Mod-5-Cohomology of Co3, a group of order 495766656000
General information on the group
- The group order factors as 210 · 37 · 53 · 7 · 11 · 23.
- The group is defined by Group([(1,245,185)(2,42,87)(3,112,266)(4,15,22)(5,131,30)(6,7,188)(8,75,111)(9,132,82)(12,187,124)(13,186,136)(14,265,213)(16,159,99)(17,256,130)(18,43,167)(19,101,31)(20,123,269)(21,134,74)(23,32,60)(24,209,67)(25,238,162)(26,35,154)(27,45,221)(28,235,270)(29,126,129)(33,66,210)(34,80,114)(36,251,229)(37,117,161)(38,206,63)(39,71,196)(40,118,180)(41,93,170)(44,271,164)(46,261,108)(47,182,49)(48,155,248)(50,230,153)(51,172,103)(52,236,165)(53,109,64)(54,191,100)(55,76,203)(56,156,260)(57,73,149)(58,116,145)(59,147,273)(61,127,189)(62,231,197)(65,122,169)(69,86,95)(70,255,192)(72,139,78)(77,252,151)(79,262,184)(81,214,242)(83,181,223)(84,174,200)(88,250,276)(89,257,244)(90,243,91)(92,158,107)(94,148,215)(96,105,125)(97,249,202)(98,263,193)(102,115,175)(106,219,150)(110,204,152)(113,225,216)(119,237,166)(120,241,234)(121,224,195)(128,190,268)(133,143,228)(135,168,201)(137,227,247)(138,217,240)(140,258,232)(141,220,205)(142,178,207)(144,146,254)(157,274,179)(160,253,176)(163,272,226)(171,233,194)(173,246,212)(177,198,211)(183,267,222)(199,208,259)(218,264,239),(1,204,123,82)(2,203,14,53)(3,33,40,118)(4,236,168,138)(6,172,188,157)(7,77,25,242)(8,76,85,264)(9,47,22,190)(10,146,50,26)(11,133,220,254)(12,224,179,58)(13,229,169,23)(15,84,148,78)(17,223)(18,228)(19,130,104,167)(20,131,90,60)(21,252,185,205)(24,263,214,81)(27,32,209,61)(28,196,67,137)(29,199,87,48)(30,65,218,112)(31,246,98,213)(34,99,165,265)(35,241)(36,198,161,89)(37,269)(38,251,42,271)(39,75,260,193)(41,176,192,57)(43,183,91,171)(44,116,173,102)(45,197,119,73)(46,238,124,162)(49,178,83,274)(51,174,244,100)(52,129)(54,153,217,151)(55,272,136,237)(56,257,154,121)(62,175,219,215)(63,106,66,259)(64,159,210,194)(68,258,221,234)(69,261,134,155)(70,164,211,266)(71,262)(72,189,114,222)(74,177,256,135)(79,226,267,202)(80,96,120,239)(86,132,216,160)(88,117,231,109)(92,201,111,276)(93,101,115,166)(94,253)(97,142,270,110)(105,243,212,225)(107,141,230,249)(113,139)(122,170,126,207)(125,163,184,158)(127,145,206,149)(140,273)(143,248,247,195)(144,180)(147,250,182,187)(152,232,208,191)(186,235,233,275)]).
- It is non-abelian.
- It has 5-Rank 2.
- The centre of a Sylow 5-subgroup has rank 1.
- Its Sylow 5-subgroup has 6 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.
Structure of the cohomology ring
This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(Normalizer(Co3,Centre(SylowSubgroup(Co3,5))); GF(5)).
General information
- The cohomology ring is of dimension 2 and depth 1.
- The depth coincides with the Duflot bound.
