Finite Group DatabasebetaFinite Group Databasebeta| order | 20058 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | D3343⋊C3 |
| unambiguous semidirect product | (C3343⋊C3)⋊C2 |
| Fitting, socle, derived, Sylow | C3343 ⋊ C2×C3 |
| Hall | C3343⋊C3 ⋊ C2 |
| Hall | D3343 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3343 | C2×C3 |
| LCS | G ▷ C3343 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 3343 |
| transitive degree | μt(G) = 3343 |
A minimal faithful permutation representation — 2 generators on 3343 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,128)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,111,112,83,84,86)(63,117,109,87,88,81)(74,133,134,101,102,169)(76,131,132,104,100,125)(79,94,130,106,107,126)(85,146,147,113,114,121)(89,153,151,118,119,116)(92,158,144,122,123,110)(103,174,175,136,137,216)(105,172,173,139,135,166)(108,129,171,141,142,167)(115,187,188,148,149,162)(120,194,195,154,155,157)(124,200,192,159,160,152)(127,205,185,163,164,145)(138,221,222,177,178,269)(140,219,220,180,176,213)(143,170,218,182,183,214)(150,234,235,189,190,209)(156,241,242,196,197,204)(161,248,246,201,202,199)(165,253,239,206,207,193)(168,258,232,210,211,186)(179,274,275,224,225,328)(181,272,273,227,223,266)(184,217,271,229,230,267)(191,287,288,236,237,262)(198,294,295,243,244,257)(203,301,302,249,250,252)(208,307,299,254,255,247)(212,312,292,259,260,240)(215,317,285,263,264,233)(226,333,334,277,278,393)(228,331,332,280,276,325)(231,270,330,282,283,326)(238,346,347,289,290,321)(245,353,354,296,297,316)(251,360,361,303,304,311)(256,367,365,308,309,306)(261,372,358,313,314,300)(265,377,351,318,319,293)(268,382,344,322,323,286)(279,398,399,336,337,464)(281,396,397,339,335,390)(284,329,395,341,342,391)(291,411,412,348,349,386)(298,418,419,355,356,381)(305,425,426,362,363,376)(310,432,433,368,369,371)(315,438,430,373,374,366)(320,443,423,378,379,359)(324,448,416,383,384,352)(327,453,409,387,388,345)(338,469,470,401,402,541)(340,467,468,404,400,461)(343,394,466,406,407,462)(350,482,483,413,414,457)(357,489,490,420,421,452)(364,496,497,427,428,447)(370,503,504,434,435,442)(375,510,508,439,440,437)(380,515,501,444,445,431)(385,520,494,449,450,424)(389,525,487,454,455,417)(392,530,480,458,459,410)(403,546,547,472,473,624)(405,544,545,475,471,538)(408,465,543,477,478,539)(415,559,560,484,485,534)(422,566,567,491,492,529)(429,573,574,498,499,524)(436,580,581,505,506,519)(441,587,588,511,512,514)(446,593,585,516,517,509)(451,598,578,521,522,502)(456,603,571,526,527,495)(460,608,564,531,532,488)(463,613,557,535,536,481)(474,629,630,549,550,713)(476,627,628,552,548,621)(479,542,626,554,555,622)(486,642,643,561,562,617)(493,649,650,568,569,612)(500,656,657,575,576,607)(507,663,664,582,583,602)(513,670,671,589,590,597)(518,677,675,594,595,592)(523,682,668,599,600,586)(528,687,661,604,605,579)(533,692,654,609,610,572)(537,697,647,614,615,565)(540,702,640,618,619,558)(551,718,719,632,633,808)(553,716,717,635,631,710)(556,625,715,637,638,711)(563,731,732,644,645,706)(570,738,739,651,652,701)(577,745,746,658,659,696)(584,752,753