Finite Group DatabasebetaFinite Group Databasebeta| order | 22105 |
| automorphism | ? |
| homology | Abelianization (H1) = C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C4421⋊C5 |
| Fitting, socle, derived, Sylow | C4421 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C4421 | C5 |
| LCS | G ▷ C4421 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 4421 |
| transitive degree | μt(G) = 4421 |
A minimal faithful permutation representation — 2 generators on 4421 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)(30,58,109,195,121)(32,62,117,211,362)(37,69,41,76,68)(39,73,132,106,189)(43,80,143,147,163)(45,84,150,173,122)(47,88,158,272,347)(51,95,169,92,164)(53,99,176,136,123)(55,103,184,313,321)(57,107,191,67,124)(59,111,199,339,279)(61,115,207,354,218)(63,119,215,370,605)(70,128,83,148,166)(72,120,98,174,126)(74,134,235,204,348)(77,138,130,94,167)(79,141,140,137,127)(81,145,250,265,322)(85,152,261,306,280)(87,156,268,332,219)(89,160,276,454,590)(96,171,291,190,323)(100,178,302,254,281)(102,182,309,295,220)(104,186,317,515,564)(108,193,328,165,282)(110,197,335,239,221)(112,201,343,556,522)(114,205,350,125,222)(116,209,358,582,460)(118,213,366,597,375)(129,227,155,266,325)(131,230,181,307,284)(133,233,196,333,224)(135,237,399,363,591)(139,242,232,192,326)(142,245,247,255,285)(144,248,258,296,225)(146,252,417,447,565)(149,256,288,168,286)(151,259,299,240,226)(153,263,432,508,523)(157,270,443,549,461)(159,274,450,575,376)(161,278,458,721,940)(170,289,175,297,229)(172,293,478,349,566)(177,300,244,241,217)(179,304,493,436,524)(183,311,504,497,462)(185,315,511,538,377)(187,319,519,804,914)(194,330,534,324,525)(198,337,545,421,463)(200,341,552,482,378)(202,345,560,865,872)(206,352,571,283,464)(208,356,578,402,379)(210,360,586,906,810)(212,364,593,223,380)(214,368,601,932,725)(216,372,609,947,614)(228,386,273,448,568)(231,390,314,509,527)(234,394,340,550,466)(236,397,355,576,382)(238,401,645,606,941)(243,406,396,351,569)(246,409,414,437,528)(249,412,429,498,467)(251,415,440,539,383)(253,419,666,714,915)(257,424,475,327,529)(260,427,490,422,468)(262,430,501,483,384)(264,434,684,797,873)(267,438,531,287,469)(269,441,542,403,385)(271,445,699,858,811)(275,452,710,899,726)(277,456,717,925,615)(290,473,308,499,471)(292,476,334,540,388)(294,480,750,592,916)(298,485,392,329,532)(301,488,411,423,472)(303,491,426,484,374)(305,495,767,703,874)(310,502,487,404,389)(312,506,782,786,812)(316,513,793,847,727)(318,517,800,888,616)(320,521,808,1193,1385)(331,536,828,567,875)(336,543,408,405,393)(338,547,843,688,813)(342,554,854,771,728)(344,558,861,832,617)(346,562,869,1275,1343)(353,573,884,526,814)(357,580,895,669,729)(359,584,902,753,618)(361,588,910,1336,1281)(365,595,921,465,730)(367,599,928,647,619)(369,603,936,1377,1197)(371,607,943,381,620)(373,611,951,1403,1090)(387,626,455,715,918)(391,631,516,798,877)(395,636,557,859,816)(398,640,583,900,732)(400,643,598,926,621)(407,652,642,594,919)(410,655,663,704,878)(413,658,681,787,817)(416,661,696,848,733)(418,664,707,889,622)(420,668,1015,1083,1386)(425,673,747,570,879)(428,676,765,689,818)(431,679,779,772,734)(433,682,790,833,623)(435,686,1036,1186,1344)(439,691,825,530,819)(442,694,840,670,735)(444,697,851,754,624)(446,701,1053,1268,1282)(449,705,881,470,736)(451,708,892,648,625)(453,712,1068,1329,1198)(457,719,1079,1370,1091)(459,723,