Finite Group DatabasebetaFinite Group Databasebeta| order | 20005 |
| automorphism | ? |
| homology | Abelianization (H1) = C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C4001⋊C5 |
| Fitting, socle, derived, Sylow | C4001 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C4001 | C5 |
| LCS | G ▷ C4001 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 4001 |
| transitive degree | μt(G) = 4001 |
A minimal faithful permutation representation — 2 generators on 4001 points:
(2,3,5,9,17)(4,7,13,25,48)(6,11,21,40,33)(8,15,29,56,105)(10,19,36,18,34)(12,23,44,82,90)(14,27,52,97,64)(16,31,60,113,203)(20,38,71,49,91)(22,42,78,75,65)(24,46,86,154,188)(26,50,93,35,66)(28,54,101,180,162)(30,58,109,195,120)(32,62,117,211,362)(37,69,41,76,68)(39,73,132,106,189)(43,80,143,147,163)(45,84,150,173,121)(47,88,158,272,347)(51,95,169,92,164)(53,99,176,136,122)(55,103,184,313,321)(57,107,191,67,123)(59,111,199,339,279)(61,115,207,354,217)(63,119,215,370,494)(70,127,83,148,166)(72,130,98,174,125)(74,134,235,204,348)(77,138,129,94,167)(79,141,140,137,126)(81,145,250,265,322)(85,152,261,306,280)(87,156,268,332,218)(89,160,276,454,590)(96,171,291,190,323)(100,178,302,254,281)(102,182,309,295,219)(104,186,317,515,564)(108,193,328,165,282)(110,197,335,239,220)(112,201,343,556,522)(114,205,350,124,221)(116,209,358,582,460)(118,213,366,597,374)(128,226,155,266,325)(131,230,181,307,284)(133,233,196,333,223)(135,237,399,363,591)(139,242,232,192,326)(142,245,247,255,285)(144,248,258,296,224)(146,252,417,447,565)(149,256,288,168,286)(151,259,299,240,225)(153,263,432,508,523)(157,270,443,549,461)(159,274,450,575,375)(161,278,458,718,763)(170,289,175,297,228)(172,293,478,349,566)(177,300,244,241,229)(179,304,493,436,524)(183,311,504,497,462)(185,315,511,538,376)(187,319,519,801,910)(194,330,534,324,525)(198,337,545,421,463)(200,341,552,482,377)(202,345,560,861,868)(206,352,571,283,464)(208,356,578,402,378)(210,360,586,902,807)(212,364,593,222,379)(214,368,601,928,722)(216,372,608,840,612)(227,385,273,448,568)(231,390,314,509,527)(234,394,340,550,466)(236,397,355,576,381)(238,401,643,605,765)(243,406,396,351,569)(246,409,414,437,528)(249,412,429,498,467)(251,415,440,539,382)(253,419,664,711,911)(257,424,475,327,529)(260,427,490,422,468)(262,430,501,483,383)(264,434,682,794,869)(267,438,531,287,469)(269,441,542,403,384)(271,445,696,854,808)(275,452,707,895,723)(277,456,714,921,613)(290,473,308,499,471)(292,476,334,540,387)(294,480,747,592,912)(298,485,392,329,532)(301,488,411,423,472)(303,491,426,484,388)(305,495,764,700,870)(310,502,487,404,389)(312,506,779,783,809)(316,513,790,843,724)(318,517,797,884,614)(320,521,805,1180,1124)(331,536,825,567,871)(336,543,408,405,393)(338,547,839,686,810)(342,554,850,768,725)(344,558,857,829,615)(346,562,865,1260,1326)(353,573,880,526,811)(357,580,891,667,726)(359,584,898,750,616)(361,