- The Poincaré series is
| 1 − 2·t + 3·t2 − 4·t3 + 4·t4 − 4·t5 + 4·t6 − 3·t7 + 3·t8 − 3·t9 + 3·t10 − 4·t11 + 4·t12 − 4·t13 + 4·t14 − 2·t15 + 2·t16 − 2·t17 + 3·t18 − 5·t19 + 5·t20 − 5·t21 + 4·t22 − 2·t23 + 2·t24 − 2·t25 + 3·t26 − 4·t27 + 4·t28 − 4·t29 + 3·t30 − 3·t31 + 3·t32 − 3·t33 + 4·t34 − 4·t35 + 4·t36 − 4·t37 + 3·t38 − 2·t39 + t40 |
|
| ( − 1 + t)2 · (1 + t2)2 · (1 − t + t2 − t3 + t4) · (1 + t4) · (1 + t + t2 + t3 + t4) · (1 − t2 + t4 − t6 + t8) · (1 − t4 + t8 − t12 + t16) |
- The a-invariants are -∞,-16,-2. They were obtained using the filter regular HSOP of the Benson test.
- The filter degree type of any filter regular HSOP is [-1, -2, -2].
Ring generators
The cohomology ring has 12 minimal generators of maximal degree 40:
- a_7_0, a nilpotent element of degree 7
- b_8_0, an element of degree 8
- a_15_1, a nilpotent element of degree 15
- a_16_1, a nilpotent element of degree 16
- a_18_0, a nilpotent element of degree 18
- a_19_0, a nilpotent element of degree 19
- a_23_1, a nilpotent element of degree 23
- a_24_1, a nilpotent element of degree 24
- a_27_1, a nilpotent element of degree 27
- b_28_0, an element of degree 28
- a_39_1, a nilpotent element of degree 39
- c_40_1, a Duflot element of degree 40
Ring relations
There are 6 "obvious" relations:
a_7_02, a_15_12, a_19_02, a_23_12, a_27_12, a_39_12
Apart from that, there are 52 minimal relations of maximal degree 67:
- a_7_0·a_15_1
- a_16_1·a_7_0
- b_8_0·a_15_1
- b_8_0·a_16_1
- a_18_0·a_7_0
- b_8_0·a_18_0 − 2·a_7_0·a_19_0
- a_7_0·a_23_1
- a_16_1·a_15_1
- a_24_1·a_7_0
- b_8_0·a_23_1
- a_16_12
- b_8_0·a_24_1
- a_18_0·a_15_1
- a_16_1·a_18_0
- a_7_0·a_27_1
- a_15_1·a_19_0
- a_16_1·a_19_0
- b_28_0·a_7_0 + 2·b_8_0·a_27_1
- a_18_02
- a_18_0·a_19_0
- a_15_1·a_23_1
- a_16_1·a_23_1
- a_24_1·a_15_1
- a_16_1·a_24_1
- a_18_0·a_23_1
- a_18_0·a_24_1
- a_15_1·a_27_1
- a_19_0·a_23_1
- a_16_1·a_27_1
- a_24_1·a_19_0
- b_28_0·a_15_1
- a_16_1·b_28_0
- a_18_0·a_27_1
- a_19_0·a_27_1 − 2·a_7_0·a_39_1
- a_18_0·b_28_0 + 2·a_7_0·a_39_1
- a_24_1·a_23_1
- b_28_0·a_19_0 + b_8_0·a_39_1
- a_24_12
- a_23_1·a_27_1
- a_24_1·a_27_1
- b_28_0·a_23_1
- a_24_1·b_28_0
- a_15_1·a_39_1
- a_16_1·a_39_1
- b_28_0·a_27_1 + b_8_0·c_40_1·a_7_0
- b_28_02 − 2·b_8_02·c_40_1
- a_18_0·a_39_1
- a_19_0·a_39_1
- a_23_1·a_39_1
- a_24_1·a_39_1
- a_27_1·a_39_1 − c_40_1·a_7_0·a_19_0
- b_28_0·a_39_1 + 2·b_8_0·c_40_1·a_19_0
Data used for Benson′s test
- Benson′s completion test succeeded in degree 67.