1086,1396,956)(474,741,510,788,821)(477,745,551,849,738)(479,748,577,890,628)(481,752,1121,942,1387)(486,757,638,572,882)(489,760,660,690,822)(492,763,678,773,739)(494,766,693,834,613)(496,769,1139,1072,1345)(500,774,743,533,823)(503,777,762,671,740)(505,780,776,755,629)(507,784,1156,1175,1283)(512,791,837,649,630)(514,795,1171,1257,1199)(518,802,1182,1318,1092)(520,806,1189,1359,957)(535,826,541,835,633)(537,830,1222,917,1346)(544,838,657,672,744)(546,841,675,756,634)(548,845,1239,1057,1284)(553,852,759,650,635)(555,856,1253,1160,1200)(559,863,1264,1243,1093)(561,867,1271,1303,958)(563,871,1279,1783,1930)(574,886,1299,876,1285)(579,893,654,651,639)(581,897,1314,1039,1201)(585,904,1325,1142,1094)(587,908,1332,1225,959)(589,912,1340,1863,1869)(596,923,1355,815,1202)(600,930,1366,1017,1095)(602,934,1373,1123,960)(604,938,1381,1924,1787)(608,945,1392,731,1096)(610,949,1399,993,961)(612,953,1407,1963,1684)(627,968,722,1084,1389)(632,973,805,1187,1348)(637,979,866,1269,1287)(641,984,907,1330,1204)(644,988,933,1371,1097)(646,991,948,1397,962)(653,999,990,944,1390)(656,1001,1012,1073,1349)(659,1004,1033,1176,1288)(662,1007,1050,1258,1205)(665,1010,1065,1319,1098)(667,1013,1076,1360,963)(674,1022,1118,920,1350)(677,1025,1137,1058,1289)(680,1028,1154,1161,1206)(683,1031,1168,1244,1099)(685,1034,1179,1304,964)(687,1038,1496,1677,1931)(692,1043,1219,880,1290)(695,1046,1237,1040,1207)(698,1048,1250,1143,1100)(700,1051,1261,1226,965)(702,1055,1515,1776,1870)(706,1060,1296,820,1208)(709,1063,1311,1018,1101)(711,1066,1322,1124,966)(713,1070,1532,1856,1788)(716,1074,1352,737,1102)(718,1077,1363,994,967)(720,1081,1547,1917,1685)(724,1088,1558,1956,1562)(742,1107,799,1177,1292)(746,1112,860,1259,1210)(749,1116,901,1320,1104)(751,1119,927,1361,969)(758,1128,986,922,1353)(761,1131,1009,1059,1293)(764,1134,1030,1162,1211)(768,1138,1062,1305,955)(770,1141,1614,1551,1932)(775,1146,1114,883,1294)(778,1149,1136,1041,1212)(781,1152,1151,1144,1105)(783,1155,1165,1227,970)(785,1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2,1501,2055,2666,3268,3777,2805,676,38,72,131,231,391,632,974,658,23,45,85,153,264,435,687,636,978,1432,1988,2608,3226,3754,2761,3343,3826,3209,3737,3747,4116,3634,4034,4273,4381,3148,3680,3736,4104,3612,4016,4260,4374,3056,3592,3723,4094,4302,4392,4416,3922,2960,3508,3712,4087,4299,4390,4415,3918,1639,2199,2799,3372,3844,4165,4335,3914,1624,242,405,650,996,1453,2011,2630,1608,227,385,624,965,1419,1976,2597,1583,2143,2750,1954,2569,2590,3211,3740,4110,4312,4152,1915,2516,2586,3204,3730,4100,4306,4145,1854,2436,2579,3193,3718,4091,4300,4137,1774,2344,2568,3181,3709,4085,4297,4134,1675,2237,2554,3167,3696,4075,4291,2920,1560,2123,2734,3324,3817,4153,4331,2907,757,1127,1601,2163,2770,3351,3830,2896,736,1101,1572,2134,2744,3332,3821,2871,3433,3887,3319,923,1356,1886,2481,3085,3621,4023,3306,897,1315,1403,1956,2572,3187,3714,3288,856,1254,1399,1949,2562,3176,3706,3275,795,1172,1392,1938,2547,3159,3689,3262,712,1069,1531,2086,2695,3289,3789,3250,603,937,1380,1923,2529,3138,3669,1722,2291,2891,3451,3900,4066,4288,3967,1696,2262,2862,3427,3884,4056,4283,3954,4223,4357,4182,2097,2707,3301,3798,4142,4328,4401,4177,2081,330,535,827,1218,1707,2273,2872,2062,304,494,605,940,1385,1930,2539,2041,263,433,683,1032,1489,2042,2654,2021,201,344,559,864,1267,1765,2335,2003,2622,3238,3763,4127,3146,3676,2