Cohomology of group number 934 of order 128

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

General information on the group

  • The group is also known as Syl2(J2), the Sylow 2-subgroup of Hall-Janko Group J_2.
  • The group has 3 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 1.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 2 and 4, respectively.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    t10  −  t9  +  2·t8  −  t6  +  t5  +  t2  +  1
    (t  +  1) · (t  −  1)4 · (t2  +  1)2 · (t4  +  1)
  • The a-invariants are -∞,-∞,-3,-5,-4. They were obtained using the filter regular HSOP of the Benson test.

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Ring generators

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

  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. a_2_4, a nilpotent element of degree 2
  5. a_2_5, a nilpotent element of degree 2
  6. b_3_8, an element of degree 3
  7. b_3_9, an element of degree 3
  8. b_4_14, an element of degree 4
  9. b_5_20, an element of degree 5
  10. b_5_21, an element of degree 5
  11. c_8_49, a Duflot regular element of degree 8
  12. b_10_83, an element of degree 10

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Ring relations

There are 33 minimal relations of maximal degree 20:

  1. b_1_0·b_1_1
  2. b_1_0·b_1_2
  3. a_2_4·b_1_1
  4. a_2_5·b_1_2
  5. a_2_5·b_1_1 + a_2_4·b_1_2
  6. a_2_52 + a_2_4·a_2_5 + a_2_42
  7. b_1_0·b_3_8 + a_2_42
  8. b_1_0·b_3_9 + a_2_52
  9. a_2_4·b_3_8
  10. a_2_5·b_3_9
  11. a_2_5·b_3_8 + a_2_4·b_3_9
  12. b_4_14·b_1_0
  13. b_3_92 + b_1_2·b_5_20 + b_1_23·b_3_9 + b_1_1·b_1_22·b_3_8 + b_1_12·b_1_2·b_3_9
       + b_1_13·b_3_9 + b_4_14·b_1_22 + b_4_14·b_1_12 + a_2_5·b_4_14
  14. b_1_0·b_5_20 + a_2_4·b_1_04 + a_2_4·a_2_5·b_1_02
  15. b_3_92 + b_1_2·b_5_21 + b_1_23·b_3_9 + b_1_1·b_5_20 + b_1_12·b_1_2·b_3_9
       + b_1_13·b_3_9 + b_4_14·b_1_22 + b_4_14·b_1_12 + a_2_5·b_4_14
  16. b_1_0·b_5_21 + a_2_5·b_1_04 + a_2_4·a_2_5·b_1_02 + a_2_42·b_1_02 + a_2_42·a_2_5
  17. b_3_82 + b_1_1·b_5_21 + b_1_1·b_5_20 + b_1_1·b_1_22·b_3_9 + b_1_1·b_1_22·b_3_8
       + b_1_12·b_1_2·b_3_9 + b_1_13·b_3_8 + b_4_14·b_1_22 + a_2_4·b_4_14
  18. a_2_5·b_5_20 + a_2_4·b_4_14·b_1_2 + a_2_4·a_2_5·b_1_03 + a_2_42·a_2_5·b_1_0
  19. a_2_5·b_5_21 + a_2_4·b_5_20 + a_2_4·a_2_5·b_1_03 + a_2_42·a_2_5·b_1_0
  20. a_2_4·b_5_21 + a_2_4·b_5_20 + a_2_4·b_4_14·b_1_2 + a_2_4·a_2_5·b_1_03
       + a_2_42·b_1_03
  21. b_5_202 + b_1_22·b_3_8·b_5_20 + b_1_24·b_3_8·b_3_9 + b_1_25·b_5_20 + b_1_27·b_3_9
       + b_1_1·b_1_2·b_3_9·b_5_20 + b_1_1·b_1_2·b_3_8·b_5_20 + b_1_1·b_1_24·b_5_20
       + b_1_1·b_1_26·b_3_9 + b_1_12·b_1_25·b_3_9 + b_1_13·b_1_22·b_5_20
       + b_1_13·b_1_24·b_3_9 + b_1_14·b_1_2·b_5_20 + b_1_14·b_1_23·b_3_8
       + b_1_15·b_1_22·b_3_9 + b_1_15·b_1_22·b_3_8 + b_4_14·b_1_2·b_5_20
       + b_4_14·b_1_23·b_3_8 + b_4_14·b_1_26 + b_4_14·b_1_1·b_1_22·b_3_9
       + b_4_14·b_1_1·b_1_22·b_3_8 + b_4_14·b_1_14·b_1_22 + a_2_5·b_4_142
       + a_2_42·b_1_06 + c_8_49·b_1_22
  22. b_5_20·b_5_21 + b_5_202 + b_1_1·b_1_23·b_3_8·b_3_9 + b_1_1·b_1_24·b_5_20
       + b_1_1·b_1_26·b_3_9 + b_1_12·b_3_9·b_5_20 + b_1_12·b_3_8·b_5_20
