Finite Group DatabasebetaFinite Group Databasebeta| order | 19655 |
| automorphism | ? |
| homology | Abelianization (H1) = C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3931⋊C5 |
| Fitting, socle, derived, Sylow | C3931 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3931 | C5 |
| LCS | G ▷ C3931 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 3931 |
| transitive degree | μt(G) = 3931 |
A minimal faithful permutation representation — 2 generators on 3931 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)(63,119,215,370,512)(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)(161,278,458,718,789)(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)(216,372,608,858,612)(227,385,273,448,568)(231,390,314,509,527)(234,394,340,550,466)(236,397,355,576,381)(238,401,643,605,791)(243,406,396,351,569)(246,409,414,437,528)(249,412,429,498,467)(251,415,440,539,382)(253,419,663,711,911)(257,424,475,327,529)(260,427,490,422,468)(262,430,501,483,383)(264,434,681,794,869)(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)(305,495,765,700,870)(310,502,487,404,389)(312,506,780,784,809)(316,513,790,843,724)(318,517,797,884,614)(320,521,805,1180,1156)(331,536,825,567,871)(336,543,408,405,393)(338,547,839,685,810)(342,554,850,769,725)(344,558,857,829,615)(346,562,865,1260,1327)(353,573,880,526,811)(357,580,891,666,726)(359,584,898,750,616)(361,588,906,1320,1266)(365,595,917,465,727)(367,599,924,645,617)(369,603,932,1361,1184)(371,606,798,380,618)(373,610,943,1317,1077)(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)(407,649,640,594,915)(410,652,660,701,874)(413,655,678,785,814)(416,658,693,844,730)(418,661,704,885,620)(420,665,1002,1070,1158)(425,670,744,570,875)(428,673,762,686,815)(431,676,777,770,731)(433,679,788,830,621)(435,683,1022,1173,1328)(439,688,822,530,816)(442,691,836,667,732)(444,694,847,751,622)(446,698,1040,1253,1267)(449,702,877,470,733)(451,705,888,646,623)(453,709,1055,1313,1185)(457,716,1066,1354,1078)(459,720,1073,1248,947)(474,738,510,786,818)(477,742,551,845,735)(479,745,577,886,626)(481,749,1108,936,1160)(486,754,636,572,878)(489,757,657,687,819)(492,760,675,771,736)(494,763,690,831,627)(496,767,1128,1059,1329)(500,772,740,533,820)(503,775,759,668,737)(505,778,774,752,628)(507,782,1146,1163,1268)(514,792,1159,1243,1186)(518,799,1169,1302,1079)(520,803,1176,1343,948)(535,823,541,832,631)(537,827,1209,913,1330)(544,834,654,669,741)(546,837,672,753,632)(548,841,1225,1044,1269)(553,848,756,647,633)(555,852,1239,1150,1187)(559,859,1249,1229,1080)(561,863,1256,1288,949)(563,867,1264,1744,1610)(574,882,1284,872,1270)(579,889,651,648,637)(581,893,1298,1025,1188)(585,900,1309,1131,1081)(587,904,1316,1212,950)(589,908,1324,1823,1829)(596,919,1339,812,1189)(600,926,1350,1004,1082)(602,930,1357,1110,951)(604,934,1365,1883,1748)(607,938,1177,728,1083)(609,941,1257,982,952)(611,945,1380,1880,1649)(625,959,719,1071,1168)(630,963,802,1174,1332)(635,968,862,1254,1272)(639,973,903,1314,1191)(642,977,929,1355,1084)(644,980,940