588,906,1319,1266)(365,595,917,465,727)(367,599,924,645,617)(369,603,932,1360,1184)(371,606,780,380,618)(373,610,943,1298,1077)(386,624,455,712,914)(391,629,516,795,873)(395,634,557,855,813)(398,638,583,896,729)(400,641,598,922,619)(407,650,640,594,915)(410,653,661,701,874)(413,656,679,784,814)(416,659,693,844,730)(418,662,704,885,620)(420,666,1004,1070,1126)(425,671,744,570,875)(428,674,762,687,815)(431,677,776,769,731)(433,680,787,830,621)(435,684,1025,1173,1327)(439,689,822,530,816)(442,691,836,668,732)(444,694,847,751,622)(446,698,1041,1253,1267)(449,702,877,470,733)(451,705,888,646,623)(453,709,1055,1312,1185)(457,716,1066,1353,1078)(459,720,1073,1223,947)(474,738,510,785,818)(477,742,551,845,735)(479,745,577,886,626)(481,749,1108,936,1128)(486,754,636,572,878)(489,757,658,688,819)(492,760,676,770,736)(496,766,1127,1059,1328)(500,771,740,533,820)(503,774,759,669,737)(505,777,773,752,627)(507,781,1144,1162,1268)(512,788,833,647,628)(514,792,1158,1242,1186)(518,799,1169,1301,1079)(520,803,1176,1342,948)(535,823,541,831,631)(537,827,1209,913,1329)(544,834,655,670,741)(546,837,673,753,632)(548,841,1224,1045,1269)(553,848,756,648,633)(555,852,1238,1148,1187)(559,859,1249,1228,1080)(561,863,1256,1287,949)(563,867,1264,1745,1574)(574,882,1284,872,1270)(579,889,652,649,637)(581,893,1297,1028,1188)(585,900,1308,1130,1081)(587,904,1315,1212,950)(589,908,1323,1823,1829)(596,919,1338,812,1189)(600,926,1349,1006,1082)(602,930,1356,1110,951)(604,934,1364,1737,1749)(607,938,1159,728,1083)(609,941,1239,982,952)(611,945,1380,1861,1649)(625,959,719,1071,1143)(630,963,802,1174,1331)(635,968,862,1254,1272)(639,973,903,1313,1191)(642,977,929,1354,1084)(644,980,940,1225,953)(651,988,979,937,1145)(654,990,1001,1060,1332)(657,993,1022,1163,1273)(660,996,1038,1243,1192)(663,999,1052,1302,1085)(665,1002,1063,1343,954)(672,1011,1105,916,1333)(675,1014,1125,1046,1274)(678,1017,1142,1149,1193)(681,1020,1155,1229,1086)(683,1023,1166,1288,955)(685,1027,1462,1642,1576)(690,1032,1206,876,1275)(692,1034,1222,1029,1194)(695,1036,1235,1131,1087)(697,1039,1246,1213,956)(699,1043,1480,1738,1830)(703,1048,1281,817,1195)(706,1050,1294,1007,1088)(708,1053,1305,1111,957)(710,1057,1496,1816,1750)(713,1061,1335,734,1089)(715,1064,1346,983,958)(717,1068,1510,1874,1650)(721,1075,1521,1787,1525)(739,1094,796,1164,1277)(743,1099,856,1244,1197)(746,1103,897,1303,1091)(748,1106,923,1344,960)(755,1115,975,918,1336)(758,1118,998,1047,1278)(761,1121,1019,1150,1198)(767,1129,1579,1514,1578)(772,1134,1101,879,1279)(775,1137,1123,1030,1199)(778,1140,1139,1132,1092)(782,1146,1598,1631,1831)(786,1151,1202,821,1200)(789,1153,1219,1008,1093)(791,1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641,3142,3538,3794,2766,757,66,122,219,376,614,948,1383,473,737,1092,1071,1515,2034,2582,1675,2202,2739,2714,714,1063,749,1109,1559,2078,2623,2162,2700,2517,