1095,1078)(998,1171,1142,1099,1100,1072)(1003,1174,1149,1104,1105,1065)(1008,1177,1148,1109,1110,1058)(1013,1175,1141,1106,1113,1051)(1017,1172,1134,1101,1116,1044)(1020,1170,1127,1096,1118,1037)(1032,1165,1166,1125,1083,1182)(1042,1183,1184,1131,1132,1181)(1049,1185,1186,1138,1139,1180)(1056,1189,1190,1145,1146,1179)(1063,1194,1195,1152,1153,1178)(1070,1199,1188,1157,1158,1176)(1076,1203,1133,1161,1162,1173)(1140,1204,1205,1187,1163,1209)(1147,1210,1211,1191,1192,1208)(1154,1212,1213,1196,1197,1207)(1159,1215,1193,1200,1201,1206)(1198,1216,1217,1214,1202,1218);(1,2,4,5,3)(6,7,10,16,24,37,56,79,108,143,184,231,284,343,408,479,556,639,728,823,924,1030,961,853,751,655,565,473,339,400,458,453,448,443,437,430,423,416,401,338,399,468,542,621,705,795,891,993,1096,928,727,820,834,841,848,855,862,869,875,880,885,890,895,900,905,909,912,914,1018,1117,1180,1208,1218,1216,1200,1157,1068,962,861,766,677,594,516,444,378,318,264,215,147,78,105,119,124,128,131,133,171,214,262,316,376,442,514,591,674,763,858,959,1065,1148,1057,951,850,755,666,583,506,435,370,307,247,193,137,75,100,122,116,101,74,99,132,170,213,261,315,375,441,512,589,672,761,856,957,1063,1154,1198,1213,1190,1137,1039,918,724,815,903,898,893,888,883,878,873,866,859,852,845,838,831,816,723,814,913,1017,1099,996,894,798,708,623,483,342,405,419,426,433,440,445,450,455,460,464,467,469,543,622,706,796,892,994,1097,1125,1031,925,824,729,640,557,480,409,344,285,232,185,144,109,80,57,38,25,9,13,18,28,42,61,84,113,148,189,236,289,348,413,484,561,644,733,828,929,1035,1080,968,860,758,662,572,488,402,280,335,387,382,377,372,365,358,351,336,279,334,397,465,538,616,700,790,886,988,1034,822,921,935,942,949,956,963,970,977,982,987,992,997,1002,1007,1012,1016,1019,1021,1119,1181,1209,1215,1212,1210,1204,1161,1074,969,868,773,683,599,521,449,383,323,268,188,107,140,154,160,165,169,172,174,218,267,321,381,447,519,597,681,770,865,966,1072,1141,1050,944,843,748,659,576,499,428,363,304,251,200,152,102,52,71,87,72,51,70,97,129,166,208,256,310,369,434,505,582,665,754,849,950,1056,1147,1193,1195,1144,1046,933,817,635,720,802,797,792,787,782,777,771,764,757,750,743,736,721,634,719,812,910,1013,1104,1001,899,803,712,560,407,476,490,497,504,511,517,522,527,532,537,541,544,546,626,711,801,897,999,1102,1173,1131,1040,934,833,738,649,566,489,418,353,294,241,194,153,118,88,63,41,15,21,29,43,62,86,115,150,191,238,291,350,415,486,563,646,735,830,931,1037,1128,1084,975,867,765,669,579,495,417,337,227,276,322,317,312,306,299,292,277,226,275,332,394,461,533,611,695,785,881,983,923,1028,1041,1048,1055,1062,1069,1075,1082,1088,1092,1095,1100,1105,1110,1113,1116,1118,1120,1121,1182,1203,1199,1194,1189,1185,1183,1165,1081,976,874,778,688,604,526,454,388,327,235,142,181,195,202,207,212,216,219,221,271,326,386,452,524,602,686,776,872,973,1078,1134,1043,937,836,741,652,569,492,421,356,297,244,197,156,117,