Finite Group DatabasebetaFinite Group Databasebeta| order | 23255 |
| automorphism | ? |
| homology | Abelianization (H1) = C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C4651⋊C5 |
| Fitting, socle, derived, Sylow | C4651 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C4651 | C5 |
| LCS | G ▷ C4651 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 4651 |
| transitive degree | μt(G) = 4651 |
A minimal faithful permutation representation — 2 generators on 4651 points:
(2,3,5,9,17)(4,7,13,25,48)(6,11,21,40,33)(8,15,29,56,104)(10,19,36,18,34)(12,23,44,81,89)(14,27,52,96,63)(16,31,60,112,200)(20,38,70,49,90)(22,42,77,74,64)(24,46,85,152,185)(26,50,92,35,65)(28,54,100,177,159)(30,58,108,192,118)(32,62,116,208,354)(37,68,41,75,67)(39,72,130,105,186)(43,79,141,145,160)(45,83,148,170,119)(47,87,156,266,339)(51,94,166,91,161)(53,98,173,134,120)(55,102,181,306,313)(57,106,188,66,121)(59,110,196,331,272)(61,114,204,346,212)(69,125,82,146,163)(71,128,97,171,123)(73,132,230,201,340)(76,136,127,93,164)(78,139,138,135,124)(80,143,244,259,314)(84,150,255,299,273)(86,154,262,324,213)(88,158,270,439,570)(95,168,284,187,315)(99,175,295,248,274)(101,179,302,288,214)(103,183,310,497,544)(107,190,320,162,275)(109,194,327,233,215)(111,198,335,537,503)(113,202,342,122,216)(115,206,350,562,443)(117,210,358,577,362)(126,221,153,260,317)(129,225,178,300,277)(131,228,193,325,218)(133,232,386,355,571)(137,236,227,189,318)(140,239,241,249,278)(142,242,252,289,219)(144,246,403,432,545)(147,250,281,165,279)(149,253,292,234,220)(151,257,417,490,504)(155,264,428,530,444)(157,268,435,555,363)(167,282,172,290,223)(169,286,461,341,546)(174,293,238,235,224)(176,297,475,421,505)(180,304,486,479,445)(182,308,493,519,364)(184,312,501,762,867)(191,322,515,316,506)(195,329,526,406,446)(197,333,533,464,365)(199,337,541,820,826)(203,344,551,276,447)(205,348,558,388,366)(207,352,566,860,766)(209,356,573,217,367)(211,360,581,885,685)(222,372,267,433,548)(226,377,307,491,508)(229,381,332,531,449)(231,384,347,556,368)(237,392,383,343,549)(240,395,400,422,509)(243,398,414,480,450)(245,401,425,520,369)(247,405,632,678,868)(251,409,458,319,510)(254,412,472,407,451)(256,415,483,465,370)(258,419,649,755,827)(261,423,512,280,452)(263,426,523,389,371)(265,430,663,813,767)(269,437,674,853,686)(271,441,681,878,585)(283,456,301,481,454)(285,459,326,521,374)(287,463,709,572,869)(291,467,379,321,513)(294,470,397,408,455)(296,473,411,466,375)(298,477,726,667,828)(303,484,469,390,376)(305,488,740,744,768)(309,495,751,802,687)(311,499,758,842,586)(323,517,784,547,829)(328,524,394,391,380)(330,528,798,652,769)(334,535,809,729,688)(336,539,816,787,587)(338,543,824,1183,1247)(345,553,838,507,770)(349,560,849,634,689)(351,564,856,711,588)(353,568,864,1241,1187)(357,575,874,448,690)(359,579,881,614,589)(361,583,889,1281,1106)(373,595,440,679,871)(378,600,498,756,831)(382,605,538,814,772)(385,609,563,854,691)(387,612,578,879,590)(393,619,611,574,872)(396,621,629,668,832)(399,624,646,745,773)(402,627,660,803,692)(404,630,671,843,591)(410,638,706,550,833)(413,641,723,653,774)(416,644,737,730,693)(418,647,748,788,592)(420,651,960,1099,1248)(424,655,781,511,775)(427,658,795,635,694)(429,661,806,712,593)(431,665,977,1176,1188)(434,669,835,453,695)(436,672,846,615,594)(438,676,991,1234,1107)(442,683,1002,1274,1006)(457,700,492,746,777)(460,704,532,804,697)(462,707,557,844,596)(468,715,607,552,836)(471,718,626,654,778)(474,721,643,731,698)(476,724,657,789,597)(478,728,1053,995,1249)(482,732,702,514,779)(485,735,720,636,699)(487,738,734,713,598)(489,742,1070,1088,1189)(494,749,792,616,599)(496,753,1084,1165,1108)(500,760