       + b_1_12·b_1_23·b_5_20 + b_1_12·b_1_25·b_3_9 + b_1_13·b_1_24·b_3_9
       + b_1_14·b_1_2·b_5_20 + b_1_14·b_1_23·b_3_9 + b_1_15·b_5_20 + b_1_15·b_1_22·b_3_8
       + b_1_16·b_1_2·b_3_9 + b_1_16·b_1_2·b_3_8 + b_4_14·b_1_1·b_5_20
       + b_4_14·b_1_1·b_1_22·b_3_8 + b_4_14·b_1_1·b_1_25 + b_4_14·b_1_12·b_1_2·b_3_9
       + b_4_14·b_1_12·b_1_2·b_3_8 + b_4_14·b_1_15·b_1_2 + a_2_5·b_4_142 + a_2_4·b_4_142
       + a_2_4·a_2_5·b_1_06 + a_2_42·b_1_06 + c_8_49·b_1_1·b_1_2
  23. b_5_212 + b_5_202 + b_1_12·b_3_9·b_5_21 + b_1_12·b_3_9·b_5_20
       + b_1_12·b_3_8·b_5_21 + b_1_12·b_1_22·b_3_8·b_3_9 + b_1_12·b_1_23·b_5_20
       + b_1_12·b_1_25·b_3_9 + b_1_13·b_1_2·b_3_8·b_3_9 + b_1_13·b_1_22·b_5_20
       + b_1_13·b_1_24·b_3_8 + b_1_14·b_1_2·b_5_20 + b_1_14·b_1_23·b_3_9 + b_1_15·b_5_21
       + b_1_15·b_5_20 + b_1_16·b_1_2·b_3_8 + b_1_17·b_3_9 + b_1_17·b_3_8
       + b_4_14·b_1_1·b_5_21 + b_4_14·b_1_1·b_5_20 + b_4_14·b_1_13·b_3_9
       + b_4_14·b_1_13·b_3_8 + b_4_14·b_1_13·b_1_23 + b_4_14·b_1_16 + a_2_4·b_4_142
       + a_2_4·a_2_5·b_1_06 + c_8_49·b_1_12
  24. b_3_8·b_3_9·b_5_20 + b_1_23·b_3_9·b_5_20 + b_1_23·b_3_8·b_5_20 + b_1_26·b_5_20
       + b_1_28·b_3_9 + b_1_28·b_3_8 + b_1_1·b_1_22·b_3_9·b_5_20
       + b_1_1·b_1_22·b_3_8·b_5_20 + b_1_1·b_1_24·b_3_8·b_3_9 + b_1_1·b_1_25·b_5_20
       + b_1_1·b_1_27·b_3_9 + b_1_12·b_1_2·b_3_8·b_5_20 + b_1_12·b_1_26·b_3_9
       + b_1_12·b_1_26·b_3_8 + b_1_13·b_1_23·b_5_20 + b_1_15·b_1_23·b_3_9
       + b_1_15·b_1_23·b_3_8 + b_1_16·b_5_20 + b_1_16·b_1_22·b_3_9
       + b_1_16·b_1_22·b_3_8 + b_1_17·b_1_2·b_3_9 + b_1_17·b_1_2·b_3_8 + b_10_83·b_1_2
       + b_4_14·b_1_2·b_3_8·b_3_9 + b_4_14·b_1_22·b_5_20 + b_4_14·b_1_24·b_3_8
       + b_4_14·b_1_27 + b_4_14·b_1_1·b_1_2·b_5_20 + b_4_14·b_1_1·b_1_23·b_3_9
       + b_4_14·b_1_1·b_1_26 + b_4_14·b_1_12·b_5_20 + b_4_14·b_1_13·b_1_2·b_3_8
       + b_4_14·b_1_14·b_1_23 + b_4_142·b_1_1·b_1_22 + b_4_142·b_1_12·b_1_2
       + a_2_4·b_4_14·b_5_20 + a_2_4·b_4_142·b_1_2 + c_8_49·b_1_23 + c_8_49·b_1_12·b_1_2
       + a_2_4·c_8_49·b_1_2
  25. b_10_83·b_1_0 + a_2_42·b_1_07 + c_8_49·b_1_03 + a_2_4·c_8_49·b_1_0
  26. b_3_8·b_3_9·b_5_21 + b_3_8·b_3_9·b_5_20 + b_1_1·b_1_22·b_3_9·b_5_20
       + b_1_1·b_1_22·b_3_8·b_5_20 + b_1_1·b_1_25·b_5_20 + b_1_1·b_1_27·b_3_9
       + b_1_1·b_1_27·b_3_8 + b_1_12·b_1_2·b_3_9·b_5_20 + b_1_12·b_1_2·b_3_8·b_5_20
       + b_1_13·b_3_9·b_5_20 + b_1_13·b_3_8·b_5_20 + b_1_13·b_1_22·b_3_8·b_3_9
       + b_1_13·b_1_23·b_5_20 + b_1_14·b_1_2·b_3_8·b_3_9 + b_1_14·b_1_22·b_5_20
       + b_1_14·b_1_24·b_3_9 + b_1_14·b_1_24·b_3_8 + b_1_16·b_5_21 + b_1_16·b_5_20
       + b_1_16·b_1_22·b_3_8 + b_1_17·b_1_2·b_3_9 + b_1_18·b_3_9 + b_1_18·b_3_8
       + b_10_83·b_1_1 + b_4_14·b_1_1·b_3_8·b_3_9 + b_4_14·b_1_1·b_1_2·b_5_20
       + b_4_14·b_1_1·b_1_23·b_3_9 + b_4_14·b_1_1·b_1_23·b_3_8 + b_4_14·b_1_1·b_1_26
       + b_4_14·b_1_12·b_5_21 + b_4_14·b_1_12·b_1_22·b_3_9 + b_4_14·b_1_13·b_1_2·b_3_8
       + b_4_14·b_1_13·b_1_24 + b_4_14·b_1_14·b_3_8 + b_4_14·b_1_14·b_1_23
       + b_4_142·b_1_12·b_1_2 + b_4_142·b_1_13 + a_2_4·b_4_14·b_5_20
       + a_2_4·b_4_142·b_1_2 + c_8_49·b_1_1·b_1_22 + c_8_49·b_1_13
