Cohomology of group number 242 of order 64

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

General information on the group

  • The group is also known as Syl2(L3(4)), the Sylow 2-subgroup of L_3(4).
  • The group has 4 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 2.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t6  −  t5  +  2·t4  +  t  +  1
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-3,-5,-4. They were obtained using the 1st, the 2nd, the second power of the 3rd, and the 4th filter regular parameter of the Benson test.

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

Ring generators

The cohomology ring has 9 minimal generators of maximal degree 6:

  1. b_1_0, an element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. b_1_3, an element of degree 1
  5. b_3_10, an element of degree 3
  6. b_3_11, an element of degree 3
  7. c_4_18, a Duflot regular element of degree 4
  8. c_4_19, a Duflot regular element of degree 4
  9. b_6_47, an element of degree 6

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

Ring relations

There are 16 minimal relations of maximal degree 12:

  1. b_1_1·b_1_2 + b_1_0·b_1_3 + b_1_0·b_1_2
  2. b_1_1·b_1_3 + b_1_12 + b_1_0·b_1_2 + b_1_0·b_1_1 + b_1_02
  3. b_1_0·b_1_22 + b_1_02·b_1_2
  4. b_1_0·b_1_2·b_1_3 + b_1_0·b_1_22 + b_1_02·b_1_3 + b_1_03
  5. b_1_0·b_1_2·b_3_10 + b_1_02·b_3_10
  6. b_1_0·b_1_3·b_3_10 + b_1_0·b_1_2·b_3_11 + b_1_0·b_1_2·b_3_10 + b_1_0·b_1_1·b_3_10
       + b_1_02·b_3_11
  7. b_1_0·b_1_3·b_3_11 + b_1_0·b_1_3·b_3_10 + b_1_0·b_1_2·b_3_11 + b_1_0·b_1_2·b_3_10
       + b_1_0·b_1_1·b_3_11 + b_1_0·b_1_1·b_3_10
  8. b_3_102 + b_1_33·b_3_10 + b_1_2·b_1_32·b_3_11 + b_1_2·b_1_32·b_3_10
       + b_1_22·b_1_3·b_3_11 + b_1_23·b_3_10 + b_1_0·b_1_12·b_3_11 + b_1_02·b_1_1·b_3_11
       + b_1_02·b_1_1·b_3_10 + b_1_02·b_1_14 + b_1_03·b_3_11 + b_1_03·b_3_10
       + c_4_19·b_1_22 + c_4_18·b_1_32 + c_4_18·b_1_22 + c_4_18·b_1_02
  9. b_3_112 + b_3_102 + b_1_33·b_3_11 + b_1_22·b_1_3·b_3_11 + b_1_23·b_3_11
       + b_1_13·b_3_11 + b_1_13·b_3_10 + b_1_0·b_1_12·b_3_11 + b_1_0·b_1_12·b_3_10
       + b_1_0·b_1_15 + b_1_02·b_1_1·b_3_11 + b_1_02·b_1_1·b_3_10 + b_1_03·b_3_10
       + b_1_06 + c_4_19·b_1_32 + c_4_18·b_1_22 + c_4_18·b_1_12 + c_4_18·b_1_02
  10. b_1_2·b_3_10·b_3_11 + b_1_2·b_1_33·b_3_10 + b_1_23·b_1_3·b_3_11
       + b_1_23·b_1_3·b_3_10 + b_1_24·b_3_11 + b_1_24·b_3_10 + b_1_0·b_3_10·b_3_11
       + b_1_02·b_1_12·b_3_11 + b_1_03·b_1_1·b_3_10 + b_1_04·b_3_11 + b_1_04·b_3_10
       + b_1_04·b_1_13 + b_1_05·b_1_12 + b_1_07 + b_6_47·b_1_2 + c_4_19·b_1_2·b_1_32
       + c_4_18·b_1_2·b_1_32 + c_4_18·b_1_22·b_1_3 + c_4_18·b_1_23
       + c_4_18·b_1_0·b_1_12 + c_4_18·b_1_02·b_1_3
