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Singular

Gap

Mod-3-Cohomology of Normalizer(G2(5),Centre(SylowSubgroup(G2(5),3))), a group of order 756000

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• Normalizer(G2(5),Centre(SylowSubgroup(G2(5),3))) is a group of order 756000.
• The group order factors as 2⁵ · 3³ · 5³ · 7.
• The group is defined by 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37,1006,3143,454,1627,892,1226,3596,1944,2295,550,1560)(452,501,3380,3340,3456,1550,2315,3611,1447,925,1767,2767)(457,2271,1124,883,1570,2442,2524,516,3400,1909,3604,1920)(462,2908,1505,1262,1244,2032,689,2495,1653,1348,3048,1699)(464,1709,3725,2125,3375,3307,552,3352,3893,696,2888,519)(465,3664,1576,480,1708,1158,2551,631,3827,2772,3734,3586)(467,1772,893,2043,1572,1545,1149,2326,3174,1994,938,1906)(470,2919,1946,2729)(471,703,3033,3759,3615,1196,1198,3322,3187,813,940,3288)(473,2108,827,3543)(477,1697,1187,3219,3297,1497,1800,3533,3706,3571,2697,1086)(482,3720,1663,1419,920,3168,1455,653,1525,3665,1317,2969)(486,825,3338,1295)(490,3211,2186,2921)(499,3308,758,3777,2432,667,3883,1643,1726,503,1233,1021)(507,1548,1074,1076,3587,3417,1814,2000,2444,2135,831,3361)(510,3396,3312,949,645,3169,997,3728,3822,1565,3061,3618)(517,2210,1063,2737,896,1417,1075,982,3755,732,2736,1665)(518,3328,555,3462,1415,3010,971,2741,2366,2138,1010,2049)(525,716,593,2223,1221,3449,1395,3156,1802,1302,3859,817)(539,616,2394,2616,1305,3747,3369,3819,904,1798,3251,2893)(541,1662)(543,2051,3294,1438,1681,3762,855,2484,1297,799,574,1406)(545,1225,2976,968,3736,3005,3473,2243,1138,2984,1490,801)(546,905,2720,1566,3450,2664,2482,3502,2770,684,2158,3139)(554,3065,3499,1883,1971,816,3475,2988,595,918,788,2068)(559,1362,1688,3467,2629,3258,1520,3362,2835,3565,3891,830)(560,3165,1175,2086,1899,2635,941,1059,3359,3567,3521,3694)(563,2648,3212,775,3289,3444,2719,2455,1266,1483,2147,2221)(566,3465,851,3164,2191,845,1474,2596,1177,2309,3129,2929)(570,2157,3671,3852,3350,1359,2153,2282,2543,2644,2082,2950)(579,1568,3080,3185,1329,590,687,1795,2630,3821,2621,3679)(582,3483,625,3214,1029,3411,3405,2453,1536,2506,2777,2075)(584,2047,1805,2669,730,3527,3179,3145,1036,2054,1396,769)(603,1774,3282,2019)(605,1747,3240,1374,865,3562,1615,1642,3659,3122,752,3800)(607,1707,3899,2845,1879,3184,1224,3177,3531,1731,2413,2500)(614,3890,3750,2428,675,1268,1315,3261,935,644,3743,3602)(615,3751,1784,792,3403,2682,2670,1398,1203,2815,3874,2695)(618,1864,3135,2201,2516,2395,3044,1236,2389,3780,1896,2909)(634,3699,3009,1017,2046,1860,2981,2619,1496,850,2978,1965)(637,1067,3641,3223,1640,3640,3233,2542,3832,1351,2470,2958)(640,2831,1796,3098,3195,3448,2782,3672,972,1373,3421,1200)(641,3882,1761,3345,1389,2576,3363,3495,1934,1698,1393,2013)(643,1553,2313,1170,2446,650,1887,1913,3017,2033,1428,3134)(651,1274,1941,3670,2072,1495,3760,3365,2475,1882,3506,2396)(657,1013,1459,3027)(682,1630,2906,2979,2526,903,3796,2518,1963,2376,765,1862)(691,2150,2242,2870,3578,1171)(692