Finite Group DatabasebetaFinite Group Databasebeta| order | 23105 |
| automorphism | ? |
| homology | Abelianization (H1) = C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C4621⋊C5 |
| Fitting, socle, derived, Sylow | C4621 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C4621 | C5 |
| LCS | G ▷ C4621 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 4621 |
| transitive degree | μt(G) = 4621 |
A minimal faithful permutation representation — 2 generators on 4621 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)(16,31,60,113,204)(20,38,71,49,91)(22,42,78,75,65)(24,46,86,155,189)(26,50,93,35,66)(28,54,101,181,163)(30,58,109,196,121)(32,62,117,212,366)(37,69,41,76,68)(39,73,133,106,190)(43,80,144,148,164)(45,84,151,174,122)(47,88,159,275,351)(51,95,170,92,165)(53,99,177,137,123)(55,103,185,317,325)(57,107,192,67,124)(59,111,200,343,283)(61,115,208,358,220)(63,119,216,374,616)(70,128,83,149,167)(72,131,98,175,126)(74,135,238,205,352)(77,139,130,94,168)(79,142,141,138,127)(81,146,253,268,326)(85,153,264,310,284)(87,157,271,336,221)(89,161,279,463,601)(96,172,295,191,327)(100,179,306,257,285)(102,183,313,299,222)(104,187,321,526,575)(108,194,332,166,286)(110,198,339,242,223)(112,202,347,567,533)(114,206,354,125,224)(116,210,362,593,471)(118,214,370,608,382)(120,218,378,624,979)(129,229,156,269,329)(132,233,182,311,288)(134,236,197,337,226)(136,240,407,367,602)(140,245,235,193,330)(143,248,250,258,289)(145,251,261,300,227)(147,255,426,456,576)(150,259,292,169,290)(152,262,303,243,228)(154,266,441,519,534)(158,273,452,560,472)(160,277,459,586,383)(162,281,467,744,964)(171,293,176,301,231)(173,297,489,353,577)(178,304,247,244,232)(180,308,504,445,535)(184,315,515,508,473)(186,319,522,549,384)(188,323,530,833,939)(195,334,545,328,536)(199,341,556,430,474)(201,345,563,493,385)(203,349,571,893,900)(207,356,582,287,475)(209,360,589,411,386)(211,364,597,932,839)(213,368,604,225,387)(215,372,612,956,752)(217,376,620,971,632)(219,380,628,987,1475)(230,393,276,457,579)(234,398,318,520,538)(237,402,344,561,477)(239,405,359,587,389)(241,409,665,617,965)(246,415,404,355,580)(249,418,423,446,539)(252,421,438,509,478)(254,424,449,550,390)(256,428,688,737,940)(260,433,486,331,540)(263,436,501,431,479)(265,439,512,494,391)(267,443,707,826,901)(270,447,542,291,480)(272,450,553,412,392)(274,454,722,887,840)(278,461,733,925,753)(280,465,740,950,633)(282,469,748,1142,1460)(294,484,312,510,482)(296,487,338,551,395)(298,491,777,603,941)(302,496,400,333,543)(305,499,420,432,483)(307,502,435,495,396)(309,506,796,726,902)(314,513,498,413,397)(316,517,811,815,841)(320,524,822,876,754)(322,528,829,914,634)(324,532,837,1262,1434)(335,547,857,578,903)(340,554,417,414,401)(342,558,872,711,842)(346,565,883,800,755)(348,569,890,861,635)(350,573,897,1344,1400)(357,584,911,537,843)(361,591,921,692,756)(363,595,928,781,636)(365,599,936,1396,1350)(369,606,946,476,757)(371,610,952,669,637)(373,614,960,1427,1266)(375,618,967,388,638)(377,622,975,1452,1150)(379,626,983,1467,995)(381,630,991,1483,600)(394,645,464,738,943)(399,651,527,827,904)(403,656,568,888,845)(406,660,594,926,759)(408,663,609,585,640)(410,667,1037,980,1461)(416,674,662,605,944)(419,677,685,727,905)(422,680,704,816,846)(425,683,719,877,760)(427,686,730,915,641)(429,690,1064,1135,1435)(434,696,774,581,906)(437,699,793,712,847)(440,702,808,801,761)(442,705,819,862,642)(444,709,1087,1255,1401)(448,714,854,541,848)(451,717,869,693,762)(453,720,880,782,643)(455,724,1105,1338,1351)(458,728,908,481,763)(460