Finite Group Databasebeta

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Q₄₀₉₆ #72210165

• p-group
• nilpotent
• metacyclic
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

Subgroups

33 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 4096 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1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61,3609)(538,2586,1562,3610)(539,2587,1563,3611)(540,2588,1564,3612)(541,2589,1565,3613)(542,2590,1566,3614)(543,2591,1567,3615)(544,2592,1568,3616)(545,2593,1569,3617)(546,2594,1570,3618)(547,2595,1571,3619)(548,2596,1572,3620)(549,2597,1573,3621)(550,2598,1574,3622)(551,2599,1575,3623)(552,2600,1576,3624)(553,2601,1577,3625)(554,2602,1578,3626)(555,2603,1579,3627)(556,2604,1580,3628)(557,2605,1581,3629)(558,2606,1582,3630)(559,2607,1583,3631)(560,2608,1584,3632)(561,2609,1585,3633)(562,2610,1586,3634)(563,2611,1587,3635)(564,2612,1588,3636)(565,2613,1589,3637)(566,2614,1590,3638)(567,2615,1591,3639)(568,2616,1592,3640)(569,2617,1593,3641)(570,2618,1594,3642)(571,2619,1595,3643)(572,2620,1596,3644)(573,2621,1597,3645)(574,2622,1598,3646)(575,2623,1599,3647)(576,2624,1600,3648)(577,2625,1601,3649)(578,2626,1602,3650)(579,2627,1603,3651)(580,2628,1604,3652)(581,2629,1605,3653)(582,2630,1606,3654)(583,2631,1607,3655)(584,2632,1608,3656)(585,2633,1609,3657)(586,2634,1610,3658)(587,2635,1611,3659)(588,2636,1612,3660)(589,2637,1613,3661)(590,2638,1614,3662)(591,2639,1615,3663)(592,2640,1616,3664)(593,2641,1617,3665)(594,2642,1618,3666)(595,2643,1619,3667)(596,2644,1620,3668)(597,2645,1621,3669)(598,2646,1622,3670)(599,2647,1623,3671)(600,2648,1624,3672)(601,2649,1625,3673)(602,2650,1626,3674)(603,2651,1627,3675)(604,2652,1628,3676)(605,2653,1629,3677)(606,2654,1630,3678)(607,2655,1631,3679)(608,2656,1632,3680)(609,2657,1633,3681)(610,2658,1634,3682)(611,2659,1635,3683)(612,2660,1636,3684)(613,2661,1637,3685)(614,2662,1638,3686)(615,2663,1639,3687)(616,2664,1640,3688)(617,2665,1641,3689)(618,2666,1642,3690)(619,2667,1643,3691)(620,2668,1644,3692)(621,2669,1645,3693)(622,2670,1646,3694)(623,2671,1647,3695)(624,2672,1648,3696)(625,2673,1649,3697)(626,2674,1650,3698)(627,2675,1651,3699)(628,2676,1652,3700)(629,2677,1653,3701)(630,2678,1654,3702)(631,2679,1655,3703)(632,2680,1656,3704)(633,2681,1657,3705)(634,2682,1658,3706)(635,2683,1659,3707)(636,2684,1660,3708)(637,2685,1661,3709)(638,2686,1662,3710)(639,2687,1663,3711)(640,2688,1664,3712)(641,2689,1665,3713)(642,2690,1666,3714)(643,2691,1667,3715)(644,2692,1668,3716)(645,2693,1669,3717)(646,2694,1670,3718)(647,2695,1671,3719)(648,2696,1672,3720)(649,2697,1673,3721)(650,2698,1674,3722)(651,2699,1675,3723)(652,2700,1676,3724)(653,2701,1677,3725)(654,2702,1678,3726)(655,2703,1679,3727)(656,2704,1680,3728)(657,2705,1681,3729)(658,2706,1682,3730)(659,2707,1683,3731)(660,2708,1684,3732)(661,2709,1685,3733)(662,2710,1686,3734)(663,2711,1687,3735)(664,2712,1688,3736)(665,2713,1689,3737)(666,2714,1690,3738)(667,2715,1691,3739)(668,2716,1692,3740)(669,2717,1693,3741)(670,2718,1694,3742)(671,2719,1695,3743)(672,2720,1696,3744)(673,2721,1697,3745)(674,2722,1698,3746)(675,2723,1699,3747)(676,2724,1700,3748)(677,2725,1701,3749)(678,2726,1702,3750)(679,2727,1703,3751)(680,2728,1704,3752)(681,2729,1705,3753)(682,2730,1706,3754)(683,2731,1707,3755)(684,2732,1708,3756)(685,2733,1709,3757)(686,2734,1710,3758)(687,2735,1711,3759)(688,2736,1712,3760)(689,2737,1713,3761)(690,2738,1714,3762)(691,2739,1715,3763)(692,2740,1716,3764)(693,2741,1717,3765)(694,2742,1718,3766)(695,2743,1719,3767)(696,2744,1720,3768)(697,2745,1721