Mod-3-Cohomology of Normalizer(SymplecticGroup(6,4),Centre(SylowSubgroup(SymplecticGroup(6,4),3))), a group of order 362880

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • Normalizer(SymplecticGroup(6,4),Centre(SylowSubgroup(SymplecticGroup(6,4),3))) is a group of order 362880.
  • The group order factors as 27 · 34 · 5 · 7.
  • The group is defined by 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86,417,1030,556,404,416)(247,1023,423,445,869,429,483,624)(250,640,683,682,886,1182,835,396)(253,1308,551,1059,732,833,956,658)(256,1214,905,597,1231,765,1054,565)(262,467,1173,1287,1158,1277,1145,292)(266,541,596,1086)(267,374,321,346,1310,781,331,575)(284,722,936,1070,286,1201,537,401)(287,1075,439,1142)(295,918,450,1004,821,1045,1227,1174)(314,684,932,1275,749,528,608,1237)(326,818,1074,798,1103,808,1078,332)(327,1328,651,1063)(343,484,1265,398,1184,554,1276,742)(344,695,967,876)(345,1171,1343,514,1105,1189,605,754)(347,545,1155,1139,444,495,611,631)(351,767,1296,1217)(354,385,809,1117,928,1357,982,1164)(360,379,955,664,1365,576,1224,1012)(362,1233,509,1316,1087,382,589,394)(369,859,463,1190,994,972,570,527)(383,879,1186,1109,769,1029,441,979)(386,1235,992,674,838,547,1081,785)(392,559,1168,1056,1228,1289,705,1294)(403,1223,1311,619)(440,974,699,896,546,1041,761,629)(480,512,693,1025)(491,504,1124,1363,858,1099,615,1136)(532,594,830,1238)(544,643,584,796)(571,1016,846,919)(595,844,807,1092,1037,891,1064,1125)(616,1202,783,1362,922,829,1330,985)(667,704,1349,1251,1338,1122,870,963)(670,1001,747,1021,1295,997,1126,825)(673,1172,1356,1291)(677,680,1218,1339,1144,993,1280,1350)(755,855,1364,1129,1282,1079,1093,1098)(826,1135,877,831)(958,1096,1249,1048)(1036,1284)]).
  • It is non-abelian.
  • It has 3-Rank 3.
  • The centre of a Sylow 3-subgroup has rank 1.
  • Its Sylow 3-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 2 and 3, respectively.

Structure of the cohomology ring

The computation was based on 18 stability conditions for H*(Syl3(A9); GF(3)).

General information

  • The cohomology ring is of dimension 3 and depth 3.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    ( − 1)·(1  −  2·t  +  3·t2  −  3·t3  +  3·t4  −  3·t5  +  4·t6  −  3·t7  +  3·t8  −  3·t9  +  3·t10  −  2·t11  +  t12)
    ( − 1  +  t)3 · (1  +  t  +  t2) · (1  +  t2)3 · (1  −  t2  +  t4)
  • The a-invariants are -∞,-∞,-∞,-3. 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, -3, -3].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 10 minimal generators of maximal degree 12:

  1. a_3_1, a nilpotent element of degree 3
  2. a_3_0, a nilpotent element of degree 3
  3. b_4_1, an element of degree 4
  4. b_4_0, an element of degree 4
  5. a_6_0, a nilpotent element of degree 6
  6. a_7_5, a nilpotent element of degree 7
  7. a_7_1, a nilpotent element of degree 7
  8. b_8_3, an element of degree 8
  9. a_11_8, a nilpotent element of degree 11
  10. c_12_6, a Duflot element of degree 12

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There are 5 "obvious" relations:
   a_3_02, a_3_12, a_7_12, a_7_52, a_11_82

Apart from that, there are 16 minimal relations of maximal degree 19:

  1. a_6_0·a_3_1
  2. a_3_1·a_7_5
  3. b_4_1·a_6_0 − a_3_1·a_7_1
  4. b_8_3·a_3_1 − b_4_1·a_7_5
  5. a_6_02
  6. a_6_0·a_7_1
  7. a_6_0·a_7_5
  8. a_7_1·a_7_5 + a_3_1·a_11_8
  9. a_6_0·b_8_3 − a_3_1·a_11_8
  10. b_8_3·a_7_1 − b_4_1·a_11_8
  11. b_8_3·a_7_5 − c_12_6·a_3_1
  12. b_8_32 − b_4_1·c_12_6
  13. a_6_0·a_11_8
  14. a_7_1·a_11_8
  15. a_7_5·a_11_8 − a_6_0·c_12_6
  16. b_8_3·a_11_8 − c_12_6·a_7_1

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 19 using the Hilbert-Poincaré criterion.
  • The completion test was perfect: It applied in the last degree in which a generator or relation was found.
  • The following is a filter regular homogeneous system of parameters:
    1. c_12_6, an element of degree 12
    2. b_4_1·b_8_3 − b_4_13 − b_4_02·b_4_1 + b_4_03, an element of degree 12
    3. b_4_1, an element of degree 4
  • A Duflot regular sequence is given by c_12_6.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 25].
  • Modifying the above filter regular HSOP, we obtained the following parameters:
    1. c_12_6, an element of degree 12
    2. b_4_0, an element of degree 4
    3. b_4_1, an element of degree 4
  • We found that there exists some HSOP over a finite extension field, in degrees 12,4,4.