,1095,1223,1007)(502,764,1102,1263,893)(516,782,522,790,602)(518,786,1130,870,1250)(525,793,623,637,703)(527,796,640,714,603)(529,800,1147,980,1190)(534,807,717,617,604)(536,811,1161,1073,1109)(540,818,1172,1150,1008)(542,822,1179,1208,894)(554,840,1205,830,1191)(559,847,620,618,608)(561,851,1219,962,1110)(565,858,1230,1055,1009)(567,862,1237,1132,895)(569,866,1245,1694,1698)(576,876,1259,771,1111)(580,883,1270,942,1010)(582,887,1277,1035,896)(584,891,1285,1752,1617)(601,907,763,1100,1252)(606,912,821,1177,1193)(610,917,861,1235,1112)(613,921,886,1275,1011)(622,929,939,996,1253)(625,931,957,1089,1194)(628,934,974,1166,1113)(631,937,988,1224,1012)(633,940,999,1264,897)(639,947,1032,873,1254)(642,949,1050,981,1195)(645,952,1067,1074,1114)(648,955,1081,1151,1013)(650,958,1092,1209,898)(656,966,1127,834,1196)(659,969,1144,963,1115)(662,972,1158,1056,1014)(664,975,1169,1133,899)(666,979,1369,1610,1699)(670,983,1202,776,1116)(673,986,1216,943,1015)(675,989,1227,1036,900)(677,993,1386,1687,1618)(680,997,1256,696,1016)(682,1000,1267,923,901)(684,1004,1400,1745,1517)(701,1021,757,1090,1198)(705,1026,815,1167,1118)(708,1030,855,1225,1017)(710,1033,880,1265,902)(716,1040,919,875,1257)(719,1042,936,982,1199)(722,1045,954,1075,1119)(725,1048,971,1152,1018)(727,1051,985,1210,903)(733,1059,1028,837,1200)(736,1062,1047,964,1120)(739,1065,1064,1057,1019)(741,1068,1078,1134,904)(743,1072,1471,1506,1700)(747,1076,1123,780,1121)(750,1079,1141,944,1020)(752,1082,1155,1037,905)(754,1086,1488,1599,1619)(759,1093,1213,924,906)(761,1097,1502,1676,1518)(765,1104,1513,1734,1404)(783,1125,805,1153,1023)(785,1128,845,1211,908)(791,1136,914,839,1203)(794,1139,933,965,1124)(797,1142,951,1058,1024)(799,1145,968,1135,909)(801,1149,1564,1389,1701)(808,1156,1044,945,1025)(810,1159,1061,1038,910)(812,1163,1581,1491,1620)(817,1170,1138,925,911)(819,1174,1595,1584,151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4328,4487,4066,2174,2676,3155,3580,3931,4202,4203,2531,718,65,120,214,364,586,893,612,920,1313,1779,2293,2818,1864,2365,2872,3349,3764,4096,4097,2176,2680,3160,3585,3936,4206,3700,1667,325,520,788,1133,1547,2013,1564,2029,2520,3005,3450,3832,3833,1889,2388,2891,3363,3774,4102,3417,1381,194,328,525,794,1140,1554,885,1274,1734,2240,2757,3245,3250,3675,4019,4274,4446,4551,4450,3119,1125,1536,2000,2493,2981,3431,2589,758,1092,1496,1956,2447,2937,2938,966,1355,1813,2318,2834,3322,2842,760,1096,1501,1961,2452,2941,2235,551,835,1198,1629,2107,2604,2609,3091,3526,3891,4176,4381,3966,2073,2565,3045,3481,3853,4148,3865,1946,417,646,954,1342,1801,2308,2309,419,650,959,1347,1806,2312,1707,2205,2716,3202,3631,3981,3300,3730,4072,4319,4479,4572,4495,3178,3604,3953,4219,4406,4525,4526,3471,1463,239,3,6,12,24,47,88,116,209,357,576,877,1262,1287,1757,2276,2807,3308,3742,3296,3726,4067,4313,4474,4568,4309,2674,807,96,170,288,464,711,1035,1040,1438,1899,2398,2900,3370,2904,917,1309,1775,2289,2815,3312,2363,564,857,1229,1668,2153,2652,1748,2259,2780,3271,3696,4036,4037,2149,2648,3127,3557,3915,4192,3923,2027,459,703,1024,1420,1880,2379,1513,1976,2470,2960,3413,3805,3806,1836,322,516,783,1126,1538,2003,1615,2093,2590,3073,3509,3876,3261,1219,1655,2137,2635,3115,3547,3548,1532,1996,2489,2978,3429,3817,3069,1090,1493,1952,2443,2933,3391,2949,977,1366,1823,2325,2838,3324,3325,1086,1489,1947,2437,2928,3387,2747,833,1195,1625,2103,2600,3083,2190,2698,3182,3610,3960,4225,3977,4241,4424,4538,4601,4633,4634,4154,2433,644,42,78,140,240,396,622,354,570,867,1247,1698,2196,2201,2712,3199,3629,3980,4244,4003,4263,4440,4549,4608,4636,4582,3602,1593,302,483,734,1061,1459,1916,1917,392,618,926,1319,1785,2298,1793,206