,1250,953)(650,986,979,937,1170)(653,988,999,1060,1333)(656,991,1019,1164,1273)(659,994,1037,1244,1192)(662,997,1052,1303,1085)(664,1000,1063,1344,954)(671,1008,1105,916,1334)(674,1011,1125,1045,1274)(677,1014,1143,1151,1193)(680,1017,1157,1230,1086)(682,1020,1167,1289,955)(684,1024,1457,1642,1612)(689,1029,1206,876,1275)(692,1032,1222,1026,1194)(695,1035,1236,1132,1087)(697,1038,1247,1213,956)(699,1042,1477,1737,1830)(703,1047,1281,817,1195)(706,1050,1295,1005,1088)(708,1053,1306,1111,957)(710,1057,1495,1816,1749)(713,1061,1336,734,1089)(715,1064,1347,983,958)(717,1068,1510,1876,1650)(721,1075,1521,1811,1525)(739,1094,796,1165,1277)(743,1099,856,1245,1197)(746,1103,897,1304,1091)(748,1106,923,1345,960)(755,1114,975,918,1337)(758,1117,996,1046,1278)(761,1120,1016,1152,1198)(764,1123,1034,1231,1092)(766,1126,1049,1290,961)(768,1130,1580,1514,1614)(773,1135,1101,879,1279)(776,1138,1122,1027,1199)(779,1141,1140,1133,1093)(781,1144,1155,1214,962)(783,114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904,3346,1348,203,347,564,868,1266,1748,2283,928,1354,1865,2422,2958,1929,1902,2469,3012,3448,3730,3872,3523,1429,111,200,342,555,853,1242,1715,1017,84,151,260,428,674,1012,1444,694,52,98,175,298,486,755,1115,450,704,1049,1486,1999,2545,1671,1645,2172,2714,3199,3564,3786,3313,1220,167,285,467,730,1085,1532,2051,848,97,173,295,482,750,1110,1560,556,854,1243,1716,2245,2779,3246,1823,2367,2897,3341,3649,2807,3269,1334,1837,2386,2920,3363,3667,2365,2894,922,1344,1852,2406,2942,3382,1881,2443,2982,3419,3707,3858,3514,1406,1378,1898,2465,3008,3444,3726,3142,994,54,102,183,312,507,783,1149,676,42,79,142,246,410,653,989,441,25,49,92,165,283,465,728,272,447,700,1044,1480,1993,1200,1180,1642,2168,2710,3195,3561,2865,834,91,163,280,461,723,1078,1525,550,844,1230,1700,2228,2763,3234,1817,2359,2890,3334,3644,2708,3193,3435,3721,3869,3921,3931,3877,2265,2798,912,1329,1831,2380,2914,3358,1814,2355,2884,3327,3394,3686,3509,3500,1359,1873,2431,2967,3403,3693,3127,968,941,1374,1894,2461,3004,3440,2649,655,23,45,85,153,264,435,684,427,19,37,70,128,227,386,625,266,437,686,1026,1460,1972,2522,1174,1633,2156,2695,3180,3547,2849,816,1194,1659,2187,2728,3210,3570,2335,540,831,1214,1108,1557,2076,2620,1808,2348,2877,2777,3243,2581,3093,3431,3719,3867,3823,3903,3875,2153,2692,3176,3544,3774,3887,3925,3741,1726,2256,2789,3253,3380,3678,3842,3491,1311,1803,2341,2868,2955,3391,3115,3099,926,1351,1860,2414,2949,3387,2629,629,606,937,1370,1890,2457,3000,2109,409,7,14,28,55,104,187,320,256,423,668,458,719,1072,1517,1165,1621,2141,1991,2538,3056,2832,2811,3274,3609,3559,3784,3892,3849,2317,525,810,1187,1652,2180,2722,3206,1795,2332,2860,2760,747,1105,1553,2071,2615,3116,3506,2242,2774,2013,2561,3075,3483,3746,3646,3822,3736,1617,2137,2674,3160,3533,3768,3885,3475,1241,1713,2241,2773,2937,3376,3674,3084,893,1299,1787,2322,2418,2952,2609,2587,595,918,1338,1843,2394,2928,2086,385,623,957,1391,1912,2478,3018,1589,242,405,647,372,609,942,1376,1153,1606,2124,1969,276,455,713,1062,1503,2019