,3769)(698,2746,1722,3770)(699,2747,1723,3771)(700,2748,1724,3772)(701,2749,1725,3773)(702,2750,1726,3774)(703,2751,1727,3775)(704,2752,1728,3776)(705,2753,1729,3777)(706,2754,1730,3778)(707,2755,1731,3779)(708,2756,1732,3780)(709,2757,1733,3781)(710,2758,1734,3782)(711,2759,1735,3783)(712,2760,1736,3784)(713,2761,1737,3785)(714,2762,1738,3786)(715,2763,1739,3787)(716,2764,1740,3788)(717,2765,1741,3789)(718,2766,1742,3790)(719,2767,1743,3791)(720,2768,1744,3792)(721,2769,1745,3793)(722,2770,1746,3794)(723,2771,1747,3795)(724,2772,1748,3796)(725,2773,1749,3797)(726,2774,1750,3798)(727,2775,1751,3799)(728,2776,1752,3800)(729,2777,1753,3801)(730,2778,1754,3802)(731,2779,1755,3803)(732,2780,1756,3804)(733,2781,1757,3805)(734,2782,1758,3806)(735,2783,1759,3807)(736,2784,1760,3808)(737,2785,1761,3809)(738,2786,1762,3810)(739,2787,1763,3811)(740,2788,1764,3812)(741,2789,1765,3813)(742,2790,1766,3814)(743,2791,1767,3815)(744,2792,1768,3816)(745,2793,1769,3817)(746,2794,1770,3818)(747,2795,1771,3819)(748,2796,1772,3820)(749,2797,1773,3821)(750,2798,1774,3822)(751,2799,1775,3823)(752,2800,1776,3824)(753,2801,1777,3825)(754,2802,1778,3826)(755,2803,1779,3827)(756,2804,1780,3828)(757,2805,1781,3829)(758,2806,1782,3830)(759,2807,1783,3831)(760,2808,1784,3832)(761,2809,1785,3833)(762,2810,1786,3834)(763,2811,1787,3835)(764,2812,1788,3836)(765,2813,1789,3837)(766,2814,1790,3838)(767,2815,1791,3839)(768,2816,1792,3840)(769,2817,1793,3841)(770,2818,1794,3842)(771,2819,1795,3843)(772,2820,1796,3844)(773,2821,1797,3845)(774,2822,1798,3846)(775,2823,1799,3847)(776,2824,1800,3848)(777,2825,1801,3849)(778,2826,1802,3850)(779,2827,1803,3851)(780,2828,1804,3852)(781,2829,1805,3853)(782,2830,1806,3854)(783,2831,1807,3855)(784,2832,1808,3856)(785,2833,1809,3857)(786,2834,1810,3858)(787,2835,1811,3859)(788,2836,1812,3860)(789,2837,1813,3861)(790,2838,1814,3862)(791,2839,1815,3863)(792,2840,1816,3864)(793,2841,1817,3865)(794,2842,1818,3866)(795,2843,1819,3867)(796,2844,1820,3868)(797,2845,1821,3869)(798,2846,1822,3870)(799,2847,1823,3871)(800,2848,1824,3872)(801,2849,1825,3873)(802,2850,1826,3874)(803,2851,1827,3875)(804,2852,1828,3876)(805,2853,1829,3877)(806,2854,1830,3878)(807,2855,1831,3879)(808,2856,1832,3880)(809,2857,1833,3881)(810,2858,1834,3882)(811,2859,1835,3883)(812,2860,1836,3884)(813,2861,1837,3885)(814,2862,1838,3886)(815,2863,1839,3887)(816,2864,1840,3888)(817,2865,1841,3889)(818,2866,1842,3890)(819,2867,1843,3891)(820,2868,1844,3892)(821,2869,1845,3893)(822,2870,1846,3894)(823,2871,1847,3895)(824,2872,1848,3896)(825,2873,1849,3897)(826,2874,1850,3898)(827,2875,1851,3899)(828,2876,1852,3900)(829,2877,1853,3901)(830,2878,1854,3902)(831,2879,1855,3903)(832,2880,1856,3904)(833,2881,1857,3905)(834,2882,1858,3906)(835,2883,1859,3907)(836,2884,1860,3908)(837,2885,1861,3909)(838,2886,1862,3910)(839,2887,1863,3911)(840,2888,1864,3912)(841,2889,1865,3913)(842,2890,1866,3914)(843,2891,1867,3915)(844,2892,1868,3916)(845,2893,1869,3917)(846,2894,1870,3918)(847,2895,1871,3919)(848,2896,1872,3920)(849,2897,1873,3921)(850,2898,1874,3922)(851,2899,1875,3923)(852,2900,1876,3924)(853,2901,1877,3925)(854,2902,1878,3926)(855,2903,1879,3927)(856,2904,1880,3928)(857,2905,1881,3929)(858,2906,1882,3930)(859,2907,1883,3931)(860,2908,1884,3932)(861,2909,1885,3933)(862,2910,1886,3934)(863,2911,1887,3935)(864,2912,1888,3936)(865,2913,1889,3937)(866,2914,1890,3938)(867,2915,1891,3939)(868,2916,1892,3940)(869,2917,1893,3941)(870,2918,1894,3942)(871,2919,1895,3943)(872,2920,1896,3944)(873,2921,1897,3945)(874,2922,1898,3946)(875,2923,1899,3947)(