  27. a_2_5·b_10_83 + a_2_42·a_2_5·b_1_06 + a_2_5·c_8_49·b_1_02 + a_2_4·a_2_5·c_8_49
  28. a_2_4·b_10_83 + a_2_4·c_8_49·b_1_02 + a_2_42·c_8_49
  29. b_1_25·b_3_9·b_5_20 + b_1_27·b_3_8·b_3_9 + b_1_28·b_5_20 + b_1_210·b_3_9
       + b_1_210·b_3_8 + b_1_1·b_1_24·b_3_9·b_5_20 + b_1_1·b_1_26·b_3_8·b_3_9
       + b_1_1·b_1_29·b_3_9 + b_1_1·b_1_29·b_3_8 + b_1_12·b_1_23·b_3_9·b_5_20
       + b_1_12·b_1_25·b_3_8·b_3_9 + b_1_12·b_1_28·b_3_9 + b_1_13·b_1_22·b_3_8·b_5_20
       + b_1_13·b_1_24·b_3_8·b_3_9 + b_1_13·b_1_25·b_5_20 + b_1_14·b_1_2·b_3_9·b_5_20
       + b_1_14·b_1_23·b_3_8·b_3_9 + b_1_14·b_1_26·b_3_9 + b_1_15·b_3_8·b_5_21
       + b_1_15·b_3_8·b_5_20 + b_1_15·b_1_25·b_3_8 + b_1_16·b_1_22·b_5_20
       + b_1_16·b_1_24·b_3_9 + b_1_16·b_1_24·b_3_8 + b_1_17·b_1_2·b_5_20 + b_1_18·b_5_21
       + b_1_19·b_1_2·b_3_9 + b_1_19·b_1_2·b_3_8 + b_1_110·b_3_9 + b_1_110·b_3_8
       + b_10_83·b_3_8 + b_10_83·b_1_23 + b_4_14·b_1_2·b_3_9·b_5_20
       + b_4_14·b_1_2·b_3_8·b_5_20 + b_4_14·b_1_23·b_3_8·b_3_9 + b_4_14·b_1_1·b_3_8·b_5_21
       + b_4_14·b_1_1·b_1_23·b_5_20 + b_4_14·b_1_1·b_1_25·b_3_8
       + b_4_14·b_1_12·b_1_22·b_5_20 + b_4_14·b_1_12·b_1_27
       + b_4_14·b_1_13·b_1_2·b_5_20 + b_4_14·b_1_13·b_1_23·b_3_9
       + b_4_14·b_1_13·b_1_26 + b_4_14·b_1_14·b_5_20 + b_4_14·b_1_14·b_1_25
       + b_4_14·b_1_15·b_1_2·b_3_8 + b_4_14·b_1_15·b_1_24 + b_4_14·b_1_16·b_3_8
       + b_4_14·b_1_16·b_1_23 + b_4_14·b_1_17·b_1_22 + b_4_14·b_1_18·b_1_2
       + b_4_14·b_1_19 + b_4_142·b_1_22·b_3_9 + b_4_142·b_1_25
       + b_4_142·b_1_1·b_1_2·b_3_8 + b_4_142·b_1_12·b_3_8 + b_4_142·b_1_12·b_1_23
       + b_4_142·b_1_13·b_1_22 + b_4_142·b_1_14·b_1_2 + b_4_142·b_1_15
       + a_2_4·b_4_142·b_3_9 + c_8_49·b_1_22·b_3_8 + c_8_49·b_1_25 + c_8_49·b_1_1·b_1_24
       + c_8_49·b_1_12·b_3_9 + c_8_49·b_1_12·b_3_8 + c_8_49·b_1_12·b_1_23
       + c_8_49·b_1_14·b_1_2 + a_2_42·c_8_49·b_1_0
  30. b_1_27·b_3_8·b_3_9 + b_1_1·b_1_29·b_3_8 + b_1_13·b_1_22·b_3_9·b_5_20
       + b_1_13·b_1_24·b_3_8·b_3_9 + b_1_13·b_1_25·b_5_20 + b_1_15·b_3_9·b_5_21
       + b_1_15·b_3_9·b_5_20 + b_1_15·b_3_8·b_5_20 + b_1_15·b_1_23·b_5_20
       + b_1_15·b_1_25·b_3_9 + b_1_15·b_1_25·b_3_8 + b_1_16·b_1_24·b_3_8
       + b_1_17·b_3_8·b_3_9 + b_1_18·b_5_21 + b_1_18·b_5_20 + b_1_18·b_1_22·b_3_9
       + b_1_18·b_1_22·b_3_8 + b_1_19·b_1_2·b_3_9 + b_1_110·b_3_8 + b_10_83·b_3_9
       + b_10_83·b_1_13 + b_4_14·b_1_2·b_3_9·b_5_20 + b_4_14·b_1_2·b_3_8·b_5_20
       + b_4_14·b_1_24·b_5_20 + b_4_14·b_1_1·b_3_9·b_5_21 + b_4_14·b_1_1·b_3_8·b_5_21
       + b_4_14·b_1_1·b_3_8·b_5_20 + b_4_14·b_1_1·b_1_22·b_3_8·b_3_9
       + b_4_14·b_1_1·b_1_23·b_5_20 + b_4_14·b_1_1·b_1_28 + b_4_14·b_1_12·b_1_22·b_5_20
       + b_4_14·b_1_12·b_1_27 + b_4_14·b_1_13·b_3_8·b_3_9 + b_4_14·b_1_13·b_1_23·b_3_9
       + b_4_14·b_1_13·b_1_23·b_3_8 + b_4_14·b_1_13·b_1_26 + b_4_14·b_1_14·b_5_21
       + b_4_14·b_1_14·b_5_20 + b_4_14·b_1_14·b_1_25 + b_4_14·b_1_15·b_1_24
       + b_4_14·b_1_16·b_3_8 + b_4_14·b_1_17·b_1_22 + b_4_14·b_1_18·b_1_2
       + b_4_14·b_1_19 + b_4_142·b_1_22·b_3_8 + b_4_142·b_1_25
       + b_4_142·b_1_1·b_1_2·b_3_9 + b_4_142·b_1_1·b_1_24 + b_4_142·b_1_12·b_3_9