419,665,1003,1436,1951,2507,3039,1944,186,318,518,800,1172,1630,2151,1406,1923,2484,2465,3009,3459,3406,3130,3531,3792,3787,1879,2441,2981,2932,3386,3706,3887,3505,1321,1820,2365,2896,3352,3680,3358,3077,865,1261,1740,2270,2801,3271,2895,855,1243,1715,2242,2775,3250,3610,2353,552,847,1232,1702,2229,2762,3239,1801,2341,2874,3333,3665,3186,3566,3424,3736,3907,3977,3997,3943,2680,689,1031,1467,1981,2532,3058,3486,2128,427,19,37,70,128,227,386,625,248,5,10,20,39,74,135,238,132,232,392,631,965,1398,1917,747,1105,1553,2070,2614,1712,2239,591,911,1327,1830,2382,2921,1160,1614,2132,2669,2997,3448,3752,3052,709,1056,1495,2009,2556,3072,2866,2509,368,602,931,1359,1873,2433,2320,525,810,1187,1652,2179,2720,3210,1765,2301,2834,2812,2452,2998,3450,3401,3717,3894,3882,2272,515,794,1162,1617,2136,2674,3166,1734,2261,2086,379,617,951,1386,1908,2475,3020,1541,2059,2605,2585,3099,3511,3783,3226,3598,3827,3818,2022,352,572,879,1280,1766,2303,2837,1491,197,336,544,835,1218,1687,2214,1039,101,181,308,500,772,1135,1587,682,1022,1456,1969,2521,3048,2436,2117,2655,3152,3544,3798,3433,3741,1868,354,575,884,1287,1776,2315,2850,1349,1857,2415,2957,3410,3723,2276,2808,3277,3629,3842,3944,3868,2243,1647,2174,2715,3205,3583,3819,3674,1703,297,484,752,1112,1562,2080,2624,1220,167,285,467,730,1085,1532,2051,823,1203,1669,1645,2170,1984,2534,2314,2849,3312,3288,3637,3549,1448,134,236,398,639,974,1407,1925,993,46,87,157,271,446,699,1044,634,967,1399,1375,1894,2459,3004,2073,2616,3123,3105,930,1357,1869,1833,2385,2924,3379,2577,586,903,1314,1808,2349,2882,3340,2013,2559,2015,2563,3078,3496,3746,1817,2360,2892,1366,1885,2451,2996,3435,1315,1809,1655,2182,2723,3213,3589,2982,3429,3301,3644,3850,3843,3760,3919,3980,3973,3996,4001,3999,3625,1620,2139,2677,3169,3554,3802,3930,3265,1153,148,255,422,668,1007,1441,1955,777,78,140,244,408,652,989,1423,493,762,1123,1573,2090,1286,1775,1695,2222,2757,3236,3602,3070,841,1225,1694,2221,2756,3235,3601,2538,506,780,1143,1595,2112,2650,3147,1970,270,444,695,1037,1474,1988,2537,1435,115,208,357,581,894,1300,1289,980,1414,1932,1904,2471,3016,2958,875,1274,1758,2294,2828,3294,3640,2401,2941,3394,3372,1260,1738,2267,2798,3268,3622,3837,2910,3362,3240,1115,1565,2083,2627,3133,3533,3793,2751,733,1088,1535,2054,2602,3114,3521,2198,2735,3221,3202,1061,1502,2018,2566,3081,3498,3776,2698,705,56,106,190,324,526,812,1190,443,693,278,459,721,1076,1524,1613,2131,1920,2483,3024,3007,3458,3258,3615,3529,3791,3927,3926,2432,2974,928,1353,1864,2423,2966,3418,1814,2356,2888,3345,3676,3867,3954,3494,1258,1735,2263,2794,3266,3620,3349,3064,803,1177,1637,2158,2697,3188,2883,845,458,719,1072,1517,2037,2586,2339,543,48,90,162,279,460,722,1077,328,531,818,1197,1663,2190,2730,1284,1771,2309,2843,3306,2664,671,1010,1444,1958,2513,3043,3477,2109,409,7,14,28,55,104,187,320,230,389,627,606,937,1369,1887,1103,1550,2066,2041,366,