,2567,1477,1988,2535,2294,2828,3287,3618,3192,3558,3783,3691,1779,2315,2846,2748,3227,3583,3797,3416,3704,3856,3816,2223,478,744,1101,1548,2066,2610,3112,1712,2239,1479,1992,2540,3058,3471,3337,3645,3465,1148,1601,2119,2656,3146,3524,3764,3063,841,1226,1695,2221,2400,2933,3372,2569,573,881,1283,1768,1861,2415,2060,2036,2585,3096,2999,3438,3724,3586,1566,221,378,616,950,1385,1907,2474,1135,1585,2102,1949,213,367,600,927,1353,1864,2421,1455,158,273,449,703,1048,1485,1998,1040,1474,1985,1753,2288,2822,3282,2706,3191,3557,3399,3690,3848,3814,2205,2744,3224,3581,3796,3897,3927,3631,1694,291,475,740,1096,1543,2061,2606,1239,1709,1024,1458,1970,2521,3045,2893,3336,3048,767,1129,1579,2096,2636,3131,3515,2546,536,826,1208,1675,1845,2396,2929,2018,2565,3077,2911,3355,1350,1858,1537,2056,2603,3108,2453,2996,3433,3237,1114,1563,2081,1934,1904,2472,3015,2783,3249,3591,3526,1432,115,208,357,581,894,1301,1791,1020,86,155,267,439,689,1030,1464,696,1037,1471,1982,2530,3049,3464,2164,2704,3189,3555,3781,3890,3625,1674,2201,2741,3221,3578,3793,3895,3314,1223,169,288,471,735,1091,1538,2057,850,1236,665,1003,1436,1950,2506,2363,2892,2526,480,748,1107,1556,2075,2619,3120,1997,2543,3059,2816,3277,1337,1841,2391,2925,3368,3671,2372,2903,924,1347,1083,1530,2050,2598,1885,2450,2991,2766,3236,3585,3516,1411,1380,1901,2468,3011,3447,3729,3145,997,58,110,198,338,548,842,1228,679,44,83,149,257,425,671,1009,443,693,1034,1468,1979,2527,3047,1638,2162,2702,3187,3553,3779,3302,1204,1670,2198,2738,3218,3575,3791,2867,837,93,166,284,466,729,1084,1531,552,847,401,644,981,1416,1935,1819,2361,1975,2524,3046,2720,3205,3437,3723,3550,3778,3889,3879,2274,2806,915,1333,1836,2385,2919,3362,1821,2364,597,921,1343,1851,2405,2941,1363,1880,2441,2980,3418,3706,3130,973,943,1377,1897,2464,3007,3443,2652,658,27,53,100,179,305,496,768,430,21,41,77,139,243,407,650,268,440,690,1031,1465,1976,2525,1176,1636,2160,2700,3185,3551,2851,820,1199,1665,2193,2734,3215,3573,2337,543,48,90,162,279,460,722,1077,339,549,843,1229,1699,2227,2762,1320,1816,2358,2889,3333,2166,2707,2997,3434,3720,3868,3920,3743,1735,2264,591,911,1328,1830,2379,2913,1318,1813,2353,2882,3326,3638,3119,3101,930,1358,1871,2428,2965,3401,2632,634,608,940,1373,1893,2460,3003,2112,412,11,22,43,81,146,253,420,259,9,18,35,67,124,222,380,154,265,436,685,1025,1459,1971,801,1173,1632,2155,2694,3179,2319,529,815,1193,1658,2186,2727,3209,1797,333,539,830,1213,1682,2211,2750,1314,1807,2347,2876,3321,2031,2580,2993,3430,3718,3866,3919,3737,1630,2152,2690,3174,3542,3773,3886,3477,1251,1725,2254,2787,2939,3379,3677,3087,900,1310,1801,2338,2866,3312,2613,2589,599,925,1349,1857,2412,2947,2089,390,370,605,936,1369,1889,2456,1592,245,3,6,12,24,47,89,161,148,255,422,667,1005,1438,1951,795,1164,1620,2140,2678,3163,2298,2277,2810,3273,3608,3807,3899,3692,1781,323,524,809,1186,1651,2179,2721,1304,1794,2331,2225,2759,1888,2455,2989,3428,3716,3642,3820,3735,1497,2012,2559,3072,3480,3744,3878,3466,1161,1616,2135,2672,3158,3532,3767,3066,852,1240,1711,2238,2402,2936