,665,666,691)(591,759,760,672,673,686)(596,766,767,678,679,681)(601,772,764,683,684,676)(606,777,757,688,689,669)(611,782,750,693,694,662)(616,787,743,698,699,655)(620,792,736,703,704,648)(623,797,729,707,708,641)(634,813,814,721,722,909)(636,811,812,724,720,805)(639,714,810,726,727,806)(646,826,827,733,734,801)(653,833,834,740,741,796)(660,840,841,747,748,791)(667,847,848,754,755,786)(674,854,855,761,762,781)(680,861,862,768,769,776)(685,868,866,773,774,771)(690,873,859,778,779,765)(695,878,852,783,784,758)(700,883,845,788,789,751)(705,888,838,793,794,744)(709,893,831,798,799,737)(712,898,824,802,803,730)(723,914,915,816,817,1016)(725,912,913,819,815,906)(728,809,911,821,822,907)(735,927,928,828,829,902)(742,934,935,835,836,897)(749,941,942,842,843,892)(756,948,949,849,850,887)(763,955,956,856,857,882)(770,962,963,863,864,877)(775,969,970,869,870,872)(780,975,967,874,875,867)(785,980,960,879,880,860)(790,985,953,884,885,853)(795,990,946,889,890,846)(800,995,939,894,895,839)(804,1000,932,899,900,832)(807,1005,925,903,904,825)(818,1021,1022,917,918,1129)(820,1019,1020,920,916,1013)(823,910,1018,922,923,1014)(830,1034,1035,929,930,1009)(837,1041,1042,936,937,1004)(844,1048,1049,943,944,999)(851,1055,1056,950,951,994)(858,1062,1063,957,958,989)(865,1069,1070,964,965,984)(871,1076,1077,971,972,979)(876,1083,1081,976,977,974)(881,1088,1074,981,982,968)(886,1093,1067,986,987,961)(891,1098,1060,991,992,954)(896,1103,1053,996,997,947)(901,1108,1046,1001,1002,940)(905,1113,1039,1006,1007,933)(908,1118,1032,1010,1011,926)(919,1134,1135,1024,1025,1248)(921,1132,1133,1027,1023,1126)(924,1017,1131,1029,1030,1127)(931,1147,1148,1036,1037,1122)(938,1154,1155,1043,1044,1117)(945,1161,1162,1050,1051,1112)(952,1168,1169,1057,1058,1107)(959,1175,1176,1064,1065,1102)(966,1182,1183,1071,1072,1097)(973,1189,1190,1078,1079,1092)(978,1196,1197,1084,1085,1087)(983,1202,1194,1089,1090,1082)(988,1207,1187,1094,1095,1075)(993,1212,1180,1099,1100,1068)(998,1217,1173,1104,1105,1061)(1003,1222,1166,1109,1110,1054)(1008,1227,1159,1114,1115,1047)(1012,1232,1152,1119,1120,1040)(1015,1237,1145,1123,1124,1033)(1026,1253,1254,1137,1138,1373)(1028,1251,1252,1140,1136,1245)(1031,1130,1250,1142,1143,1246)(1038,1266,1267,1149,1150,1241)(1045,1273,1274,1156,1157,1236)(1052,1280,1281,1163,1164,12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427,2260,2099,1944,1795,1652,1515,1384,1259,1140,1027,920,819,724,635,552,475,404,339,280,227,180,139,104,75,52,33,20,4,8,13,23,37,56,79,108,143,184,231,284,343,408,479,556,639,728,823,924,1031,1144,1263,1388,1519,1656,1799,1948,2103,2264,2431,2602,2641,2478,2311,2150,1995,1846,1703,1566,1435,1310,1191,1078,971,869,773,683,599,521,449,383,322,230,137,176,190,197,202,207,211,214,216,266,321,381,447,519,597,681,771,867,968,1075,1188,1307,1432,1563,1700,1843,1992,2147,2308,2475,2638,2788,2914,3027,3122,3143,3058,2955,2836,2688,2524,2356,2194,2038,1888,1744,1606,1474,1348,1228,1114,1006,903,727,550,631,645,652,659,666,673,679,684,689,694,699,704,708,711,713,805,902,1004,1112,1226,1346,1472,1604,1742,1886,2036,2192,2354,2522,2686