598,923,1345,1853,2411,2953,1510,2026,1600,2118,2656,3153,3545,3194,3572,3480,1043,1481,1995,2542,2991,3440,3745,3038,588,907,1322,1822,2369,2902,2852,2495,3032,3473,3468,3767,3921,3899,2297,2831,3296,3641,2995,3446,3749,3713,3892,3969,3962,2800,2379,2918,3373,3696,3881,3963,3875,2259,509,784,1149,1602,2120,2658,3154,1721,309,501,773,1136,1588,2105,2644,1238,1708,2235,2768,3244,2569,3084,2877,3336,3668,3862,3952,3809,2004,330,535,824,1205,1672,2199,2737,1473,178,303,492,761,1122,1572,2089,1020,84,151,260,428,675,1015,1449,662,29,57,108,194,331,537,828,399,640,975,1408,1926,2487,1855,348,565,869,1267,1750,2286,2821,1336,1841,2395,2377,2914,3367,2164,2703,3193,3571,3758,3915,3864,2227,1523,2044,2593,3106,3517,3786,3663,1686,282,463,725,1080,1527,2047,2596,1200,1665,2191,2168,2708,3198,3576,2846,3308,3648,3635,1636,272,447,700,1045,1483,1998,2546,1169,1624,1424,62,118,214,369,604,935,1367,963,1396,1915,1890,2456,3002,3455,2612,3121,3525,3514,1351,1861,2419,2961,3414,3727,3900,3088,900,1309,1800,2340,2873,3332,3664,2557,558,858,1248,1719,2246,2779,3253,1994,317,516,796,1165,1621,2141,1807,1462,1975,2527,3054,3484,3771,3428,1306,195,332,538,829,1212,1681,2209,891,1294,1783,2322,2856,3316,3653,2420,2962,3415,3728,3901,3618,1605,2123,2661,3157,3548,3800,3929,3254,1137,138,241,404,647,984,1418,1936,760,69,126,224,382,620,954,1388,476,741,1097,1073,1518,1211,1680,1678,2206,2744,2716,3206,3060,766,1128,1578,2095,2635,3137,3535,2519,434,683,1024,1458,1971,2523,3050,1947,201,344,559,860,1252,1726,2254,1411,1928,2488,2469,3013,3462,3408,1270,1753,2289,2823,1882,2445,2986,2937,3390,3708,3693,3508,1323,1824,2372,2906,3359,3685,3360,1254,1728,2256,2788,3261,3617,3834,2898,857,1246,1083,1530,2050,2599,3112,2355,2887,2733,3220,3594,3579,3816,3740,3909,3855,3951,3987,3985,3187,1048,1486,2001,2549,3068,3491,3773,2682,691,50,94,168,287,470,734,1090,430,21,41,77,139,243,407,651,250,414,658,995,1428,1943,2500,1127,1577,2094,2634,3136,2345,2878,922,1343,1851,2409,2951,3405,1725,2252,593,914,1331,1835,2388,2927,1171,1628,2147,2684,3175,3557,3339,3055,716,1067,1509,2025,2573,3087,2869,831,372,609,942,1376,1896,2462,2323,529,815,1193,1658,2185,2725,3214,1769,2306,2839,2815,801,1173,1631,2152,2690,3181,3561,2274,2804,2645,650,987,1421,1939,2498,3034,3474,2088,385,623,957,1391,1912,2478,3021,1546,2062,2607,2587,595,918,1337,1843,2398,2938,3392,2024,356,579,890,1293,1782,2321,2855,1494,199,340,551,846,1231,1701,2228,1041,1477,867,1265,1748,2284,2820,2672,3164,3029,3472,3768,3765,3920,2438,582,895,1301,1791,2331,2864,3324,1870,2427,2969,3421,3733,3904,2912,3366,3691,3877,3960,3989,3957,2789,2283,2818,3287,3636,3847,3947,3869,2245,499,770,1132,1583,2100,2639,3140,1706,300,17,33,64,120,217,374,612,169,288,471,735,1091,1538,2057,825,1206,1673,2200,2738,1985,293,479,746,1104,1552,2069,2613,1451,145,251,416,660,997,1430,1945,996,54,102