       + b_4_142·b_1_12·b_3_8 + b_4_142·b_1_12·b_1_23 + b_4_142·b_1_13·b_1_22
       + b_4_142·b_1_14·b_1_2 + b_4_142·b_1_15 + c_8_49·b_1_22·b_3_9
       + c_8_49·b_1_22·b_3_8 + c_8_49·b_1_25 + c_8_49·b_1_1·b_1_24 + c_8_49·b_1_12·b_3_9
       + c_8_49·b_1_12·b_1_23 + c_8_49·b_1_13·b_1_22 + c_8_49·b_1_15
       + a_2_4·c_8_49·b_3_9 + a_2_4·a_2_5·c_8_49·b_1_0 + a_2_42·c_8_49·b_1_0
  31. b_1_27·b_3_8·b_5_20 + b_1_1·b_1_211·b_3_8 + b_1_12·b_1_210·b_3_9
       + b_1_13·b_1_24·b_3_9·b_5_20 + b_1_13·b_1_27·b_5_20 + b_1_14·b_1_25·b_3_8·b_3_9
       + b_1_14·b_1_28·b_3_9 + b_1_14·b_1_28·b_3_8 + b_1_15·b_1_24·b_3_8·b_3_9
       + b_1_15·b_1_27·b_3_9 + b_1_15·b_1_27·b_3_8 + b_1_16·b_1_24·b_5_20
       + b_1_16·b_1_26·b_3_9 + b_1_17·b_3_8·b_5_20 + b_1_17·b_1_22·b_3_8·b_3_9
       + b_1_17·b_1_23·b_5_20 + b_1_17·b_1_25·b_3_8 + b_1_19·b_1_23·b_3_8
       + b_1_110·b_5_20 + b_1_110·b_1_22·b_3_9 + b_1_111·b_1_2·b_3_9
       + b_1_111·b_1_2·b_3_8 + b_10_83·b_5_20 + b_10_83·b_1_1·b_1_2·b_3_9
       + b_10_83·b_1_1·b_1_2·b_3_8 + b_10_83·b_1_13·b_1_22 + b_10_83·b_1_14·b_1_2
       + b_4_14·b_1_23·b_3_8·b_5_20 + b_4_14·b_1_25·b_3_8·b_3_9 + b_4_14·b_1_26·b_5_20
       + b_4_14·b_1_1·b_1_25·b_5_20 + b_4_14·b_1_12·b_1_26·b_3_8
       + b_4_14·b_1_13·b_3_8·b_5_20 + b_4_14·b_1_13·b_1_25·b_3_8
       + b_4_14·b_1_14·b_1_2·b_3_8·b_3_9 + b_4_14·b_1_14·b_1_24·b_3_9
       + b_4_14·b_1_15·b_1_23·b_3_9 + b_4_14·b_1_15·b_1_26
       + b_4_14·b_1_16·b_1_22·b_3_8 + b_4_14·b_1_17·b_1_2·b_3_8 + b_4_14·b_1_17·b_1_24
       + b_4_14·b_1_19·b_1_22 + b_4_142·b_1_22·b_5_20 + b_4_142·b_1_24·b_3_9
       + b_4_142·b_1_24·b_3_8 + b_4_142·b_1_27 + b_4_142·b_1_1·b_1_23·b_3_9
       + b_4_142·b_1_12·b_1_22·b_3_8 + b_4_142·b_1_12·b_1_25
       + c_8_49·b_1_2·b_3_8·b_3_9 + c_8_49·b_1_22·b_5_20 + c_8_49·b_1_24·b_3_9
       + c_8_49·b_1_24·b_3_8 + c_8_49·b_1_27 + c_8_49·b_1_12·b_5_20
       + c_8_49·b_1_12·b_1_22·b_3_9 + c_8_49·b_1_12·b_1_22·b_3_8
       + c_8_49·b_1_12·b_1_25 + c_8_49·b_1_13·b_1_2·b_3_9 + c_8_49·b_1_13·b_1_2·b_3_8
       + c_8_49·b_1_15·b_1_22 + b_4_14·c_8_49·b_1_23 + b_4_14·c_8_49·b_1_1·b_1_22
       + b_4_14·c_8_49·b_1_12·b_1_2 + a_2_4·c_8_49·b_5_20 + a_2_4·c_8_49·b_1_05
       + a_2_4·b_4_14·c_8_49·b_1_2 + a_2_4·a_2_5·c_8_49·b_1_03
  32. b_1_1·b_1_26·b_3_8·b_5_20 + b_1_12·b_1_210·b_3_8 + b_1_13·b_1_29·b_3_9
       + b_1_14·b_1_23·b_3_9·b_5_20 + b_1_14·b_1_26·b_5_20 + b_1_15·b_1_24·b_3_8·b_3_9
       + b_1_15·b_1_27·b_3_9 + b_1_15·b_1_27·b_3_8 + b_1_16·b_1_23·b_3_8·b_3_9
       + b_1_16·b_1_26·b_3_9 + b_1_16·b_1_26·b_3_8 + b_1_17·b_3_8·b_5_21
       + b_1_17·b_3_8·b_5_20 + b_1_17·b_1_23·b_5_20 + b_1_17·b_1_25·b_3_9
       + b_1_18·b_1_2·b_3_8·b_3_9 + b_1_18·b_1_22·b_5_20 + b_1_18·b_1_24·b_3_8
       + b_1_19·b_1_23·b_3_9 + b_1_19·b_1_23·b_3_8 + b_1_110·b_5_21
       + b_1_110·b_1_22·b_3_9 + b_1_111·b_1_2·b_3_9 + b_1_112·b_3_9 + b_1_112·b_3_8
       + b_10_83·b_5_21 + b_10_83·b_5_20 + b_10_83·b_1_1·b_1_2·b_3_8 + b_10_83·b_1_12·b_3_9
       + b_10_83·b_1_12·b_3_8 + b_10_83·b_1_14·b_1_2 + b_10_83·b_1_15
       + b_4_14·b_1_1·b_1_22·b_3_8·b_5_20 + b_4_14·b_1_1·b_1_24·b_3_8·b_3_9
       + b_4_14·b_1_1·b_1_25·b_5_20 + b_4_14·b_1_12·b_1_24·b_5_20