,2834,2952,3055,3140,3213,3272,3313,3336,3326,3295,3246,3179,3094,2990,2871,2723,2559,2391,2229,2073,1923,1778,1518,1257,1380,1394,1401,1408,1415,1422,1429,1436,1443,1450,1455,1460,1465,1470,1475,1480,1485,1490,1495,1499,1502,1504,1638,1777,1921,2071,2227,2389,2557,2721,2869,2988,3092,3113,3109,3105,3101,3097,3093,3088,3083,3078,3073,3068,3063,3056,3049,3042,3038,3034,3018,2899,2601,2258,2423,2437,2444,2451,2458,2465,2472,2479,2486,2493,2500,2507,2514,2520,2525,2530,2535,2540,2545,2550,2555,2560,2565,2570,2575,2580,2584,2587,2589,2755,2780,2620,2442,2268,2092,1790,1935,2077,2072,2067,2062,2057,2052,2047,2042,2037,2032,2027,2021,2014,2007,2000,1993,1986,1979,1972,1965,1958,1951,1936,1789,1934,2087,2245,2408,2576,2740,2886,3005,3108,3193,3260,3284,3288,3292,3296,3300,3304,3307,3278,3219,3142,3050,2940,2815,2661,2491,2317,2149,1987,1831,1681,1537,1399,1267,1133,915,1018,1118,1113,1108,1103,1098,1093,1088,1083,1076,1069,1062,1055,1048,1041,1034,1019,914,1017,1128,1244,1365,1491,1623,1761,1905,2055,2211,2373,2541,2705,2853,2972,3076,3162,3230,3263,3204,3127,3032,2923,2805,2658,2495,2328,2167,2012,1863,1720,1583,1448,1316,1190,1070,956,848,746,650,560,468,334,395,453,448,443,438,432,425,418,411,396,333,394,463,537,616,700,790,886,988,1096,1210,1330,1456,1588,1726,1869,2018,2173,2334,2501,2664,2811,2929,2905,2775,2617,2446,2279,2118,1963,1814,1671,1534,1403,1278,1159,1046,939,838,743,654,571,494,423,358,299,246,195,147,97,47,66,82,67,46,65,92,124,161,203,251,305,364,429,500,577,660,749,844,945,1052,1165,1284,1409,1540,1677,1820,1969,2124,2285,2452,2614,2443,2276,2115,1960,1811,1668,1531,1400,1275,1156,1043,936,835,740,651,568,491,420,355,296,243,196,154,118,87,55,18,29,39,41,61,86,116,152,193,240,293,352,417,488,565,648,737,832,933,1040,1153,1272,1397,1528,1665,1808,1957,2112,2273,2440,2611,2771,2904,2802,2655,2492,2325,2164,2009,1860,1717,1580,1449,1323,1203,1089,981,879,783,693,609,531,458,342,225,276,290,297,304,309,314,319,323,326,328,390,457,529,607,691,781,877,979,1087,1201,1321,1446,1577,1714,1857,2006,2161,2322,2489,2652,2799,2920,3031,3126,3203,3223,3155,3069,2965,2846,2698,2534,2366,2204,2048,1898,1754,1616,1484,1358,1238,1123,923,722,815,829,836,843,850,857,864,870,875,880,885,890,895,900,904,907,909,1013,1122,1236,1356,1482,1614,1752,1896,2046,2202,2364,2532,2696,2844,2963,3067,3153,3221,3276,3317,3339,3341,3340,3333,3303,3254,3187,3102,2999,2880,2733,2569,2401,2239,2082,1798,1513,1648,1662,1669,1676,1683,1690,1697,1704,1711,1718,1725,1730,1735,1740,1745,1750,1755,1760,1765,1770,1775,1779,1782,1784,1930,2081,2237,2399,2567,2731,2595,2760,2895,2891,2887,2883,2879,2875,2870,2865,2860,2855,2850,2845,2840,2835,2828,2821,2814,2807,2800,2793,2789,2597,2763,2769,2774,2772,2767,2761,2594,2759,2796,2803,2810,2817,2824,2831,2837,2842,2847,2852,2857,2862,2867,2872,2877,2881,2885,2889,2893,2897,2900,2902,2786,2630,2456,2282,2114,1952,1788,1510,1643,1773,1768,1763,1758,1753,1748,1743,1738,1733,1728,1722,1715,1708,1701,1694,1687,1680,1673,1666,1659,1644,1509,1642,1783,1929,2080,2236,2398,2566,2730,2878,2997,3100,3185,3252,3301,3332,3