999,3543,3965,4230,4361,4400,3135,3664,2974,3520,3948,4220,4356,4395,3122,3654,4048,4281,4384,3396,3861,4175,3108,3642,4040,4276,4382,3386,1043,1502,2056,2667,3269,3778,4132,3377,1025,73,133,234,395,637,980,1435,1004,46,87,157,271,446,702,1056,979,1433,1989,2609,3227,1924,2530,3139,3670,4058,3738,4107,4117,1920,2523,3129,3660,4052,3681,4067,4105,1913,2512,3117,3650,4044,3593,4000,4095,1902,2499,3105,3640,4038,3509,3939,4088,1887,2482,3086,3622,4024,2200,2800,3373,3845,4166,4256,4372,4200,2184,406,651,997,1454,2012,2631,3244,2169,386,625,966,1420,1977,2598,3219,2144,2751,3336,2570,3184,3212,3741,4111,4313,4396,4330,2517,3123,3205,932,1370,1906,2505,3110,2437,3032,3194,928,1363,1896,2493,3099,2345,2941,3182,921,1352,1880,2473,3076,2238,2835,3168,3697,4076,3601,4006,3472,2124,2736,3327,3819,4155,3591,3997,3461,1128,1602,2164,2771,3352,3581,3991,3454,1102,1573,2135,2745,3333,3570,3983,3434,3888,4188,3812,1357,1888,2484,3089,3626,4028,4269,3801,1316,1829,1957,2573,3188,3715,4089,3788,1255,1749,1950,362,590,914,1343,1869,1173,1648,1939,2548,3160,2457,3058,3594,1070,1533,2089,2697,3291,2453,3051,3585,938,1382,1927,2536,3149,2446,3041,2292,2892,3452,3901,4195,4289,2435,3029,2263,2863,3428,3885,4187,4284,2422,3018,3559,3976,4235,2708,3302,3799,2410,3008,3551,3970,4232,2690,536,829,1221,1711,2278,2877,3438,2672,495,768,941,1386,1931,2540,3152,2653,434,685,1035,1492,2045,2657,3260,2637,345,561,868,1274,1775,1336,1856,2439,3035,3572,3985,4240,3677,1332,1849,2429,3025,3565,3980,4238,3665,1325,1838,2417,3015,3556,3973,4233,3655,1314,1826,2407,3006,3549,3968,4231,3643,1299,1809,2387,2985,3530,1503,2057,2668,3270,3779,3938,4213,3847,1482,134,236,398,641,985,1441,1998,1460,88,159,275,453,713,1071,1535,1434,1990,2610,3228,3755,2531,3140,3671,4059,4285,4108,4311,4316,2524,582,899,1318,1832,2412,3010,3552,2513,578,892,1308,1820,2401,3000,3544,2500,571,881,1292,1800,2377,2975,3521,2483,3087,3623,2958,3504,2801,3374,3846,4167,4336,4373,2949,3496,2786,652,998,1455,2013,2632,2940,3488,2774,626,967,1421,1978,2599,2929,3478,2752,3337,3823,3185,3713,3742,2916,3468,3910,4199,4348,3124,3657,3731,1371,1907,2506,3111,3644,3033,3571,3719,1364,203,347,564,872,1281,1787,2363,1353,1881,2474,1783,2356,2836,3403,3698,4077,4292,4007,1779,2349,2737,3328,3820,4156,4333,3998,1772,2340,1603,2165,2772,3353,3831,3992,1762,2330,1574,2136,2746,3334,3822,3984,1748,2316,2913,3466,3908,1889,2486,3092,1733,2301,2900,3457,3903,1830,2411,2574,1715,2283,2883,3443,3894,1750,2319,2563,591,915,1344,1870,2461,1649,2207,2549,3161,3690,3059,3595,4001,1534,2091,2700,3293,3792,3052,865,1268,1383,1928,2537,3150,3682,3042,861,1261,1756,2325,2922,3473,3912,3030,854,1250,1743,2312,2909,3462,3905,3019,843,1237,1730,2299,2898,3455,3902,3009,828,1219,1708,2274,2873,830,1223,1714,2282,2882,3417,3877,3272,769,1140,1387,1932,2541,3153,3684,3257,686,1037,1495,2048,2660,3263,3773,3246,562,870,1278,1782,2355,1857,2440,3036,3573,3986,4241,4364,4062,1850,339,549,847,1243,1736,2305,2903,1839,335,542,837,1231,1724,2293,2893,1827,328,531,821,1210,1698,2264,2864,1810,2388,2986,2247,2846,2058,2669,3271,3780,4133,4214,2236,2833,2036,237,400,644,989,1446,2004,2623,2016,160,277,457,720,1082,1550,2110,1991,2611,3229,3756,4123,3141,2193,2793,3368,3841,4163,4334,4398,3130,900,1319,1833,2413,3011,3553,3971,3118,893,105,188,321,522,810,1197,1684,882,1293,1801,1193,1677