       + b_4_14·b_1_13·b_3_8·b_5_21 + b_4_14·b_1_13·b_3_8·b_5_20
       + b_4_14·b_1_13·b_1_25·b_3_8 + b_4_14·b_1_14·b_1_24·b_3_8
       + b_4_14·b_1_15·b_3_8·b_3_9 + b_4_14·b_1_15·b_1_23·b_3_8 + b_4_14·b_1_16·b_5_20
       + b_4_14·b_1_16·b_1_22·b_3_8 + b_4_14·b_1_16·b_1_25 + b_4_14·b_1_18·b_3_8
       + b_4_14·b_1_110·b_1_2 + b_4_142·b_1_1·b_1_2·b_5_20 + b_4_142·b_1_1·b_1_23·b_3_9
       + b_4_142·b_1_1·b_1_23·b_3_8 + b_4_142·b_1_1·b_1_26
       + b_4_142·b_1_12·b_1_22·b_3_9 + b_4_142·b_1_13·b_1_2·b_3_8
       + b_4_142·b_1_13·b_1_24 + b_4_142·b_1_14·b_1_23 + a_2_4·b_4_142·b_5_20
       + a_2_4·b_4_143·b_1_2 + a_2_42·a_2_5·b_1_09 + c_8_49·b_1_1·b_3_8·b_3_9
       + c_8_49·b_1_1·b_1_2·b_5_20 + c_8_49·b_1_1·b_1_23·b_3_9 + c_8_49·b_1_1·b_1_23·b_3_8
       + c_8_49·b_1_1·b_1_26 + c_8_49·b_1_12·b_5_21 + c_8_49·b_1_12·b_5_20
       + c_8_49·b_1_13·b_1_2·b_3_9 + c_8_49·b_1_13·b_1_24 + c_8_49·b_1_14·b_3_9
       + c_8_49·b_1_14·b_3_8 + c_8_49·b_1_16·b_1_2 + b_4_14·c_8_49·b_1_1·b_1_22
       + b_4_14·c_8_49·b_1_12·b_1_2 + b_4_14·c_8_49·b_1_13 + a_2_5·c_8_49·b_1_05
       + a_2_4·c_8_49·b_1_05 + a_2_4·a_2_5·c_8_49·b_1_03 + a_2_42·a_2_5·c_8_49·b_1_0
  33. b_10_83·b_1_22·b_3_9·b_5_20 + b_10_83·b_1_24·b_3_8·b_3_9 + b_10_83·b_1_25·b_5_20
       + b_10_83·b_1_27·b_3_9 + b_10_83·b_1_27·b_3_8 + b_10_83·b_1_1·b_1_24·b_5_20
       + b_10_83·b_1_1·b_1_26·b_3_9 + b_10_83·b_1_12·b_3_8·b_5_21
       + b_10_83·b_1_12·b_3_8·b_5_20 + b_10_83·b_1_12·b_1_23·b_5_20
       + b_10_83·b_1_12·b_1_25·b_3_8 + b_10_83·b_1_12·b_1_28
       + b_10_83·b_1_13·b_1_2·b_3_8·b_3_9 + b_10_83·b_1_13·b_1_22·b_5_20
       + b_10_83·b_1_13·b_1_24·b_3_8 + b_10_83·b_1_13·b_1_27
       + b_10_83·b_1_14·b_3_8·b_3_9 + b_10_83·b_1_14·b_1_2·b_5_20
       + b_10_83·b_1_14·b_1_23·b_3_9 + b_10_83·b_1_14·b_1_23·b_3_8
       + b_10_83·b_1_15·b_5_21 + b_10_83·b_1_15·b_1_25 + b_10_83·b_1_16·b_1_2·b_3_8
       + b_10_83·b_1_17·b_3_9 + b_10_83·b_1_17·b_1_23 + b_10_83·b_1_18·b_1_22
       + b_10_83·b_1_19·b_1_2 + b_10_832 + b_4_14·b_1_213·b_3_8
       + b_4_14·b_1_1·b_1_212·b_3_9 + b_4_14·b_1_12·b_1_214
       + b_4_14·b_1_13·b_1_25·b_3_8·b_5_20 + b_4_14·b_1_13·b_1_28·b_5_20
       + b_4_14·b_1_13·b_1_210·b_3_8 + b_4_14·b_1_14·b_1_24·b_3_9·b_5_20
       + b_4_14·b_1_14·b_1_24·b_3_8·b_5_20 + b_4_14·b_1_14·b_1_29·b_3_8
       + b_4_14·b_1_14·b_1_212 + b_4_14·b_1_15·b_1_28·b_3_8 + b_4_14·b_1_15·b_1_211
       + b_4_14·b_1_16·b_1_22·b_3_9·b_5_20 + b_4_14·b_1_16·b_1_24·b_3_8·b_3_9
       + b_4_14·b_1_16·b_1_27·b_3_9 + b_4_14·b_1_16·b_1_210
       + b_4_14·b_1_17·b_1_2·b_3_9·b_5_20 + b_4_14·b_1_17·b_1_2·b_3_8·b_5_20
       + b_4_14·b_1_17·b_1_23·b_3_8·b_3_9 + b_4_14·b_1_17·b_1_26·b_3_9
       + b_4_14·b_1_17·b_1_29 + b_4_14·b_1_18·b_3_9·b_5_21 + b_4_14·b_1_18·b_3_8·b_5_20
       + b_4_14·b_1_18·b_1_23·b_5_20 + b_4_14·b_1_18·b_1_25·b_3_8
       + b_4_14·b_1_18·b_1_28 + b_4_14·b_1_19·b_1_22·b_5_20
       + b_4_14·b_1_110·b_1_2·b_5_20 + b_4_14·b_1_110·b_1_26 + b_4_14·b_1_111·b_5_21
       + b_4_14·b_1_112·b_1_2·b_3_8 + b_4_14·b_1_112·b_1_24 + b_4_14·b_1_115·b_1_2
       + b_4_14·b_10_83·b_1_2·b_5_20 + b_4_14·b_10_83·b_1_23·b_3_9