,3375,2572,580,892,1297,1784,2320,2850,2064,2038,364,594,916,1335,1839,2389,1569,226,384,622,956,1390,1911,2477,1138,138,241,404,215,371,607,939,786,1152,1605,1457,1968,2519,3043,2269,2803,3266,3177,3545,3775,3682,1758,2293,2827,3286,3617,3811,3900,3400,1291,1778,2314,2208,2747,3226,3582,2978,3415,3703,3634,1696,2222,1326,1828,2377,2912,3357,3330,3640,3459,1042,1478,1990,2537,3055,3469,3738,3050,782,1147,1599,2116,2653,3143,3521,2549,547,840,1224,1692,1847,2399,2932,2021,352,572,879,1280,1764,2299,1541,1516,2035,2584,2454,2998,3436,3238,1117,123,220,377,615,949,1384,1906,772,1134,1584,1435,117,212,365,596,920,1342,1850,1022,1454,1965,2516,3040,3460,3734,2691,3175,3543,3385,1270,1752,2287,2821,3281,3614,3809,2962,3398,3689,3626,1677,2204,2743,3223,3580,3795,3896,3317,1225,1693,867,1265,1747,2282,2817,2885,3328,3036,683,1023,1456,1967,2518,3042,3462,2528,495,766,1127,1576,2093,2633,3128,2000,330,535,824,1205,1339,1844,2395,1502,2017,2564,2376,2910,3353,3230,1089,1536,2055,2602,1886,2452,2994,2767,754,1113,1562,1415,1933,2494,3028,2249,2782,3248,3148,1000,60,114,206,353,574,883,1287,681,1019,1451,1962,2513,3037,3458,2151,2689,3173,2945,3384,3681,3624,3611,3808,3713,3863,3918,3930,3929,3303,1207,1673,2200,2740,3220,3577,3792,2870,839,1222,521,806,1183,1648,2177,2354,2883,2509,419,664,1001,1433,1947,2503,3033,1977,293,479,746,1104,1552,2070,2614,1484,1996,2542,2281,2815,917,1336,1840,2390,2924,3367,1825,2371,2901,2753,727,1082,1529,2049,1365,1884,2448,2231,2765,3235,3133,977,1410,1928,2490,3026,3455,3733,2655,661,29,57,108,194,331,537,828,432,678,1016,1448,1959,2510,3035,1628,2149,2687,3171,3540,3771,3294,3276,3610,3411,3701,3855,3914,3912,2852,823,1203,1669,2197,2737,3217,3574,2340,545,836,278,459,721,1076,1524,1812,2352,1955,237,400,642,978,1412,1930,2492,1463,1974,2523,2176,2719,3203,3567,3183,3549,3777,3745,1742,2273,593,914,1332,1835,2384,2918,1322,1820,2362,2214,2752,3229,3123,3104,932,1362,1878,2439,2979,3417,2635,638,972,1405,1923,2486,3023,3453,2115,415,13,26,51,96,172,294,481,261,429,675,1013,1445,1956,2508,1169,1626,2147,2685,3169,3538,2836,2814,3275,2957,3395,3688,3847,3850,2321,532,819,1198,1664,2192,2733,3214,1800,335,542,119,216,373,611,946,1316,1810,1441,1953,2507,2045,2595,2995,3432,3166,3537,3770,3739,1640,2165,2705,3190,3556,3782,3891,3479,1258,1734,362,590,910,1327,1829,2378,904,1317,1811,2351,2881,3325,2618,2592,601,929,1356,1868,2426,2963,2092,394,633,967,1400,1919,2483,3021,1595,248,5,10,20,39,74,135,238,150,258,426,672,1010,1442,1954,797,1167,1624,2145,2683,3167,2302,2279,2812,2404,2940,3381,3680,3694,1783,326,528,814,1192,1657,2185,2726,1307,195,332,538,829,1212,1681,2210,902,1313,1806,2346,2875,1512,2030,2451,2992,3429,3717,3865,3468,1171,1629,2150,2688,3172,3541,3772,3068,859,1250,1723,2252,2786,3252,3593,2575,584,899,1308,1798,2336,2864,2069,2041,366,598,923,1346,1855,2410,1572,230,389,628,962,1396,1916,2481,1141,141)Polycyclic presentation:
5,3931|b^a=b^3250791 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19655, 1);