  11. b_1_3·b_3_10·b_3_11 + b_1_34·b_3_10 + b_1_22·b_1_32·b_3_11 + b_1_22·b_1_32·b_3_10
       + b_1_23·b_1_3·b_3_11 + b_1_23·b_1_3·b_3_10 + b_1_1·b_3_10·b_3_11
       + b_1_0·b_3_10·b_3_11 + b_1_0·b_1_13·b_3_11 + b_1_02·b_1_12·b_3_11
       + b_1_02·b_1_12·b_3_10 + b_1_03·b_1_1·b_3_11 + b_1_03·b_1_14 + b_1_04·b_3_11
       + b_1_04·b_3_10 + b_1_05·b_1_12 + b_1_06·b_1_1 + b_1_07 + b_6_47·b_1_3
       + c_4_19·b_1_33 + c_4_19·b_1_02·b_1_2 + c_4_19·b_1_03 + c_4_18·b_1_33
       + c_4_18·b_1_2·b_1_32 + c_4_18·b_1_22·b_1_3 + c_4_18·b_1_13 + c_4_18·b_1_03
  12. b_1_0·b_1_13·b_3_10 + b_1_02·b_1_12·b_3_11 + b_1_02·b_1_12·b_3_10
       + b_1_03·b_1_1·b_3_11 + b_1_03·b_1_1·b_3_10 + b_1_04·b_3_11 + b_1_04·b_1_13
       + b_1_05·b_1_12 + b_1_07 + b_6_47·b_1_0 + c_4_19·b_1_0·b_1_12
       + c_4_19·b_1_02·b_1_3 + c_4_19·b_1_02·b_1_1 + c_4_18·b_1_02·b_1_3
       + c_4_18·b_1_02·b_1_1 + c_4_18·b_1_03
  13. b_1_14·b_3_10 + b_1_0·b_1_13·b_3_11 + b_1_0·b_1_13·b_3_10 + b_1_02·b_1_12·b_3_11
       + b_1_02·b_1_12·b_3_10 + b_1_03·b_1_1·b_3_11 + b_1_03·b_1_14 + b_1_04·b_1_13
       + b_1_06·b_1_1 + b_6_47·b_1_1 + c_4_19·b_1_13 + c_4_19·b_1_02·b_1_2
       + c_4_19·b_1_02·b_1_1 + c_4_19·b_1_03
  14. b_1_2·b_1_35·b_3_10 + b_1_22·b_1_34·b_3_10 + b_1_24·b_1_32·b_3_11
       + b_1_25·b_1_3·b_3_11 + b_1_25·b_1_3·b_3_10 + b_1_26·b_3_10
       + b_1_0·b_1_12·b_3_10·b_3_11 + b_1_0·b_1_15·b_3_11 + b_1_02·b_1_1·b_3_10·b_3_11
       + b_1_02·b_1_17 + b_1_03·b_3_10·b_3_11 + b_1_04·b_1_12·b_3_11
       + b_1_04·b_1_12·b_3_10 + b_1_04·b_1_15 + b_1_05·b_1_1·b_3_10 + b_1_06·b_3_11
       + b_1_06·b_3_10 + b_1_06·b_1_13 + b_1_08·b_1_1 + b_1_09 + b_6_47·b_3_10
       + b_6_47·b_1_33 + b_6_47·b_1_2·b_1_32 + b_6_47·b_1_22·b_1_3
       + b_6_47·b_1_0·b_1_12 + b_6_47·b_1_02·b_1_1 + c_4_19·b_1_32·b_3_10
       + c_4_19·b_1_35 + c_4_19·b_1_22·b_3_11 + c_4_19·b_1_22·b_1_33
       + c_4_19·b_1_23·b_1_32 + c_4_19·b_1_25 + c_4_19·b_1_0·b_1_2·b_3_11
       + c_4_19·b_1_03·b_1_12 + c_4_19·b_1_05 + c_4_18·b_1_32·b_3_11
       + c_4_18·b_1_32·b_3_10 + c_4_18·b_1_2·b_1_3·b_3_10 + c_4_18·b_1_2·b_1_34
       + c_4_18·b_1_22·b_3_11 + c_4_18·b_1_22·b_3_10 + c_4_18·b_1_23·b_1_32
       + c_4_18·b_1_25 + c_4_18·b_1_12·b_3_11 + c_4_18·b_1_12·b_3_10 + c_4_18·b_1_15
       + c_4_18·b_1_0·b_1_2·b_3_11 + c_4_18·b_1_0·b_1_1·b_3_10 + c_4_18·b_1_02·b_3_11
       + c_4_18·b_1_02·b_3_10 + c_4_18·b_1_04·b_1_1 + c_4_18·b_1_05
  15. b_1_36·b_3_10 + b_1_2·b_1_35·b_3_11 + b_1_2·b_1_35·b_3_10 + b_1_22·b_1_34·b_3_11
       + b_1_22·b_1_34·b_3_10 + b_1_23·b_1_33·b_3_10 + b_1_24·b_1_32·b_3_10
       + b_1_25·b_1_3·b_3_11 + b_1_26·b_3_11 + b_1_13·b_3_10·b_3_11
       + b_1_0·b_1_12·b_3_10·b_3_11 + b_1_02·b_1_1·b_3_10·b_3_11 + b_1_02·b_1_14·b_3_11
       + b_1_02·b_1_17 + b_1_03·b_1_16 + b_1_04·b_1_12·b_3_11 + b_1_04·b_1_12·b_3_10