       + b_4_14·b_10_83·b_1_1·b_1_22·b_3_9 + b_4_14·b_10_83·b_1_12·b_1_2·b_3_9
       + b_4_14·b_10_83·b_1_12·b_1_2·b_3_8 + b_4_14·b_10_83·b_1_12·b_1_24
       + b_4_14·b_10_83·b_1_13·b_3_9 + b_4_142·b_1_24·b_3_9·b_5_20
       + b_4_142·b_1_24·b_3_8·b_5_20 + b_4_142·b_1_29·b_3_9
       + b_4_142·b_1_1·b_1_25·b_3_8·b_3_9 + b_4_142·b_1_1·b_1_26·b_5_20
       + b_4_142·b_1_1·b_1_211 + b_4_142·b_1_12·b_1_22·b_3_9·b_5_20
       + b_4_142·b_1_12·b_1_24·b_3_8·b_3_9 + b_4_142·b_1_12·b_1_27·b_3_9
       + b_4_142·b_1_13·b_1_2·b_3_9·b_5_20 + b_4_142·b_1_13·b_1_23·b_3_8·b_3_9
       + b_4_142·b_1_13·b_1_26·b_3_8 + b_4_142·b_1_13·b_1_29
       + b_4_142·b_1_14·b_3_9·b_5_21 + b_4_142·b_1_14·b_3_8·b_5_20
       + b_4_142·b_1_14·b_1_22·b_3_8·b_3_9 + b_4_142·b_1_14·b_1_23·b_5_20
       + b_4_142·b_1_14·b_1_25·b_3_9 + b_4_142·b_1_15·b_1_2·b_3_8·b_3_9
       + b_4_142·b_1_15·b_1_24·b_3_9 + b_4_142·b_1_16·b_3_8·b_3_9
       + b_4_142·b_1_16·b_1_23·b_3_8 + b_4_142·b_1_17·b_5_20
       + b_4_142·b_1_17·b_1_22·b_3_8 + b_4_142·b_1_18·b_1_2·b_3_8
       + b_4_142·b_1_19·b_3_8 + b_4_142·b_1_110·b_1_22 + b_4_143·b_1_23·b_5_20
       + b_4_143·b_1_25·b_3_8 + b_4_143·b_1_28 + b_4_143·b_1_1·b_1_24·b_3_9
       + b_4_143·b_1_1·b_1_24·b_3_8 + b_4_143·b_1_1·b_1_27 + b_4_143·b_1_13·b_5_21
       + b_4_143·b_1_13·b_5_20 + b_4_143·b_1_13·b_1_25 + b_4_143·b_1_14·b_1_2·b_3_9
       + b_4_143·b_1_15·b_3_9 + b_4_143·b_1_16·b_1_22 + b_4_144·b_1_24
       + b_4_144·b_1_14 + c_8_49·b_1_24·b_3_9·b_5_20 + c_8_49·b_1_24·b_3_8·b_5_20
       + c_8_49·b_1_27·b_5_20 + c_8_49·b_1_29·b_3_9 + c_8_49·b_1_29·b_3_8
       + c_8_49·b_1_1·b_1_23·b_3_9·b_5_20 + c_8_49·b_1_1·b_1_23·b_3_8·b_5_20
       + c_8_49·b_1_1·b_1_25·b_3_8·b_3_9 + c_8_49·b_1_1·b_1_26·b_5_20
       + c_8_49·b_1_1·b_1_211 + c_8_49·b_1_12·b_1_22·b_3_9·b_5_20
       + c_8_49·b_1_12·b_1_22·b_3_8·b_5_20 + c_8_49·b_1_13·b_1_2·b_3_8·b_5_20
       + c_8_49·b_1_13·b_1_26·b_3_9 + c_8_49·b_1_13·b_1_29
       + c_8_49·b_1_14·b_3_9·b_5_20 + c_8_49·b_1_14·b_3_8·b_5_21
       + c_8_49·b_1_14·b_3_8·b_5_20 + c_8_49·b_1_14·b_1_22·b_3_8·b_3_9
       + c_8_49·b_1_14·b_1_23·b_5_20 + c_8_49·b_1_14·b_1_25·b_3_8
       + c_8_49·b_1_14·b_1_28 + c_8_49·b_1_15·b_1_2·b_3_8·b_3_9
       + c_8_49·b_1_15·b_1_24·b_3_9 + c_8_49·b_1_15·b_1_27
       + c_8_49·b_1_16·b_1_2·b_5_20 + c_8_49·b_1_16·b_1_26 + c_8_49·b_1_17·b_5_21
       + c_8_49·b_1_17·b_5_20 + c_8_49·b_1_19·b_3_9 + c_8_49·b_1_19·b_3_8
       + c_8_49·b_1_110·b_1_22 + c_8_49·b_1_111·b_1_2
       + b_4_14·c_8_49·b_1_22·b_3_8·b_3_9 + b_4_14·c_8_49·b_1_28
       + b_4_14·c_8_49·b_1_1·b_1_24·b_3_8 + b_4_14·c_8_49·b_1_12·b_1_23·b_3_8
       + b_4_14·c_8_49·b_1_12·b_1_26 + b_4_14·c_8_49·b_1_13·b_5_21
       + b_4_14·c_8_49·b_1_13·b_5_20 + b_4_14·c_8_49·b_1_13·b_1_22·b_3_9
       + b_4_14·c_8_49·b_1_13·b_1_22·b_3_8 + b_4_14·c_8_49·b_1_13·b_1_25
       + b_4_14·c_8_49·b_1_14·b_1_2·b_3_8 + b_4_14·c_8_49·b_1_15·b_3_9
       + b_4_14·c_8_49·b_1_15·b_3_8 + b_4_14·c_8_49·b_1_16·b_1_22
       + b_4_14·c_8_49·b_1_17·b_1_2 + b_4_14·c_8_49·b_1_18
       + b_4_142·c_8_49·b_1_1·b_1_23 + c_8_492·b_1_24 + c_8_492·b_1_12·b_1_22
       + c_8_492·b_1_14 + c_8_492·b_1_04 + a_2_42·c_8_492