- 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:
- c_40_1, an element of degree 40
- b_8_0, an element of degree 8
- A Duflot regular sequence is given by c_40_1.
- The Raw Filter Degree Type of that HSOP is [-1, 24, 46].
Restriction maps
- a_7_0 → a_7_0
- b_8_0 → b_8_0
- a_15_1 → a_15_1
- a_16_1 → a_16_1
- a_18_0 → a_18_0
- a_19_0 → a_19_0
- a_23_1 → a_23_1
- a_24_1 → a_24_1
- a_27_1 → a_27_1
- b_28_0 → b_28_0
- a_39_1 → a_39_1
- c_40_1 → c_40_1
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1
- a_7_0 → 0, an element of degree 7
- b_8_0 → 0, an element of degree 8
- a_15_1 → 0, an element of degree 15
- a_16_1 → 0, an element of degree 16
- a_18_0 → 0, an element of degree 18
- a_19_0 → 0, an element of degree 19
- a_23_1 → 0, an element of degree 23
- a_24_1 → 0, an element of degree 24
- a_27_1 → 0, an element of degree 27
- b_28_0 → 0, an element of degree 28
- a_39_1 → 0, an element of degree 39
- c_40_1 → − 2·c_2_020, an element of degree 40
Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup
- a_7_0 → 2·c_2_23·a_1_1, an element of degree 7
- b_8_0 → − c_2_24, an element of degree 8
- a_15_1 → 0, an element of degree 15
- a_16_1 → 0, an element of degree 16
- a_18_0 → c_2_1·c_2_27·a_1_0·a_1_1 − c_2_15·c_2_23·a_1_0·a_1_1, an element of degree 18
- a_19_0 → − c_2_1·c_2_28·a_1_0 + c_2_12·c_2_27·a_1_1 + c_2_15·c_2_24·a_1_0
− c_2_16·c_2_23·a_1_1, an element of degree 19
- a_23_1 → 0, an element of degree 23
- a_24_1 → 0, an element of degree 24
- a_27_1 → c_2_12·c_2_211·a_1_1 − 2·c_2_16·c_2_27·a_1_1 + c_2_110·c_2_23·a_1_1, an element of degree 27
- b_28_0 → c_2_12·c_2_212 − 2·c_2_16·c_2_28 + c_2_110·c_2_24, an element of degree 28
- a_39_1 → − c_2_13·c_2_216·a_1_0 + c_2_14·c_2_215·a_1_1 − 2·c_2_17·c_2_212·a_1_0
+ 2·c_2_18·c_2_211·a_1_1 + 2·c_2_111·c_2_28·a_1_0 − 2·c_2_112·c_2_27·a_1_1
+ c_2_115·c_2_24·a_1_0 − c_2_116·c_2_23·a_1_1, an element of degree 39
- c_40_1 → − 2·c_2_14·c_2_216 − 2·c_2_18·c_2_212 − 2·c_2_112·c_2_28 − 2·c_2_116·c_2_24
− 2·c_2_120, an element of degree 40
Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup
- a_7_0 → 2·c_2_23·a_1_1, an element of degree 7
- b_8_0 → − c_2_24, an element of degree 8
- a_15_1 → 0, an element of degree 15
- a_16_1 → 0, an element of degree 16
- a_18_0 → c_2_1·c_2_27·a_1_0·a_1_1 − c_2_15·c_2_23·a_1_0·a_1_1, an element of degree 18
- a_19_0 → − c_2_1·c_2_28·a_1_0 + c_2_12·c_2_27·a_1_1 + c_2_15·c_2_24·a_1_0
− c_2_16·c_2_23·a_1_1, an element of degree 19