,942,1732,3285)(702,3730,3188,1092,2274,3235,3228,1529,811,2073,3600,1931)(707,3271,2897,1078,1602,3096,1921,1238,3160,2461,1868,773)(715,1820,1532,1380,2257,1190,3773,2830,2610,1926,3566,872)(721,3262,2593,1822,2538,2140,2464,3508,2689,3846,2303,3320)(731,1056,3788,1919,2129,3577,921,2521,3103,1559,3754,2031)(734,2924,2961,2739,2572,2189,1284,1984,821,1358,1139,3180)(747,1999,2623,3440,1206,1100,989,1706,2062,3757,2636,1182)(751,3170,3681,2166,2529,1589,3709,1095,888,2537,887,2788)(755,1538,3293,1842,2554,2100,2903,2344,3608,1081,3785,3115)(783,3193,3660,3229,3273,2582,1242,3645,1918,3381,2550,1363)(787,3202,2647,1682,3022,2779,3778,2222,3770,1760,2144,3570)(795,3173,1674,1741,3197,3536,3535,1090,2083,3704,2687,3564)(796,1743,1102,1510,1112,1458,1892,2882,3241,1331,1007,2061)(797,2025,1791,2926,3348,3356,2352,3545,1228,3084,1116,1835)(804,1349,2351,912,2748,2239,1546,3707,3399,2805,3301,2451)(847,1283,3457,1945,3004,3386,1949,3079,3808,2468,3776,2718)(859,1710,1390,1783,2214,2832,1671,2982,2397,3805,1173,2918)(875,2625,2418,955,3250,1832)(877,1437,3376,1140)(881,2143,3595,1606,2367,3120,2747,2101,2527,978,2751,1974)(882,3327,2448,2617,1826,2604,2009,3501,2878,1907,3270,3897)(890,1028,2764,3391,2391,2561,1873,1911,3062,2076,1824,2927)(891,3689,2152,2980,1391,2010,1695,3085,1853,1120,2069,1830)(897,1724,2228,2735,1344,3343,2847,3443,981,3590,2113,3076)(899,2858,1652,1841,1803,3711,2410,2659,1940,3137,3109,1245)(906,3705,2053,2193,3555,3482,3559,3642,1473,1214,2213,3811)(914,2259,1310,3624)(929,2278,1278,1613,2329,2297,3225,2134,2491,2577,1034,3117)(947,2248,2149,2884)(965,3877,3278)(973,3263,2599,3830,3470,2874,1481,1658,3530,2839,3026,2109)(974,3833,2311,2730,2034,2438,1012,3110,1103,2294,2855,3560)(984,2626,2807,3680,1003,3383,1603,1039,3360,2965,2694,2262)(990,1264,2844,2341,1874,3682,2485,2104,2544,3702,3058,1804)(1001,1633,3410,2842,3724,2945,3367,1185,1062,3300,1954,3722)(1077,3194,1492,2841)(1169,1910,3461,2387)(1215,2575,1377,3357,2220,1937,1220,2301,1687,1670,2509,2934)(1217,1998,1249,3880,1405,2335,3106,1753,2040,2583,2813,1701)(1219,3253,3529,2316,3358,1634,2165,2771,3092,3712,3148,3803)(1231,1429,3471,1267)(1261,2390,3858,3036,3332,3598,2954,2498,3093,2132,3373,2306)(1296,1456,3203,2688,1902,2102,2178,1430,2460,2265,1316,2762)(1327,3442,3149,2305,2796,1355,2781,2643,2854,3425,3451,2048)(1379,1884,3107,1771,3840,2029)(1480,3894,2539,1778,2133,2417,2620,1583,2949,3584,1972,3104)(1635,2263,3591,3626)(1703,3503,2508,3631,3407,1780,2319,2866,2360,2642,1715,3113)(1925,1996,2915,3741)(2001,3031,3131,2819)(2015,3667,2989,3804)(2126,3260,3013,2793)(2172,3685,2638)(2218,3688,3182,2399)(2339,3406,2383,2343)]).
• It is non-abelian.
• It has 3-Rank 2.
• The centre of a Sylow 3-subgroup has rank 1.
• Its Sylow 3-subgroup has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 2 and depth 2.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  1  −  t  +  t2  −  t3  +  t4  −  t5  +  t6  −  t7  +  t8  −  t9  +  t10