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 20.
  • 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_8_49, a Duflot regular element of degree 8
    2. b_1_24 + b_1_1·b_3_9 + b_1_12·b_1_22 + b_1_14 + b_1_04 + b_4_14, an element of degree 4
    3. b_1_1·b_1_22·b_3_9 + b_1_12·b_1_2·b_3_9 + b_1_12·b_1_24 + b_1_13·b_3_9
         + b_1_14·b_1_22 + b_4_14·b_1_22 + b_4_14·b_1_1·b_1_2 + b_4_14·b_1_12, an element of degree 6
    4. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 13, 16].
  • 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 128

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  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. a_2_4 → 0, an element of degree 2
  5. a_2_5 → 0, an element of degree 2
  6. b_3_8 → 0, an element of degree 3
  7. b_3_9 → 0, an element of degree 3
  8. b_4_14 → 0, an element of degree 4
  9. b_5_20 → 0, an element of degree 5
  10. b_5_21 → 0, an element of degree 5
  11. c_8_49 → c_1_08, an element of degree 8
  12. b_10_83 → 0, an element of degree 10

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

  1. b_1_0 → c_1_1, 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. a_2_4 → 0, an element of degree 2
  5. a_2_5 → 0, an element of degree 2
  6. b_3_8 → 0, an element of degree 3
  7. b_3_9 → 0, an element of degree 3
  8. b_4_14 → 0, an element of degree 4
  9. b_5_20 → 0, an element of degree 5
  10. b_5_21 → 0, an element of degree 5
  11. c_8_49 → c_1_04·c_1_14 + c_1_08, an element of degree 8
  12. b_10_83 → c_1_04·c_1_16 + c_1_08·c_1_12, an element of degree 10