       + b_1_04·b_1_15 + b_1_06·b_3_11 + b_1_07·b_1_12 + b_1_09 + b_6_47·b_3_11
       + b_6_47·b_1_2·b_1_32 + b_6_47·b_1_22·b_1_3 + b_6_47·b_1_13
       + b_6_47·b_1_0·b_1_12 + c_4_19·b_1_32·b_3_11 + c_4_19·b_1_32·b_3_10
       + c_4_19·b_1_2·b_1_34 + c_4_19·b_1_22·b_3_10 + c_4_19·b_1_22·b_1_33
       + c_4_19·b_1_24·b_1_3 + c_4_19·b_1_12·b_3_10 + c_4_19·b_1_15 + c_4_19·b_1_04·b_1_1
       + c_4_19·b_1_05 + c_4_18·b_1_32·b_3_11 + c_4_18·b_1_32·b_3_10 + c_4_18·b_1_35
       + c_4_18·b_1_2·b_1_3·b_3_11 + c_4_18·b_1_22·b_3_11 + c_4_18·b_1_23·b_1_32
       + c_4_18·b_1_24·b_1_3 + c_4_18·b_1_25 + c_4_18·b_1_12·b_3_11
       + c_4_18·b_1_12·b_3_10 + c_4_18·b_1_15 + c_4_18·b_1_0·b_1_2·b_3_11
       + c_4_18·b_1_0·b_1_1·b_3_11 + c_4_18·b_1_0·b_1_14 + c_4_18·b_1_02·b_3_10
       + c_4_18·b_1_04·b_1_1 + c_4_18·b_1_05
  16. b_6_47·b_1_33·b_3_11 + b_6_47·b_1_36 + b_6_47·b_1_2·b_1_32·b_3_11
       + b_6_47·b_1_2·b_1_32·b_3_10 + b_6_47·b_1_2·b_1_35 + b_6_47·b_1_22·b_1_3·b_3_11
       + b_6_47·b_1_22·b_1_3·b_3_10 + b_6_47·b_1_22·b_1_34 + b_6_47·b_1_23·b_1_33
       + b_6_47·b_1_24·b_1_32 + b_6_47·b_1_25·b_1_3 + b_6_47·b_1_26
       + b_6_47·b_1_13·b_3_11 + b_6_47·b_1_13·b_3_10 + b_6_47·b_1_16
       + b_6_47·b_1_0·b_1_12·b_3_11 + b_6_47·b_1_0·b_1_15 + b_6_47·b_1_02·b_1_1·b_3_11
       + b_6_47·b_1_02·b_1_14 + b_6_47·b_1_05·b_1_1 + b_6_472 + c_4_19·b_1_35·b_3_11
       + c_4_19·b_1_38 + c_4_19·b_1_2·b_1_34·b_3_10 + c_4_19·b_1_22·b_1_33·b_3_11
       + c_4_19·b_1_22·b_1_33·b_3_10 + c_4_19·b_1_23·b_1_32·b_3_11
       + c_4_19·b_1_24·b_1_3·b_3_10 + c_4_19·b_1_25·b_3_11 + c_4_19·b_1_25·b_1_33
       + c_4_19·b_1_26·b_1_32 + c_4_19·b_1_28 + c_4_19·b_1_15·b_3_11 + c_4_19·b_1_18
       + c_4_19·b_1_0·b_1_14·b_3_11 + c_4_19·b_1_03·b_1_12·b_3_11
       + c_4_19·b_1_04·b_1_1·b_3_11 + c_4_19·b_1_04·b_1_14 + c_4_19·b_1_05·b_3_10
       + c_4_19·b_6_47·b_1_12 + c_4_19·b_6_47·b_1_0·b_1_1 + c_4_19·b_6_47·b_1_02
       + c_4_18·b_1_35·b_3_10 + c_4_18·b_1_2·b_1_34·b_3_11 + c_4_18·b_1_23·b_1_32·b_3_11
       + c_4_18·b_1_23·b_1_32·b_3_10 + c_4_18·b_1_24·b_1_3·b_3_10 + c_4_18·b_1_25·b_3_11
       + c_4_18·b_1_25·b_3_10 + c_4_18·b_1_28 + c_4_18·b_1_02·b_1_13·b_3_11
       + c_4_18·b_1_03·b_1_12·b_3_11 + c_4_18·b_1_03·b_1_15
       + c_4_18·b_1_04·b_1_1·b_3_11 + c_4_18·b_1_04·b_1_1·b_3_10 + c_4_18·b_1_04·b_1_14
       + c_4_18·b_1_05·b_1_13 + c_4_18·b_6_47·b_1_12 + c_4_18·b_6_47·b_1_02
       + c_4_192·b_1_34 + c_4_192·b_1_22·b_1_32 + c_4_192·b_1_24 + c_4_192·b_1_14
       + c_4_192·b_1_0·b_1_13 + c_4_192·b_1_02·b_1_12 + c_4_192·b_1_03·b_1_1
       + c_4_192·b_1_04 + c_4_18·c_4_19·b_1_34 + c_4_18·c_4_19·b_1_22·b_1_32
       + c_4_18·c_4_19·b_1_24 + c_4_18·c_4_19·b_1_02·b_1_12 + c_4_182·b_1_24
       + c_4_182·b_1_04

