Finite Group DatabasebetaFinite Group Databasebeta| order | 24594 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | D4099⋊C3 |
| unambiguous semidirect product | (C4099⋊C3)⋊C2 |
| Fitting, socle, derived, Sylow | C4099 ⋊ C2×C3 |
| Hall | C4099⋊C3 ⋊ C2 |
| Hall | D4099 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C4099 | C2×C3 |
| LCS | G ▷ C4099 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 4099 |
| transitive degree | μt(G) = 4099 |
A minimal faithful permutation representation — 2 generators on 4099 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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59,2763,2766,2768,2950,3133,3310,3473,3615,3734,3729,3724,3719,3714,3709,3704,3699,3694,3689,3683,3676,3669,3662,3655,3648,3641,3634,3627,3622,3618,3616,3612,3466,3302,3125,2943,2762,2430,2097,2256,2270,2277,2284,2291,2298,2305,2312,2319,2326,2333,2340,2347,2352,2357,2362,2367,2372,2377,2382,2387,2392,2397,2402,2407,2411,2414,2416,2586,2761,2941,3123,3300,3464,3610,3737,3843,3926,3921,3916,3911,3906,3901,3896,3890,3883,3876,3869,3862,3855,3850,3846,3844,3840,3730,3602,3456,3292,3115,2933,2753,2579,2410,2102,1793,1940,1954,1961,1968,1975,1982,1989,1996,2003,2010,2017,2024,2029,2034,2039,2044,2049,2054,2059,2064,2069,2074,2079,2083,2086,2088,2246,2409,2577,2751,2931,3113,3290,3454,3600,3728,3838,3929,3999,4046,4041,4036,4031,4025,4018,4011,4006,4002,4000,3996,3922,3830,3720,3592,3446,3282,3105,2923,2743,2569,2401,2239,2082,1798,1513,1648,1662,1669,1676,1683,1690,1697,1704,1711,1718,1725,1730,1735,1740,1745,1750,1755,1760,1765,1770,1775,1779,1782,1784,1930,2081,2237,2399,2567,2741,2921,3103,3280,3444,3590,3718,3828,3920,3994,4049,4083,4094,4090,4086,4084,4080,4042,3986,3912,3820,3710,3582,3436,3272,3095,2913,2733,2559,2391,2229,2073,1923,1778,1518,1257,1380,1394,1401,1408,1415,1422,1429,1436,1443,1450,1455,1460,1465,1470,1475,1480,1485,1490,1495,1499,1502,1504,1638,1777,1921,2071,2227,2389,2557,2731,2911,3093,3270,3434,3580,3708,3818,3910,3984,4040,4078,4097,4091,4070,4032,3976,3902,3810,3700,3572,3426,3262,3085,2903,2723,2549,2381,2219,2063,1913,1769,1631,1498,1262,1025,1136,1150,1157,1164,1171,1178,1185,1192,1199,1204,1209,1214,1219,1224,1229,1234,1239,1243,1246,1248,1370,1497,1629,1767,1911,2061,2217,2379,2547,2721,2901,3083,3260,3424,3570,3698,3808,3900,3974,4030,4068,4059,4020,3965,3892,3800,3690,3562,3416,3252,3075,2893,2713,2539,2371,2209,2053,1903,1759,1621,1489,1363,1242,1030,817,916,930,937,944,951,958,965,972,977,982,987,992,997,1002,1007,1011,1014,1016,1126,1241,1361,1487,1619,1757,1901,2051,2207,2369,2537,2711,2891,3073,3250,3414,3560,3688,3798,3889,3962,4017,4008,3951,3878,3787,3678,3551,3406,3242,3065,2883,2703,2529,2361,2199,2043,1893,1749,1611,1479,1353,1233,1119,1010,822,633,720,734,741,748,755,762,769,774,779,784,789,794,799,803,806,808,906,1009,1117,1231,1351,1477,1609,1747,1891,2041,2197,2359,2527,2701,2881,3063,3240,3403,3548,3675,3784,3875,3948,3939,3864,3773,3664,3537,3392,3229,3053,2872,2693,2519,2351,2189,2033,1883,1739,1601,1469,1343,1223,1109,1001,899,802,638,473,548,562,569,576,583,590,595,600,605,610,615,619,622,624,710,801,897,999,1107,1221,1341,1467,1599,1737,1881,2031,2187,2349,2517,2690,2869,3050,3226,3389,3534,3661,3770,3861,3852,3759,3650,3523,3378,3215,3039,2858,2679,2506,2339,2178,2023,1873,1729,1591,1459,1333,1213,1099,991,889,793,703,618,478,337,400,414,421,428,435,440,445,450,455,459,462,464,538,617,701,791,887,989,1097,1211,1331,1457,1589,1727,1871,2020,2175,2336,2503,2676,2855,3036,3212,3375,3520,3647,3756,3747,3636,3509,3364,3201,3025,2844,2665,2492,2325,2164,2009,1860,1717,1580,1449,1323,1203,1089,981,879,783,693,609,531,458,342,225,276,290,297,304,309,314,319,323,326,328,390,457,529,607,691,781,877,979,1087,1201,1321,1446,1577,1714,1857,2006,2161,2322,2489,2662,2841,3022,3198,3361,3506,3633,3624,3495,3350,3187,3011,2830,2651,2478,2311,2150,1995,1846,1703,1566,1435,1310,1191,1078,971,869,773,683,599,521,449,383,322,230,137,176,190,197,202,207,211,214,216,266,321,381,447,519,597,681,771,867,968,1075,1188,1307,1432,1563,1700,1843,1992,2147,2308,2475,2648,2827,3008,3184,3347,3492,3483,3336,3173,2997,2816,2637,2464,2297,2136,1981,1832,1689,1552,1421,1296,1177,1064,957,856,761,672,589,511,439,373,313,259,210,142,73,100,114,119,123,126,128,166,209,257,311,371,437,509,586,669,758,853,954,1061,1174,1293,1418,1549,1686,1829,1978,2133,2294,2461,2634,2813,2994,3170,3333,3324,3159,2983,2802,2623,2450,2283,2122,1967,1818,1675,1538,1407,1282,1163,1050,943,842,747,658,575,498,427,362,303,249,201,159,122,78,31,48,59,62,64,90,121,157,199,247,300,359,424,495,572,655,744,839,940,1047,1160,1279,1404,1535,1672,1815,1964,2119,2280,2447,2620,2799,2980,3156,3147,2969,2788,2609,2436,2269,2108,1953,1804,1661,1524,1393,1268,1149,1036,929,828,733,644,561,484,413,348,289,236,189,148,113,83,58,36,10,16,24,38,57,81,110,145,186,233,286,345,410,481,558,641,730,825,926,1033,1146,1265,1390,1521,1658,1801,1950,2105,2266,2433,2606,2785,2961,2776,2597,2424,2257,2096,1941,1792,1649,1512,1381,1256,1137,1024,917,816,721,632,549,472,401,336,277,224,177,136,101,72,49,30,17,9,3)Polycyclic presentation:
2,3,4099|c^a=c^4098,c^b=c^2017689 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24594, 1);