,731,918,670,644)(462,735,1120,1390,1267)(466,742,1131,1421,1151)(468,746,1138,1445,996)(470,750,1146,1671,2090)(485,768,521,817,850)(488,772,562,878,765)(490,775,588,916,647)(492,779,1183,966,1436)(497,785,658,583,909)(500,788,682,713,851)(503,791,701,802,766)(505,794,716,863,648)(507,798,1206,1124,1402)(511,803,770,544,852)(514,806,790,694,767)(516,809,805,783,649)(518,813,1225,1244,1352)(523,820,866,671,650)(525,824,1240,1329,1268)(529,831,1251,1379,1152)(531,835,1258,1413,997)(546,855,552,864,653)(548,859,1291,942,1403)(555,867,679,695,771)(557,870,698,784,654)(559,874,1310,1109,1353)(564,881,787,672,655)(566,885,1325,1229,1269)(570,892,1336,1314,1153)(572,895,1341,1366,998)(574,899,1348,1928,2028)(590,919,676,673,659)(592,923,1376,1090,1270)(596,930,1386,1210,1154)(598,934,1393,1295,999)(607,948,1411,844,1271)(611,954,1417,1068,1155)(613,958,1423,1187,1000)(615,962,1431,2024,1932)(619,969,1441,758,1156)(621,973,1448,1041,1001)(623,977,1456,2058,1825)(625,981,1463,639,1002)(627,985,1471,2082,1679)(629,989,1479,2096,1491)(631,993,1487,2111,2121)(646,1010,745,1136,1438)(652,1017,834,1256,1404)(657,1023,894,1339,1354)(661,1028,933,1391,1273)(664,1032,957,1113,1158)(666,1035,972,1446,1004)(668,1039,1541,1476,2091)(675,1047,1034,968,1439)(678,1050,1061,1125,1405)(681,1053,1084,1245,1355)(684,1056,1102,1330,1274)(687,1059,1117,1380,1159)(689,1062,1128,913,1005)(691,1066,1572,1664,2065)(697,1073,1180,945,1406)(700,1076,1203,1110,1356)(703,1079,1222,1230,1275)(706,1082,1237,1315,1160)(708,1085,1248,1367,1006)(710,1089,1599,1818,2029)(715,1094,1288,907,1357)(718,1097,1307,1091,1276)(721,1100,1322,1211,1161)(723,1103,1333,1296,1007)(725,1107,1619,1922,1990)(729,1112,1362,849,1277)(732,1115,1373,1069,1162)(734,1118,1383,1188,1008)(736,1122,1635,1984,1933)(739,1126,1408,764,1163)(741,1129,1415,1042,1009)(743,1133,1649,2019,1826)(747,1140,1660,2052,1680)(749,1144,1667,2075,1492)(751,1148,1675,2317,938)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225,3484,979,1460,315,516,810,1221,1763,2418,3122,2464,3164,3804,4275,4305,4535,3461,4030,2539,3242,3867,2491,3194,3826,3670,4174,2648,551,863,1297,1863,2536,3240,3866,2703,3391,3982,3059,3720,4220,3931,4350,4556,3885,4321,3254,772,1173,1705,2352,3056,3718,1312,1881,2555,3258,3879,4316,2799,3488,1630,198,340,555,868,1303,1870,2543,3246,3407,3992,4380,3883,4318,2798,3487,4052,3454,4024,1927,2609,3314,3928,4348,1977,2657,3357,2136,2843,3535,4094,4228,4501,4600,4549,2388,393,644,1008,1505,2132,2839,3531,2574,502,36,68,126,226,389,640,1004,1501,2128,2836,3529,4089,3822,1129,113,205,353,578,663,1031,1530,2155,2729,602,940,1401,1990,2105,2808,3500,4064,4123,1398,1988,2671,3368,3565,4113,4443,4257,1552,2183,2889,3573,4120,4448,4583,4342,1759,245,414,672,1044,1548,2179,2885,1911,313,512,805,1216,1758,2413,3117,3769,4256,4519,2194,202,348,570,744,1135,1653,2285,2426,426,685,1058,1563,2053,2736,3422,4002,4195,4482,2088,2783,3471,3646,960,1428,2020,2700,2909,378,625,982,1466,2074,582,908,1358,1940,2623,3330,1232,1776,2433,3137,3785,4266,4522,2823,1341,1917,2595,1089,1600,2235,2939,3616,4148,2789,3474,1591,153,265,440,703,1080,1589,2223,1788,2445,3148,3795,4037,4398,2741,3427,1830,2497,3200,1662,2296,2987,3002,3656,1954,327,535,841,1268,1827,2495,3199,2015,2695,3385,3979,4081,4426,3307,3923,4346,3106,3758,2522,3222,3847,2321,3024,3685,3629,798,1207,1746,2398,3100,3752,2096,2793,1100,99,178,305,500,789,1196,1734,2386,3091,3748,4241,3234,3858,2080,2769,3453,2760,3444,1260,1815,2477,3175,3810,1381,1964,2645,3349,3960