- a_23_1 → 0, an element of degree 23
- a_24_1 → 0, an element of degree 24
- a_27_1 → c_2_12·c_2_211·a_1_1 − 2·c_2_16·c_2_27·a_1_1 + c_2_110·c_2_23·a_1_1, an element of degree 27
- b_28_0 → c_2_12·c_2_212 − 2·c_2_16·c_2_28 + c_2_110·c_2_24, an element of degree 28
- a_39_1 → − c_2_13·c_2_216·a_1_0 + c_2_14·c_2_215·a_1_1 − 2·c_2_17·c_2_212·a_1_0
+ 2·c_2_18·c_2_211·a_1_1 + 2·c_2_111·c_2_28·a_1_0 − 2·c_2_112·c_2_27·a_1_1
+ c_2_115·c_2_24·a_1_0 − c_2_116·c_2_23·a_1_1, an element of degree 39
- c_40_1 → − 2·c_2_14·c_2_216 − 2·c_2_18·c_2_212 − 2·c_2_112·c_2_28 − 2·c_2_116·c_2_24
− 2·c_2_120, an element of degree 40
Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup
- a_7_0 → 2·c_2_23·a_1_1, an element of degree 7
- b_8_0 → − c_2_24, an element of degree 8
- a_15_1 → 0, an element of degree 15
- a_16_1 → 0, an element of degree 16
- a_18_0 → − c_2_1·c_2_27·a_1_0·a_1_1 + c_2_15·c_2_23·a_1_0·a_1_1, an element of degree 18
- a_19_0 → c_2_1·c_2_28·a_1_0 − c_2_12·c_2_27·a_1_1 − c_2_15·c_2_24·a_1_0
+ c_2_16·c_2_23·a_1_1, an element of degree 19
- a_23_1 → 0, an element of degree 23
- a_24_1 → 0, an element of degree 24
- a_27_1 → − c_2_12·c_2_211·a_1_1 + 2·c_2_16·c_2_27·a_1_1 − c_2_110·c_2_23·a_1_1, an element of degree 27
- b_28_0 → − c_2_12·c_2_212 + 2·c_2_16·c_2_28 − c_2_110·c_2_24, an element of degree 28
- a_39_1 → − c_2_13·c_2_216·a_1_0 + c_2_14·c_2_215·a_1_1 − 2·c_2_17·c_2_212·a_1_0
+ 2·c_2_18·c_2_211·a_1_1 + 2·c_2_111·c_2_28·a_1_0 − 2·c_2_112·c_2_27·a_1_1
+ c_2_115·c_2_24·a_1_0 − c_2_116·c_2_23·a_1_1, an element of degree 39
- c_40_1 → − 2·c_2_14·c_2_216 − 2·c_2_18·c_2_212 − 2·c_2_112·c_2_28 − 2·c_2_116·c_2_24
− 2·c_2_120, an element of degree 40
Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup
- a_7_0 → 2·c_2_23·a_1_1, an element of degree 7
- b_8_0 → − c_2_24, an element of degree 8
- a_15_1 → 0, an element of degree 15
- a_16_1 → 0, an element of degree 16
- a_18_0 → − c_2_1·c_2_27·a_1_0·a_1_1 + c_2_15·c_2_23·a_1_0·a_1_1, an element of degree 18
- a_19_0 → c_2_1·c_2_28·a_1_0 − c_2_12·c_2_27·a_1_1 − c_2_15·c_2_24·a_1_0
+ c_2_16·c_2_23·a_1_1, an element of degree 19
- a_23_1 → 0, an element of degree 23
- a_24_1 → 0, an element of degree 24
- a_27_1 → − c_2_12·c_2_211·a_1_1 + 2·c_2_16·c_2_27·a_1_1 − c_2_110·c_2_23·a_1_1, an element of degree 27
- b_28_0 → − c_2_12·c_2_212 + 2·c_2_16·c_2_28 − c_2_110·c_2_24, an element of degree 28
- a_39_1 → − c_2_13·c_2_216·a_1_0 + c_2_14·c_2_215·a_1_1 − 2·c_2_17·c_2_212·a_1_0