73,33,48,49,32,47,68,94,125,161,203,250,303,362,427,498,575,658,747,842,943,1049,1140,1188,1151,1053,940,832,722,552,631,707,702,697,692,687,682,675,668,661,654,647,632,551,630,717,809,906,1008,1109,1006,904,807,643,478,553,567,574,581,588,595,600,605,610,615,620,624,627,629,715,806,902,1004,1107,1176,1187,1138,1047,941,840,745,656,573,496,425,360,301,248,201,159,123,92,60,23,34,44,46,66,91,121,157,198,245,298,357,422,493,570,653,742,837,938,1044,1135,1077,972,871,772,676,586,502,424,352,278,180,223,263,258,253,246,239,224,179,222,273,329,390,456,528,606,690,780,876,978,1083,1132,1139,1146,1153,1158,1162,1166,1027,1122,1170,1172,1175,1177,1174,1171,1169,1168,1167,1164,1160,1155,1150,1143,1136,1129,1123,1026,1087,981,879,783,693,609,531,459,392,288,183,228,242,249,255,260,265,269,272,274,330,391,457,529,607,691,781,877,979,1085,1127,1036,930,829,734,645,562,485,414,349,290,237,190,149,114,85,50,20,31,19,30,45,65,90,120,155,196,243,296,355,420,491,568,651,740,835,936,1042,1133,1156,1060,947,839,737,633,475,548,618,613,608,603,598,592,585,578,571,564,549,474,547,628,714,805,901,1003,1106,1011,908,732,555,636,650,657,664,671,678,684,689,694,699,704,709,713,716,718,810,907,1009,1111,1178,1206,1191,1145,1054,948,847,752,663,580,503,432,367,308,254,206,164,127,83,36,53,64,67,69,95,126,162,204,252,305,364,429,500,577,660,749,844,945,1051,1142,1071,965,864,769,680,593,509,431,359,293,225,139,176,210,205,199,192,177,138,175,220,270,325,385,451,523,601,685,775,870,971,1076,1163,1192,1197,1201,1205,1184,1124,920,1023,1115,1112,1108,1103,1098,1093,1090,1086,1079,1073,1066,1059,1052,1045,1038,1024,919,1022,1091,986,884,788,698,614,536,463,347,230,281,295,302,309,314,319,324,328,331,333,395,462,534,612,696,786,882,984,1089,1033,927,826,731,642,559,482,411,346,287,234,187,146,111,82,59,40,27,12,11,17,26,39,58,81,110,145,186,233,286,345,410,481,558,641,730,825,926,1032,1126,1067,954,846,744,648,550,404,471,535,530,525,520,515,508,501,494,487,472,403,470,545,625,710,800,896,998,1101,1015,827,638,725,739,746,753,760,767,774,779,784,789,794,799,804,808,811,813,911,1014,1114,1179,1207,1214,1196,1152,1061,955,854,759,670,587,510,439,373,313,259,211,168,112,55,76,89,93,96,98,130,167,209,257,311,371,436,507,584,667,756,851,952,1058,1149,1064,958,857,762,673,590,513,438,366,300,240,178,104,135,163,158,151,136,103,134,173,217,266,320,380,446,518,596,679,768,863,964,1070,1159,1202,1217,1211,1186,1130,1025,819,916,1010,1005,1000,995,990,985,980,974,967,960,953,946,939,932,917,818,915,1020,1094,991,889,793,703,619,540,412,283,340,354,361,368,374,379,384,389,393,396,398,466,539,617,701,791,887,989,1029,922,821,726,637,554,477,406,341,282,229,182,141,106,77,54,35,22,14,8)

Polycyclic presentation:

2,3,5,1213|c^a=c^4,d^a=d^1212,d^b=d^217

Conjugacy classes & character table

1022 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36390, 12);