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(Syl3(A9); GF(3))

  1. a_3_1 → b_2_2·a_1_1
  2. a_3_0 → a_3_4 − b_2_0·a_1_0
  3. b_4_1 → b_2_22
  4. b_4_0 → b_4_6 − b_2_02
  5. a_6_0 → a_1_1·a_5_8
  6. a_7_5 → c_6_11·a_1_1
  7. a_7_1 → b_2_2·a_5_8
  8. b_8_3 → b_2_2·c_6_11
  9. a_11_8 → c_6_11·a_5_8 + b_2_0·c_6_11·a_3_3
  10. c_12_6 → c_6_112

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

  1. a_3_1 → 0, an element of degree 3
  2. a_3_0 → 0, an element of degree 3
  3. b_4_1 → 0, an element of degree 4
  4. b_4_0 → 0, an element of degree 4
  5. a_6_0 → 0, an element of degree 6
  6. a_7_5 → 0, an element of degree 7
  7. a_7_1 → 0, an element of degree 7
  8. b_8_3 → 0, an element of degree 8
  9. a_11_8 → 0, an element of degree 11
  10. c_12_6 → 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_1 → 0, an element of degree 3
  2. a_3_0 →  − c_2_2·a_1_1, an element of degree 3
  3. b_4_1 → 0, an element of degree 4
  4. b_4_0 →  − c_2_22, an element of degree 4
  5. a_6_0 → 0, an element of degree 6
  6. a_7_5 → 0, an element of degree 7
  7. a_7_1 → 0, an element of degree 7
  8. b_8_3 → 0, an element of degree 8
  9. a_11_8 → 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
  10. c_12_6 → 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 3 in a Sylow subgroup

  1. a_3_1 → c_2_5·a_1_2, an element of degree 3
  2. a_3_0 →  − c_2_5·a_1_1 + c_2_5·a_1_0 − c_2_4·a_1_2 − c_2_4·a_1_1 + c_2_3·a_1_2, an element of degree 3
  3. b_4_1 → c_2_52, an element of degree 4
  4. b_4_0 → c_2_4·c_2_5 − c_2_42 − c_2_3·c_2_5, an element of degree 4
  5. a_6_0 →  − c_2_4·c_2_5·a_1_0·a_1_2 + c_2_42·a_1_0·a_1_2 − c_2_3·c_2_5·a_1_1·a_1_2
       + c_2_3·c_2_5·a_1_0·a_1_2 − c_2_3·c_2_4·a_1_1·a_1_2, an element of degree 6
  6. a_7_5 → c_2_3·c_2_4·c_2_5·a_1_2 − c_2_3·c_2_42·a_1_2 + c_2_32·c_2_5·a_1_2 + c_2_33·a_1_2, an element of degree 7
  7. a_7_1 → c_2_4·c_2_52·a_1_0 − c_2_42·c_2_5·a_1_0 + c_2_3·c_2_52·a_1_1 − c_2_3·c_2_52·a_1_0
       + c_2_3·c_2_4·c_2_5·a_1_2 + c_2_3·c_2_4·c_2_5·a_1_1 + c_2_32·c_2_5·a_1_2, an element of degree 7
  8. b_8_3 → c_2_3·c_2_4·c_2_52 − c_2_3·c_2_42·c_2_5 + c_2_32·c_2_52 + c_2_33·c_2_5, an element of degree 8
  9. a_11_8 → c_2_3·c_2_42·c_2_52·a_1_0 + c_2_3·c_2_43·c_2_5·a_1_0 + c_2_3·c_2_44·a_1_0
       + c_2_32·c_2_4·c_2_52·a_1_1 + c_2_32·c_2_42·c_2_5·a_1_2 − c_2_32·c_2_43·a_1_2
       − c_2_32·c_2_43·a_1_1 + c_2_33·c_2_52·a_1_1 − c_2_33·c_2_52·a_1_0
       − c_2_33·c_2_4·c_2_5·a_1_2 + c_2_33·c_2_4·c_2_5·a_1_1 + c_2_33·c_2_4·c_2_5·a_1_0
       − c_2_33·c_2_42·a_1_2 − c_2_33·c_2_42·a_1_0 + c_2_34·c_2_5·a_1_2
       + c_2_34·c_2_5·a_1_1 − c_2_34·c_2_5·a_1_0 + c_2_34·c_2_4·a_1_2 + c_2_34·c_2_4·a_1_1
       + c_2_35·a_1_2, an element of degree 11
  10. c_12_6 → c_2_32·c_2_42·c_2_52 + c_2_32·c_2_43·c_2_5 + c_2_32·c_2_44
       − c_2_33·c_2_4·c_2_52 + c_2_33·c_2_42·c_2_5 + c_2_34·c_2_52
       − c_2_34·c_2_4·c_2_5 + c_2_34·c_2_42 − c_2_35·c_2_5 + c_2_36, an element of degree 12

About the group Ring generators Ring relations Completion information Restriction maps


Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2

Last change: 14.12.2010