,2240,2838,3088,3624,4025,3505,1189,1670,2230,2828,3398,3862,4176,3497,1182,1659,999,1456,2014,2633,3245,3489,1171,1646,968,1422,1979,2600,3220,3479,1156,1630,2190,2791,3366,3839,4112,3469,1139,1612,2173,2777,3356,3832,4101,1908,1121,1594,2156,2763,3345,3827,4092,1897,348,565,873,1282,1788,2364,2962,1882,2475,3077,2357,2952,3404,3867,4078,4293,4386,4252,2350,515,797,1175,1651,2209,2808,3379,2341,511,790,1165,1641,2202,2802,3375,2331,504,779,1151,1626,2186,2787,3363,2317,493,765,1136,1610,2171,2775,3354,2302,478,747,1114,1585,2146,2753,3189,2284,2884,3444,2735,3325,2320,2917,3177,916,1345,1871,2462,3065,2208,2807,3162,3691,4071,3596,2701,3294,2092,2702,3296,3794,4139,3586,1269,1766,1929,2538,3151,3683,4068,3577,1262,180,306,497,771,1142,1616,2176,1251,176,299,487,759,1130,1604,2166,1238,169,288,471,738,1104,1575,2137,1220,1709,2275,1558,2119,1224,1716,2285,2886,3447,3878,1547,2104,1141,1615,1933,2542,3154,3685,1532,2087,1038,1497,2051,2663,3266,3776,1515,2068,871,1280,1786,2362,2961,2441,1496,2049,2661,3264,3774,4131,4287,2430,550,848,1244,1737,2306,2904,3459,2418,543,48,90,162,279,460,725,1090,532,822,1211,721,1083,1551,2111,2389,2987,3531,2847,717,1076,1541,2099,2710,3303,3800,2834,710,1065,401,646,992,1450,2009,2629,699,1050,278,459,724,1089,1561,2125,684,1033,1490,2043,2655,3258,3672,2794,666,1012,1468,2023,2639,3247,3661,1320,645,990,1447,2005,2624,3239,3651,1309,189,322,523,811,1198,1685,2251,1294,1802,2378,1678,2241,2839,3405,3625,4026,4266,3934,1671,272,447,703,1057,1518,2073,2683,1660,268,440,693,1045,1505,2059,2670,1647,261,429,678,1027,1484,2037,2650,1631,250,414,660,1006,1462,2017,2634,1613,235,396,638,981,1436,1992,2507,1595,2157,2764,1967,2591,3213,3743,2494,566,874,1283,1789,2365,2963,3510,2476,3078,3613,2953,1867,2458,3060,3597,4002,4250,4369,2944,798,1176,1652,2210,2809,3380,3848,2936,791,82,147,254,421,669,1017,1474,780,78,140,244,408,654,1000,1457,766,71,130,229,388,628,969,1423,748,1115,1586,951,1404,1958,2575,2885,3445,3895,3326,936,1378,1919,2522,1346,1872,2463,3066,910,1337,1858,2442,3037,3574,3987,3295,869,1276,1778,2348,2943,3491,3923,1767,808,1194,1679,2243,2842,3409,3872,1757,307,498,772,1143,1617,2177,2779,1744,300,17,33,64,121,218,375,614,289,472,739,370,606,942,1388,1710,2276,2874,2120,366,598,927,1362,1895,2492,3098,2105,358,583,901,1321,1834,2414,3012,2088,343,557,860,1260,1755,2324,2921,2069,317,516,799,1178,1655,2214,2814,2050,276,455,716,1075,1540,2098,2709,849,215,371,608,946,1395,1944,2557,838,91,163,280,461,726,1091,1562,823,1212,1699,1084,1552,2112,2722,2988,3532,3955,3413,1077,113,204,349,567,876,1286,1793,1066,109,196,334,541,836,1230,1723,1051,101,181,308,500,775,1147,1622,1034,86,155,267,439,692,1044,1504,1013,60,114,206,353,574,887,1302,991,1448,2006,2625,3240,3764,4045,1821,323,524,812,1199,1686,2252,2853,1803,2379,2976,2242,2840,3406,3868,4027,4267,4378,4211,2231,448,704,1058,1519,2074,2684,3280,2218,441,25,49,92,165,283,465,731,430,21,41,77,139,243,407,653,415,13,26,51,96,172,294,481,397,639,982,1437,1993,2612,3112,2158,2765,3346,2592,3215,3746,4115,3100,875,1284,1790,2366,2964,3511,3940,3079,3614,4017,3500,2459,3062,3600,4005,4251,4370,4409,3492,1177,1653,2211,2810,3381,3849,4168,3486,1166,148,255,422,670,1018,1475,2029,1152,141)Polycyclic presentation:
5,4421|b^a=b^301889 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22105, 1);