  ( − 1  +  t)2 · (1  +  t  +  t2) · (1  +  t2)2 · (1  −  t2  +  t4)

• The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
• The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 4 minimal generators of maximal degree 12:

1. a_3_0, a nilpotent element of degree 3
2. b_4_0, an element of degree 4
3. a_11_1, a nilpotent element of degree 11
4. c_12_1, a Duflot element of degree 12

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 2 "obvious" relations:
   a_3_0², a_11_1²

Apart from that, there are no relations.

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 14 using the Hilbert-Poincaré criterion.
• However, the last relation was already found in degree 0 and the last generator in degree 12.
• The following is a filter regular homogeneous system of parameters:
  1. c_12_1, an element of degree 12
  2. b_4_0, an element of degree 4
• A Duflot regular sequence is given by c_12_1.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 14].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(E27; GF(3))

1. a_3_0 → b_2_3·a_1_1 − b_2_1·a_1_1 + b_2_0·a_1_0
2. b_4_0 → b_2_3² − b_2_0·b_2_2 − b_2_0·b_2_1 + b_2_0² + a_1_0·a_3_5 − a_1_0·a_3_4
3. a_11_1 → b_2_0³·b_2_1·a_3_5 + b_2_0⁴·a_3_5 − b_2_0⁴·a_3_4 − b_2_0⁴·b_2_1·a_1_1
      + b_2_0⁵·a_1_1 − b_2_3·c_6_8·a_3_5 + b_2_1·c_6_8·a_3_5 − b_2_0·c_6_8·a_3_4
      − b_2_0·b_2_1·c_6_8·a_1_1 + b_2_0²·c_6_8·a_1_1
4. c_12_1 → b_2_0⁵·b_2_2 − b_2_0⁴·a_1_1·a_3_5 + b_2_0⁴·a_1_0·a_3_5 + b_2_0⁴·a_1_0·a_3_4
      − b_2_0²·b_2_2·c_6_8 + b_2_0·c_6_8·a_1_1·a_3_5 + b_2_0·c_6_8·a_1_0·a_3_5
      + b_2_0·c_6_8·a_1_0·a_3_4 − c_6_8²

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

1. a_3_0 → 0, an element of degree 3
2. b_4_0 → 0, an element of degree 4
3. a_11_1 → 0, an element of degree 11
4. c_12_1 →  − c_2_0⁶, an element of degree 12

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

1. a_3_0 → c_2_2·a_1_1, an element of degree 3
2. b_4_0 → c_2_2², an element of degree 4
3. a_11_1 → c_2_1·c_2_2⁴·a_1_0 − c_2_1²·c_2_2³·a_1_1 − c_2_1³·c_2_2²·a_1_0
      + c_2_1⁴·c_2_2·a_1_1, an element of degree 11
4. c_12_1 →  − c_2_1²·c_2_2⁴ − c_2_1⁴·c_2_2² − c_2_1⁶, an element of degree 12

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

1. a_3_0 → c_2_2·a_1_1, an element of degree 3
2. b_4_0 → c_2_2², an element of degree 4
3. a_11_1 → c_2_1·c_2_2⁴·a_1_0 − c_2_1²·c_2_2³·a_1_1 − c_2_1³·c_2_2²·a_1_0
      + c_2_1⁴·c_2_2·a_1_1, an element of degree 11
4. c_12_1 →  − c_2_1²·c_2_2⁴ − c_2_1⁴·c_2_2² − c_2_1⁶, an element of degree 12

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

1. a_3_0 → c_2_2·a_1_1, an element of degree 3
2. b_4_0 → c_2_2², an element of degree 4
3. a_11_1 → c_2_1·c_2_2⁴·a_1_0 − c_2_1²·c_2_2³·a_1_1 − c_2_1³·c_2_2²·a_1_0
      + c_2_1⁴·c_2_2·a_1_1, an element of degree 11
4. c_12_1 →  − c_2_1²·c_2_2⁴ − c_2_1⁴·c_2_2² − c_2_1⁶, an element of degree 12

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

1. a_3_0 → c_2_2·a_1_1, an element of degree 3
2. b_4_0 → c_2_2², an element of degree 4
3. a_11_1 → c_2_1·c_2_2⁴·a_1_0 − c_2_1²·c_2_2³·a_1_1 − c_2_1³·c_2_2²·a_1_0
      + c_2_1⁴·c_2_2·a_1_1, an element of degree 11
4. c_12_1 →  − c_2_1²·c_2_2⁴ − c_2_1⁴·c_2_2² − c_2_1⁶, an element of degree 12

About the group

Ring generators

Ring relations

Completion information

Restriction maps