,351,565,859,1233,1675,1283,1749,2261,2782,3273,3698,2797,3291,3719,4059,4305,4467,4468,3150,1153,1569,2035,2525,3009,3453,3021,1051,130,227,379,602,908,1300,1305,1771,2286,2813,3311,3743,2867,851,1220,1657,2139,2637,3117,2257,2777,3267,3692,4032,4283,4048,2160,2659,3137,3564,3919,4194,4195,2515,700,1020,1416,1876,2376,2881,1974,433,668,981,1372,1830,2331,1847,2347,2856,3337,3756,4092,4093,2083,2578,3059,3494,3863,4155,3682,1653,318,509,773,1113,1523,1987,1183,1610,2085,2580,3062,3498,3503,3871,4162,4372,4508,4588,4512,3401,1364,179,303,485,736,1063,1461,881,1267,1724,2228,2743,3231,3232,1191,1621,2099,2597,3081,3518,3107,3541,3904,4186,4388,4517,4239,2575,751,1081,1482,1939,2429,2921,2922,949,72,131,229,382,606,913,683,1003,1399,1860,2361,2868,2218,2731,3218,3646,3994,4254,3739,4082,4329,4488,4578,4623,4624,3948,2059,481,731,1057,1455,1912,2407,1928,403,629,936,1326,1789,2300,2301,337,542,823,1182,1609,2084,1743,2252,2771,3259,3683,4024,3289,1235,1677,2164,2665,3144,3570,3161,3586,3937,4207,4398,4521,4522,3459,1445,225,376,598,904,1296,1763,1000,112,201,341,547,830,1192,1207,1642,2124,2624,3108,3543,3194,3625,3976,4240,4423,4537,4293,2657,796,92,163,277,449,691,1011,1016,1412,1873,2374,2880,3355,2465,2955,3409,3803,4118,4347,4126,2345,539,817,1171,1591,2060,2550,1741,2248,2766,3254,3678,4020,4021,2132,506,769,1109,1519,1984,2479,2011,2505,2993,3442,3827,4134,3514,1502,1962,2453,2942,3397,3793,3794,1818,297,476,725,1049,1446,1904,1265,1721,2224,2739,3227,3656,3243,1205,1638,2118,2616,3098,3532,3537,3900,4183,4386,4516,4591,4419,3052,1079,146,249,407,635,943,1332,960,1348,1807,2313,2830,3320,3321,993,1387,1846,2346,2855,3336,2728,3214,3642,3990,4251,4430,4080,2183,537,813,1165,1584,2052,2542,2547,3027,3467,3843,4142,4360,4146,2417,627,27,53,99,176,298,478,358,578,880,1266,1723,2227,2239,2756,3244,3670,4015,4271,3714,4054,4301,4465,4562,4613,4570,3584,1579,295,472,720,1044,1441,1900,1901,372,594,900,1293,1761,2278,1392,1852,2353,2861,3341,3759,3353,1243,1691,2184,2689,3171,3597,2786,3277,3702,4041,4289,4455,4456,3132,1139,161,274,445,687,1007,1404,1033,1431,1892,2391,2894,3365,2476,2966,3419,3810,4122,4349,4350,2851,811,1162,1580,2046,2535,3016,2246,556,843,1209,1644,2126,2625,2143,2641,3121,3552,3911,4190,4191,2501,2989,3438,3824,4132,4355,3878,1960,426,25,49,91,162,276,448,463,710,1034,1433,1895,2395,1981,2477,2968,3421,3812,4124,3666,1636,2115,2612,3094,3528,3892,3190,1179,1603,2075,2568,3048,3484,3485,1477,1934,2425,2919,3381,3783,3389,1346,154,263,427,659,970,1359,874,1256,1709,2208,2719,3205,3210,3638,3987,4249,4429,4541,4324,2685,3166,3592,3942,4211,4401,4223,2558,740,1067,1465,1921,2413,2909,2910,931,46,86,155,265,431,666,575,875,1258,1712,2212,2724,2275,2805,3304,3735,4078,4325,3725,4065,4311,4472,4566,4615,4616,3932,2042,470,34,64,119,213,363,585,386,611,919,1312,1778,2292,2297,2822,3318,3748,4090,4337,3765,1685,2175,2678,3157,3582,3933,3275,1228,192,324,519,787,1132,1546,1551,2017,2511,2999,3447,3831,3451,1428,1888,2387,2890,3362,3773,2964,989,108,193,326,522,791,1137,1149,1565,2031,2522,3007,3452,2761,3249,3674,4018,4273,4445,4281,2639,782,1124,1535,1999,2492,2980,2092,2588,3071,3507,3874,4163,4164,2448,655,965,1354,1812,2317,2833,2329,499,759,1094,1498,1958,2449,1730,2234,2749,3236,3662,4008,4009,2111,2608,3090,3525,3890,4175,3616,1601,2072,2564,3044,3480,3852,3496,1488,1945,2435,2926,3385,3785,3786,1802,257,418,648,956,1344,1803,1254,1706,2204,2715,3201,3630,2802,3299,3729,4071,4318,4478,4335,4494,4584,4628,4645,4650,4651,4407,3034,1065,139)Polycyclic presentation:
5,4651|b^a=b^3722935 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23255, 1);