+ 2·c_2_18·c_2_211·a_1_1 + 2·c_2_111·c_2_28·a_1_0 − 2·c_2_112·c_2_27·a_1_1
+ c_2_115·c_2_24·a_1_0 − c_2_116·c_2_23·a_1_1, an element of degree 39
- c_40_1 → − 2·c_2_14·c_2_216 − 2·c_2_18·c_2_212 − 2·c_2_112·c_2_28 − 2·c_2_116·c_2_24
− 2·c_2_120, an element of degree 40
Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup
- a_7_0 → 2·c_2_23·a_1_1, an element of degree 7
- b_8_0 → − c_2_24, an element of degree 8
- a_15_1 → 0, an element of degree 15
- a_16_1 → 0, an element of degree 16
- a_18_0 → c_2_1·c_2_27·a_1_0·a_1_1 − c_2_15·c_2_23·a_1_0·a_1_1, an element of degree 18
- a_19_0 → − c_2_1·c_2_28·a_1_0 + c_2_12·c_2_27·a_1_1 + c_2_15·c_2_24·a_1_0
− c_2_16·c_2_23·a_1_1, an element of degree 19
- a_23_1 → 0, an element of degree 23
- a_24_1 → 0, an element of degree 24
- a_27_1 → c_2_12·c_2_211·a_1_1 − 2·c_2_16·c_2_27·a_1_1 + c_2_110·c_2_23·a_1_1, an element of degree 27
- b_28_0 → c_2_12·c_2_212 − 2·c_2_16·c_2_28 + c_2_110·c_2_24, an element of degree 28
- a_39_1 → − c_2_13·c_2_216·a_1_0 + c_2_14·c_2_215·a_1_1 − 2·c_2_17·c_2_212·a_1_0
+ 2·c_2_18·c_2_211·a_1_1 + 2·c_2_111·c_2_28·a_1_0 − 2·c_2_112·c_2_27·a_1_1
+ c_2_115·c_2_24·a_1_0 − c_2_116·c_2_23·a_1_1, an element of degree 39
- c_40_1 → − 2·c_2_14·c_2_216 − 2·c_2_18·c_2_212 − 2·c_2_112·c_2_28 − 2·c_2_116·c_2_24
− 2·c_2_120, an element of degree 40
Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup
- a_7_0 → 2·c_2_23·a_1_1, an element of degree 7
- b_8_0 → − c_2_24, an element of degree 8
- a_15_1 → 0, an element of degree 15
- a_16_1 → 0, an element of degree 16
- a_18_0 → − c_2_1·c_2_27·a_1_0·a_1_1 + c_2_15·c_2_23·a_1_0·a_1_1, an element of degree 18
- a_19_0 → c_2_1·c_2_28·a_1_0 − c_2_12·c_2_27·a_1_1 − c_2_15·c_2_24·a_1_0
+ c_2_16·c_2_23·a_1_1, an element of degree 19
- a_23_1 → 0, an element of degree 23
- a_24_1 → 0, an element of degree 24
- a_27_1 → − c_2_12·c_2_211·a_1_1 + 2·c_2_16·c_2_27·a_1_1 − c_2_110·c_2_23·a_1_1, an element of degree 27
- b_28_0 → − c_2_12·c_2_212 + 2·c_2_16·c_2_28 − c_2_110·c_2_24, an element of degree 28
- a_39_1 → − c_2_13·c_2_216·a_1_0 + c_2_14·c_2_215·a_1_1 − 2·c_2_17·c_2_212·a_1_0
+ 2·c_2_18·c_2_211·a_1_1 + 2·c_2_111·c_2_28·a_1_0 − 2·c_2_112·c_2_27·a_1_1
+ c_2_115·c_2_24·a_1_0 − c_2_116·c_2_23·a_1_1, an element of degree 39
- c_40_1 → − 2·c_2_14·c_2_216 − 2·c_2_18·c_2_212 − 2·c_2_112·c_2_28 − 2·c_2_116·c_2_24
− 2·c_2_120, an element of degree 40
Created with the Sage
cohomology package
written by Simon King
and David Green
Last change: 11.09.2015