343,3337,3314,3273,3216,3146,3061,2954,2829,2675,2505,2331,2163,2001,1845,1695,1551,1413,1281,1155,1035,913,719,810,898,893,888,883,878,873,868,861,854,847,840,833,826,811,718,809,908,1012,1121,1235,1355,1481,1613,1751,1895,2045,2201,2363,2531,2695,2843,2962,3066,3152,3200,3123,3028,2915,2791,2644,2481,2314,2153,1998,1849,1706,1569,1438,1313,1194,1081,970,862,760,664,574,490,412,332,222,271,317,312,307,301,294,287,272,221,270,327,389,456,528,606,690,780,876,978,1086,1200,1319,1444,1575,1712,1855,2004,2159,2320,2487,2650,2797,2765,2603,2432,2265,2104,1949,1800,1657,1520,1389,1264,1145,1032,925,824,729,640,557,480,409,344,285,232,185,144,109,80,45,15,26,14,25,40,60,85,115,150,191,238,291,350,415,486,563,646,735,830,931,1038,1151,1270,1395,1526,1663,1806,1955,2110,2271,2438,2609,2457,2290,2129,1974,1825,1682,1545,1414,1289,1170,1057,950,849,754,665,582,505,434,368,308,254,206,163,107,50,71,84,88,91,93,125,162,204,252,306,366,431,502,579,662,751,846,947,1054,1167,1286,1411,1542,1679,1822,1971,2126,2287,2454,2623,2779,2908,3021,2934,2816,2669,2506,2339,2178,2023,1873,1729,1591,1459,1333,1213,1099,991,889,793,703,618,478,337,400,414,421,428,435,440,445,450,455,459,462,464,538,617,701,791,887,989,1097,1211,1331,1457,1589,1727,1871,2020,2175,2336,2503,2666,2813,2931,3037,3130,3207,3266,3283,3233,3165,3079,2975,2856,2708,2544,2376,2214,2058,1908,1764,1626,1494,1367,1143,918,1023,1037,1044,1051,1058,1065,1072,1079,1085,1090,1095,1100,1105,1110,1115,1120,1124,1127,1129,1245,1366,1492,1624,1762,1906,2056,2212,2374,2542,2706,2854,2973,3077,3163,3231,3281,3302,3298,3294,3290,3286,3282,3280,3277,3261,3195,3110,3007,2888,2743,2579,2410,2102,1793,1940,1954,1961,1968,1975,1982,1989,1996,2003,2010,2017,2024,2029,2034,2039,2044,2049,2054,2059,2064,2069,2074,2079,2083,2086,2088,2246,2409,2577,2741,2768,2593,2255,2418,2578,2573,2568,2563,2558,2553,2548,2543,2538,2533,2528,2523,2518,2511,2504,2497,2490,2483,2476,2469,2462,2455,2448,2441,2434,2419,2254,2417,2588,2754,2898,3017,3045,3052,3059,3065,3070,3075,3080,3085,3090,3095,3099,3103,3107,3111,3115,3118,3035,2922,2794,2640,2470,2296,2128,1966,1810,1660,1508,1254,1375,1493,1488,1483,1478,1473,1468,1463,1458,1453,1447,1440,1433,1426,1419,1412,1405,1398,1391,1376,1253,1374,1503,1637,1776,1920,2070,2226,2388,2556,2720,2868,2987,3091,3177,3244,3293,3324,3310,3269,3210,3135,3047,2944,2826,2679,2516,2345,2177,2015,1859,1709,1565,1427,1295,1169,1049,935,827,717,547,626,702,697,692,687,682,677,670,663,656,649,642,627,546,625,712,804,901,1003,1111,1225,1345,1471,1603,1741,1885,2035,2191,2353,2521,2685,2832,2950,3053,3119,3024,2911,2784,2632,2467,2300,2139,1984,1835,1692,1555,1424,1299,1180,1067,960,859,764,675,588,504,426,354,288,220,134,171,205,200,194,187,172,133,170,215,265,320,380,446,518,596,680,770,865,966,1073,1186,1305,1430,1561,1698,1841,1990,2145,2306,2473,2424,2257,2096,1941,1792,1649,1512,1381,1256,1137,1024,917,816,721,632,549,472,401,336,277,224,177,136,101,72,49,30,17,9,3)Polycyclic presentation:
2,3,3343|c^a=c^3342,c^b=c^1918563 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20058, 1);