876,2924,1900,3948)(877,2925,1901,3949)(878,2926,1902,3950)(879,2927,1903,3951)(880,2928,1904,3952)(881,2929,1905,3953)(882,2930,1906,3954)(883,2931,1907,3955)(884,2932,1908,3956)(885,2933,1909,3957)(886,2934,1910,3958)(887,2935,1911,3959)(888,2936,1912,3960)(889,2937,1913,3961)(890,2938,1914,3962)(891,2939,1915,3963)(892,2940,1916,3964)(893,2941,1917,3965)(894,2942,1918,3966)(895,2943,1919,3967)(896,2944,1920,3968)(897,2945,1921,3969)(898,2946,1922,3970)(899,2947,1923,3971)(900,2948,1924,3972)(901,2949,1925,3973)(902,2950,1926,3974)(903,2951,1927,3975)(904,2952,1928,3976)(905,2953,1929,3977)(906,2954,1930,3978)(907,2955,1931,3979)(908,2956,1932,3980)(909,2957,1933,3981)(910,2958,1934,3982)(911,2959,1935,3983)(912,2960,1936,3984)(913,2961,1937,3985)(914,2962,1938,3986)(915,2963,1939,3987)(916,2964,1940,3988)(917,2965,1941,3989)(918,2966,1942,3990)(919,2967,1943,3991)(920,2968,1944,3992)(921,2969,1945,3993)(922,2970,1946,3994)(923,2971,1947,3995)(924,2972,1948,3996)(925,2973,1949,3997)(926,2974,1950,3998)(927,2975,1951,3999)(928,2976,1952,4000)(929,2977,1953,4001)(930,2978,1954,4002)(931,2979,1955,4003)(932,2980,1956,4004)(933,2981,1957,4005)(934,2982,1958,4006)(935,2983,1959,4007)(936,2984,1960,4008)(937,2985,1961,4009)(938,2986,1962,4010)(939,2987,1963,4011)(940,2988,1964,4012)(941,2989,1965,4013)(942,2990,1966,4014)(943,2991,1967,4015)(944,2992,1968,4016)(945,2993,1969,4017)(946,2994,1970,4018)(947,2995,1971,4019)(948,2996,1972,4020)(949,2997,1973,4021)(950,2998,1974,4022)(951,2999,1975,4023)(952,3000,1976,4024)(953,3001,1977,4025)(954,3002,1978,4026)(955,3003,1979,4027)(956,3004,1980,4028)(957,3005,1981,4029)(958,3006,1982,4030)(959,3007,1983,4031)(960,3008,1984,4032)(961,3009,1985,4033)(962,3010,1986,4034)(963,3011,1987,4035)(964,3012,1988,4036)(965,3013,1989,4037)(966,3014,1990,4038)(967,3015,1991,4039)(968,3016,1992,4040)(969,3017,1993,4041)(970,3018,1994,4042)(971,3019,1995,4043)(972,3020,1996,4044)(973,3021,1997,4045)(974,3022,1998,4046)(975,3023,1999,4047)(976,3024,2000,4048)(977,3025,2001,4049)(978,3026,2002,4050)(979,3027,2003,4051)(980,3028,2004,4052)(981,3029,2005,4053)(982,3030,2006,4054)(983,3031,2007,4055)(984,3032,2008,4056)(985,3033,2009,4057)(986,3034,2010,4058)(987,3035,2011,4059)(988,3036,2012,4060)(989,3037,2013,4061)(990,3038,2014,4062)(991,3039,2015,4063)(992,3040,2016,4064)(993,3041,2017,4065)(994,3042,2018,4066)(995,3043,2019,4067)(996,3044,2020,4068)(997,3045,2021,4069)(998,3046,2022,4070)(999,3047,2023,4071)(1000,3048,2024,4072)(1001,3049,2025,4073)(1002,3050,2026,4074)(1003,3051,2027,4075)(1004,3052,2028,4076)(1005,3053,2029,4077)(1006,3054,2030,4078)(1007,3055,2031,4079)(1008,3056,2032,4080)(1009,3057,2033,4081)(1010,3058,2034,4082)(1011,3059,2035,4083)(1012,3060,2036,4084)(1013,3061,2037,4085)(1014,3062,2038,4086)(1015,3063,2039,4087)(1016,3064,2040,4088)(1017,3065,2041,4089)(1018,3066,2042,4090)(1019,3067,2043,4091)(1020,3068,2044,4092)(1021,3069,2045,4093)(1022,3070,2046,4094)(1023,3071,2047,4095)(1024,3072,2048,4096)

Polycyclic presentation:

2^12|a^2=l,b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=hi,h^2=ij,i^2=jk,j^2=kl,k^2=l,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,h^a=hi,i^a=ij,j^a=jk,k^a=kl

Conjugacy classes & character table

1027 conjugacy classes. The character table is too large to display.

Relationships

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.12);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7*F.8);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11*F.12);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);