,183,312,507,782,1147,638,972,1404,1378,1898,2463,2410,2077,2621,3128,3108,932,1361,1876,1838,2391,2930,3384,2579,3092,2689,3180,3560,3804,3914,2371,2904,3357,2031,2580,3094,3506,3748,1819,2363,2211,1382,1905,2473,3019,3437,3743,3654,1661,2188,2728,3217,3592,3824,3939,3304,3646,3851,3844,2155,2693,3184,3564,3807,3933,3983,3627,1622,266,437,687,1029,1465,1979,2530,1156,150,258,426,673,1013,1447,1961,779,1142,1594,2111,2649,1849,2407,2238,2771,3247,3608,3830,3501,1299,1789,2328,2862,3322,3657,3856,3071,852,1239,1709,2236,2769,3245,3606,2540,513,791,1157,1610,2129,2667,3161,1973,274,451,706,1051,1490,2005,1793,1438,117,212,365,596,920,1341,1292,189,322,523,808,1185,1650,2177,878,1278,1763,1745,2278,2810,3279,2404,2945,3399,3374,3697,3611,1586,2103,2642,3143,3539,3795,3928,3242,1118,123,220,377,615,949,1384,1906,738,1093,1539,1516,2035,2583,3097,2203,2740,3224,3204,1064,113,204,349,567,872,1271,1755,707,1052,666,1005,1439,1954,2510,2148,2685,2501,319,520,804,1179,1641,2165,2705,1924,2485,3025,3010,3460,1360,1874,2434,2976,3426,3737,2442,2983,3430,3387,3707,3888,3967,3780,1821,2367,2899,3354,3682,3870,3684,3495,1262,1742,2273,2803,3273,3626,3351,1244,805,1181,1644,2169,2710,3201,2885,848,97,173,295,482,750,1110,1560,545,836,1219,1688,2215,2750,3231,1788,2327,2861,3321,3656,3172,1032,1468,1982,2533,3059,3487,3772,2666,674,38,72,131,231,391,630,964,412,11,22,43,81,146,253,420,233,393,632,608,940,1373,1892,1106,1554,2071,2043,2591,2240,2773,912,1328,1831,2383,2922,3378,1615,2134,2671,3163,3449,3753,3913,3482,1057,1497,2012,2558,2993,3442,3326,3041,603,933,1363,1880,2443,2985,2854,811,1188,1653,2180,2721,3211,3587,2302,2835,3299,3281,2999,3452,3757,3718,3895,3971,3964,2802,795,1163,1618,2137,2675,3167,3552,2262,511,787,618,952,1387,1909,2476,1723,2249,2060,2606,3117,3100,3512,3347,3677,3599,3828,3940,3937,2570,573,881,1283,1770,2308,2842,3305,2006,337,546,838,1221,1690,2217,2752,1475,182,310,503,775,1138,1590,2107,1023,86,155,267,439,690,1033,1470,664,1001,1434,1949,2505,3037,2426,576,885,1288,1777,2316,2851,3313,1858,350,568,873,1272,1756,2292,2826,1338,1844,2399,2939,3393,2776,2175,2717,3207,3584,3820,3938,3865,2230,485,753,1113,1563,2081,2625,3131,1689,286,468,731,1086,1533,2052,2600,1204,1670,2196,2171,454,711,1059,1500,2016,2564,3079,1638,2159,1962,237,400,642,978,1412,1930,2491,1426,88,159,275,453,710,1058,1499,968,1400,1918,1895,2460,3005,3456,2617,3124,3527,3516,1358,1871,2429,2386,2925,3380,3702,3090,904,1316,1811,2351,2884,3342,3673,2560,560,862,1255,1730,2258,2790,2361,1996,2543,1886,2453,3000,3454,3742,1810,339,549,843,1228,1698,2225,2759,1308,1798,2338,2871,3330,3662,3858,2973,3423,3735,3906,3976,3839,2140,2678,3170,3555,3803,3931,3982,3619,1607,256,423,669,1008,1442,1956,2511,1140,141)Polycyclic presentation:
5,4001|b^a=b^1401805 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20005, 1);