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 → c_1_1, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_2_5 → 0, an element of degree 2
  6. b_3_8 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
  7. b_3_9 → c_1_2·c_1_32 + c_1_22·c_1_3 + 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
  8. b_4_14 → c_1_34 + c_1_22·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_13·c_1_3 + c_1_0·c_1_1·c_1_22
       + c_1_02·c_1_1·c_1_2, an element of degree 4
  9. b_5_20 → c_1_23·c_1_32 + c_1_24·c_1_3 + c_1_1·c_1_23·c_1_3 + c_1_12·c_1_22·c_1_3
       + c_1_0·c_1_24 + c_1_0·c_1_1·c_1_23 + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2
       + c_1_02·c_1_1·c_1_22 + c_1_04·c_1_2, an element of degree 5
  10. b_5_21 → c_1_23·c_1_32 + c_1_24·c_1_3 + c_1_1·c_1_23·c_1_3 + c_1_13·c_1_2·c_1_3
       + c_1_0·c_1_24 + c_1_0·c_1_13·c_1_2 + c_1_0·c_1_14 + c_1_02·c_1_1·c_1_22
       + c_1_04·c_1_2 + c_1_04·c_1_1, an element of degree 5
  11. c_8_49 → c_1_1·c_1_22·c_1_35 + c_1_1·c_1_23·c_1_34 + c_1_1·c_1_24·c_1_33
       + c_1_1·c_1_26·c_1_3 + c_1_12·c_1_36 + c_1_12·c_1_2·c_1_35
       + c_1_12·c_1_24·c_1_32 + c_1_12·c_1_25·c_1_3 + c_1_13·c_1_35
       + c_1_13·c_1_22·c_1_33 + c_1_13·c_1_23·c_1_32 + c_1_13·c_1_24·c_1_3
       + c_1_14·c_1_34 + c_1_15·c_1_33 + c_1_15·c_1_2·c_1_32 + c_1_15·c_1_22·c_1_3
       + c_1_0·c_1_23·c_1_34 + c_1_0·c_1_25·c_1_32 + c_1_0·c_1_12·c_1_25
       + c_1_0·c_1_13·c_1_22·c_1_32 + c_1_0·c_1_13·c_1_23·c_1_3
       + c_1_0·c_1_14·c_1_2·c_1_32 + c_1_0·c_1_14·c_1_22·c_1_3 + c_1_0·c_1_14·c_1_23
       + c_1_0·c_1_15·c_1_32 + c_1_0·c_1_16·c_1_3 + c_1_0·c_1_16·c_1_2
       + c_1_02·c_1_24·c_1_32 + c_1_02·c_1_25·c_1_3 + c_1_02·c_1_1·c_1_2·c_1_34
       + c_1_02·c_1_1·c_1_23·c_1_32 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_12·c_1_34
       + c_1_02·c_1_12·c_1_22·c_1_32 + c_1_02·c_1_14·c_1_2·c_1_3
       + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_15·c_1_3 + c_1_02·c_1_16
       + c_1_03·c_1_13·c_1_22 + c_1_03·c_1_15 + c_1_04·c_1_34
       + c_1_04·c_1_23·c_1_3 + c_1_04·c_1_24 + c_1_04·c_1_12·c_1_32
       + c_1_04·c_1_12·c_1_2·c_1_3 + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14
       + c_1_05·c_1_12·c_1_2 + c_1_05·c_1_13 + c_1_06·c_1_1·c_1_2 + c_1_06·c_1_12
       + c_1_08, an element of degree 8
  12. b_10_83 → c_1_22·c_1_38 + c_1_24·c_1_36 + c_1_25·c_1_35 + c_1_26·c_1_34
       + c_1_27·c_1_33 + c_1_29·c_1_3 + c_1_1·c_1_22·c_1_37 + c_1_1·c_1_27·c_1_32
       + c_1_12·c_1_38 + c_1_12·c_1_2·c_1_37 + c_1_12·c_1_23·c_1_35
       + c_1_12·c_1_27·c_1_3 + c_1_13·c_1_2·c_1_36 + c_1_13·c_1_22·c_1_35
       + c_1_13·c_1_23·c_1_34 + c_1_13·c_1_24·c_1_33 + c_1_13·c_1_26·c_1_3
       + c_1_14·c_1_36 + c_1_14·c_1_2·c_1_35 + c_1_14·c_1_22·c_1_34
       + c_1_14·c_1_23·c_1_33 + c_1_14·c_1_24·c_1_32 + c_1_15·c_1_35
       + c_1_15·c_1_2·c_1_34 + c_1_15·c_1_23·c_1_32 + c_1_15·c_1_24·c_1_3
       + c_1_16·c_1_34 + c_1_16·c_1_2·c_1_33 + c_1_17·c_1_33 + c_1_19·c_1_3
       + c_1_0·c_1_23·c_1_36 + c_1_0·c_1_24·c_1_35 + c_1_0·c_1_25·c_1_34
       + c_1_0·c_1_26·c_1_33 + c_1_0·c_1_27·c_1_32 + c_1_0·c_1_28·c_1_3
       + c_1_0·c_1_1·c_1_26·c_1_32 + c_1_0·c_1_1·c_1_27·c_1_3
       + c_1_0·c_1_12·c_1_2·c_1_36 + c_1_0·c_1_12·c_1_22·c_1_35
       + c_1_0·c_1_12·c_1_26·c_1_3 + c_1_0·c_1_13·c_1_36
       + c_1_0·c_1_13·c_1_22·c_1_34 + c_1_0·c_1_13·c_1_25·c_1_3
       + c_1_0·c_1_13·c_1_26 + c_1_0·c_1_14·c_1_35 + c_1_0·c_1_14·c_1_2·c_1_34
       + c_1_0·c_1_14·c_1_22·c_1_33 + c_1_0·c_1_14·c_1_23·c_1_32
       + c_1_0·c_1_14·c_1_25 + c_1_0·c_1_15·c_1_22·c_1_32
       + c_1_0·c_1_15·c_1_23·c_1_3 + c_1_0·c_1_15·c_1_24 + c_1_0·c_1_16·c_1_33
       + c_1_0·c_1_16·c_1_2·c_1_32 + c_1_0·c_1_16·c_1_22·c_1_3 + c_1_0·c_1_16·c_1_23
       + c_1_0·c_1_17·c_1_22 + c_1_0·c_1_18·c_1_3 + c_1_0·c_1_18·c_1_2
       + c_1_02·c_1_22·c_1_36 + c_1_02·c_1_23·c_1_35 + c_1_02·c_1_25·c_1_33
       + c_1_02·c_1_27·c_1_3 + c_1_02·c_1_1·c_1_2·c_1_36
       + c_1_02·c_1_1·c_1_22·c_1_35 + c_1_02·c_1_1·c_1_25·c_1_32
       + c_1_02·c_1_12·c_1_36 + c_1_02·c_1_12·c_1_22·c_1_34
       + c_1_02·c_1_12·c_1_24·c_1_32 + c_1_02·c_1_12·c_1_26
       + c_1_02·c_1_13·c_1_35 + c_1_02·c_1_14·c_1_2·c_1_33
       + c_1_02·c_1_14·c_1_23·c_1_3 + c_1_02·c_1_15·c_1_33
       + c_1_02·c_1_15·c_1_22·c_1_3 + c_1_02·c_1_15·c_1_23 + c_1_02·c_1_17·c_1_3
       + c_1_03·c_1_25·c_1_32 + c_1_03·c_1_26·c_1_3 + c_1_03·c_1_27
       + c_1_03·c_1_1·c_1_22·c_1_34 + c_1_03·c_1_1·c_1_24·c_1_32
       + c_1_03·c_1_1·c_1_26 + c_1_03·c_1_12·c_1_2·c_1_34
       + c_1_03·c_1_12·c_1_24·c_1_3 + c_1_03·c_1_12·c_1_25 + c_1_03·c_1_13·c_1_24
       + c_1_03·c_1_15·c_1_32 + c_1_03·c_1_15·c_1_22 + c_1_03·c_1_16·c_1_3
       + c_1_03·c_1_16·c_1_2 + c_1_03·c_1_17 + c_1_04·c_1_22·c_1_34
       + c_1_04·c_1_25·c_1_3 + c_1_04·c_1_1·c_1_2·c_1_34
       + c_1_04·c_1_1·c_1_22·c_1_33 + c_1_04·c_1_1·c_1_24·c_1_3
       + c_1_04·c_1_1·c_1_25 + c_1_04·c_1_12·c_1_2·c_1_33
       + c_1_04·c_1_12·c_1_22·c_1_32 + c_1_04·c_1_13·c_1_22·c_1_3
       + c_1_04·c_1_14·c_1_32 + c_1_04·c_1_15·c_1_3 + c_1_04·c_1_15·c_1_2
       + c_1_05·c_1_23·c_1_32 + c_1_05·c_1_24·c_1_3 + c_1_05·c_1_25
       + c_1_05·c_1_1·c_1_24 + c_1_05·c_1_12·c_1_2·c_1_32
       + c_1_05·c_1_12·c_1_22·c_1_3 + c_1_05·c_1_12·c_1_23 + c_1_05·c_1_13·c_1_32
       + c_1_05·c_1_14·c_1_3 + c_1_05·c_1_14·c_1_2 + c_1_05·c_1_15
       + c_1_06·c_1_22·c_1_32 + c_1_06·c_1_23·c_1_3 + c_1_06·c_1_24
       + c_1_06·c_1_1·c_1_2·c_1_32 + c_1_06·c_1_1·c_1_22·c_1_3
       + c_1_06·c_1_12·c_1_32 + c_1_06·c_1_13·c_1_3 + c_1_06·c_1_13·c_1_2
       + c_1_06·c_1_14 + c_1_07·c_1_1·c_1_22 + c_1_07·c_1_12·c_1_2 + c_1_08·c_1_22
       + c_1_08·c_1_1·c_1_2 + c_1_08·c_1_12, an element of degree 10

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128


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