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 12.
  • 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_4_18, a Duflot regular element of degree 4
    2. c_4_19, a Duflot regular element of degree 4
    3. b_1_32 + b_1_2·b_1_3 + b_1_22, an element of degree 2
    4. b_1_3, an element of degree 1
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 5, 7].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. b_3_10 → 0, an element of degree 3
  6. b_3_11 → 0, an element of degree 3
  7. c_4_18 → c_1_14, an element of degree 4
  8. c_4_19 → c_1_14 + c_1_04, an element of degree 4
  9. b_6_47 → 0, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 4

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. b_3_10 → c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  6. b_3_11 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_32 + c_1_0·c_1_22 + c_1_02·c_1_3
       + c_1_02·c_1_2, an element of degree 3
  7. c_4_18 → c_1_1·c_1_33 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_1·c_1_23
       + c_1_12·c_1_2·c_1_3 + c_1_14 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3
       + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  8. c_4_19 → c_1_1·c_1_23 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_14 + c_1_0·c_1_33
       + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  9. b_6_47 → c_1_1·c_1_23·c_1_32 + c_1_1·c_1_24·c_1_3 + c_1_12·c_1_22·c_1_32
       + c_1_12·c_1_23·c_1_3 + c_1_12·c_1_24 + c_1_13·c_1_2·c_1_32
       + c_1_13·c_1_22·c_1_3 + c_1_14·c_1_22 + c_1_0·c_1_35 + c_1_0·c_1_2·c_1_34
       + c_1_0·c_1_25 + c_1_0·c_1_1·c_1_34 + c_1_0·c_1_1·c_1_22·c_1_32
       + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_12·c_1_33 + c_1_0·c_1_12·c_1_22·c_1_3
       + c_1_0·c_1_12·c_1_23 + c_1_02·c_1_1·c_1_33 + c_1_02·c_1_1·c_1_2·c_1_32
       + c_1_02·c_1_1·c_1_23 + c_1_02·c_1_12·c_1_32 + c_1_02·c_1_12·c_1_2·c_1_3
       + c_1_02·c_1_12·c_1_22 + c_1_03·c_1_2·c_1_32 + c_1_03·c_1_22·c_1_3
       + c_1_04·c_1_32 + c_1_04·c_1_2·c_1_3 + c_1_04·c_1_22, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 4

  1. b_1_0 → c_1_3, an element of degree 1
  2. b_1_1 → c_1_3 + c_1_2, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. b_1_3 → c_1_2, an element of degree 1
  5. b_3_10 → c_1_1·c_1_32 + c_1_1·c_1_22 + c_1_12·c_1_3 + c_1_12·c_1_2 + c_1_0·c_1_32
       + c_1_02·c_1_3, an element of degree 3
  6. b_3_11 → c_1_2·c_1_32 + c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_32 + c_1_0·c_1_22
       + c_1_02·c_1_3 + c_1_02·c_1_2, an element of degree 3
  7. c_4_18 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_1·c_1_23
       + c_1_12·c_1_32 + c_1_14 + c_1_0·c_1_2·c_1_32 + c_1_02·c_1_2·c_1_3, an element of degree 4
  8. c_4_19 → c_1_34 + c_1_2·c_1_33 + c_1_23·c_1_3 + c_1_1·c_1_33 + c_1_1·c_1_2·c_1_32
       + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_14
       + c_1_0·c_1_2·c_1_32 + c_1_02·c_1_32 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  9. b_6_47 → c_1_36 + c_1_23·c_1_33 + c_1_25·c_1_3 + c_1_1·c_1_35 + c_1_1·c_1_22·c_1_33
       + c_1_1·c_1_23·c_1_32 + c_1_1·c_1_25 + c_1_12·c_1_34 + c_1_12·c_1_22·c_1_32
       + c_1_12·c_1_23·c_1_3 + c_1_14·c_1_22 + c_1_0·c_1_2·c_1_34
       + c_1_0·c_1_23·c_1_32 + c_1_0·c_1_24·c_1_3 + c_1_02·c_1_22·c_1_32
       + c_1_02·c_1_24 + c_1_04·c_1_22, an element of degree 6

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64


Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2

Last change: 25.08.2009