,4368,4561,4015,4390,4567,4313,1701,2348,3053,3716,4218,4497,2532,3235,1875,293,483,766,1166,1697,2344,547,858,1290,1853,2523,3224,3849,3174,731,56,106,191,328,537,844,1272,1831,2034,2715,3402,3988,4182,1477,2094,2790,3476,3528,895,1342,1919,2597,3301,3919,4344,3747,1002,1498,2124,2831,3525,4087,4429,3897,1195,124,223,385,635,998,1494,2120,1323,177,303,498,787,1194,1732,2384,3089,3746,4239,1527,2153,2858,2487,463,737,1124,1638,1745,238,404,658,1025,1446,2042,2725,3412,3684,4189,4478,4590,4605,2977,597,933,1392,1978,2170,2874,3559,3115,3766,2002,2684,3378,3975,4072,4418,3293,803,1213,1754,2408,3111,3763,4250,3370,890,1333,1907,532,838,1265,1824,2492,3196,3829,3776,1056,54,102,184,316,518,814,1228,1225,1767,2422,3126,3464,4031,2017,2698,3388,3981,4128,977,1457,2061,2295,2985,1357,1939,2622,1572,2202,2905,3585,4006,1417,2007,2688,3381,3511,4075,2593,3298,3917,2207,2912,3593,4134,4454,1487,2112,2817,2928,428,689,1063,1568,2198,2901,1475,2090,702,42,79,143,249,419,678,1051,1556,2188,2895,3580,4125,4164,1450,2049,2732,3418,3615,742,1132,1648,2278,2973,916,1368,1951,2634,3341,3953,4362,3432,4009,4387,3845,4296,4533,3681,4187,4476,4466,1722,2372,1286,1847,2515,3215,3842,4294,2245,297,490,776,1179,1713,2361,3063,2459,450,25,49,92,166,287,476,758,1157,1687,2334,3039,3703,3666,981,1464,2070,2756,2958,528,830,1250,1799,2457,3158,3801,3048,3711,4214,4495,4538,4612,4040,4402,3248,763,1162,1692,2339,3044,3707,4210,3355,870,93,167,288,477,759,1158,1688,2335,3040,3704,4208,4490,4353,1812,275,456,726,1109,1181,1715,2363,3064,3449,965,1435,2029,2710,3017,3674,4179,4470,4508,2268,347,568,889,1332,1906,2585,3292,2382,3086,3742,4234,4505,4601,3495,4061,2568,496,784,1190,1727,2378,3082,3738,2663,563,880,1319,1890,2567,3272,3893,4327,4548,4617,3096,656,1022,1520,1262,1818,2482,3182,3285,796,1203,1741,2393,2773,3456,1388,1973,2654,3356,3557,4107,4440,4286,1649,2279,2974,3638,3756,1037,1538,2165,2868,3419,946,1408,1997,2679,2813,3505,1905,2583,3289,1148,1676,2320,3022,3682,1987,2669,3366,2190,161,280,466,743,1134,1652,2284,2420,421,11,22,43,81,147,256,429,691,1067,1575,2208,2914,3596,954,1418,2008,2689,2903,372,613,959,1426,2018,577,902,1352,1934,2618,3326,3940,2739,3424,4004,4385,4484,4594,2815,3509,4073,4144,1039,1542,2171,2876,3562,4111,4442,4270,1585,135,239,406,661,1029,1528,2154,1783,262,9,18,35,67,125,225,388,639,1003,1500,2127,2835,2997,618,968,1440,2035,2231,277,460,732,1116,1628,2261,2964,3634,4079,4424,4574,4401,4568,3438,4016,2511,3211,3839,2315,3013,3669,3623,4152,2640,543,851,1280,1840,2507,3208,1094,1605,2240,2944,3620,4150,4461,3927,1249,155,268,445,711,1090,1601,2236,2940,2755,3440,1242,1789,2447,2311,3007,3663,4170,4207,1617,185,318,521,818,1233,1778,2436,1686,2332,3036,3699,4204,4488,2780,3466,1869,286,474,755,1153,1682,2328,3032,1970,339,553,866,1301,1868,2541,3244,3869,4312,4541,2360,3062,3721,3320,833,1255,1807,2467,2554,489,774,1176,1709,2076,2762,885,1326,1898,2575,3280,3901,4332,3814,1120,1632,2265,2966,3074,3730,4223,3892,4326,2722,3409,3994,4381,4475,2100,2800,1317,1887,2563,1348,1929,2612,3318,3932,1393,1979,2659,1516,2144,2852,3544,4101,4437,4578,4333,1739,233,397,649,1014,1512,2140,2848,1899,2576,3281,3902,4059,4410,2972,591,922,1375,1957,2163,2865,3551,3193,3824,1996,2678,3373,2401,3102,3754,4242,4395,2048,2117,2825,3520,4084,4428,3365,3968,4372,3579,380,629,990,1482,2103,2806,3498,3772,1050,31,61,116,211,365,600,938,1220,142)Polycyclic presentation:
5,4621|b^a=b^1153929 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23105, 1);