Small group number 854 of order 128

G is the group 128gp854

G has 3 minimal generators, rank 4 and exponent 8. The centre has rank 1.

There are 4 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3, 4, 4.

This cohomology ring calculation is complete.

Ring structure | Completion information | Koszul information | Restriction information | Poincaré series

Ring structure

The cohomology ring has 14 generators:

y₁ in degree 1, a nilpotent element
y₂ in degree 1
y₃ in degree 1
x₁ in degree 2
x₂ in degree 2
w₁ in degree 3
w₂ in degree 3
w₃ in degree 3
v in degree 4
u₁ in degree 5
u₂ in degree 5
t in degree 6
s in degree 7
r in degree 8, a regular element

There are 51 minimal relations:

y₁.y₃ = 0
y₁² = 0
y₃.x₂ = y₁.y₂²
y₂.y₃² = y₂².y₃ + y₁.y₂²
y₁.x₁ = y₁.y₂²
x₁.x₂ = y₂³.y₃ + y₂⁴ + y₁.y₂³
y₃.w₁ = y₁.w₂ + y₁.y₂³
y₂.y₃.x₁ = y₂².x₁ + y₂³.y₃ + y₂⁴ + y₁.w₂
y₂².x₂ = y₂³.y₃ + y₂⁴ + y₁.w₁
y₁.w₃ = 0
x₂.w₃ = y₁.y₂⁴
x₂.w₂ = y₂².w₁ + y₂⁴.y₃ + y₂⁵ + y₁.y₂.w₂
x₁.w₁ = y₂².w₁ + y₁.y₂.w₂
y₂.y₃.w₃ = y₂².w₃ + y₁.y₂⁴
y₂.y₃.w₂ = y₂².w₂ + y₂².w₁ + y₂⁴.y₃ + y₂⁵
y₁.v = y₁.y₂.w₂ + y₁.y₂.w₁
w₃² = x₁³ + y₃.x₁.w₃ + y₃².v + y₃³.w₂ + y₂³.w₃ + y₂³.w₂ + y₂³.w₁ + y₂⁵.y₃ + y₁.y₂⁵
w₂² = x₁.v + y₃.u₁ + y₃².v + y₃³.w₂ + y₂.x₁.w₃ + y₂.x₁.w₂ + y₂³.w₃ + y₂³.w₂ + y₂⁴.x₁ + y₁.y₂².w₁ + y₁.y₂⁵
w₁.w₃ = y₁.y₂².w₁
w₁.w₂ = y₂.y₃.v + y₂².v + y₂³.w₁
w₁² = x₂.v + y₂.x₂.w₁ + y₂³.w₁ + y₁.x₂.w₁ + y₁.y₂².w₁
y₃.u₂ = y₃.x₁.w₂ + y₃².v + y₂⁴.x₁ + y₂⁶ + y₁.y₂⁵
y₁.u₂ = y₁.x₂.w₁ + y₁.y₂².w₁ + y₁.y₂⁵
y₁.u₁ = y₁.x₂.w₁ + y₁.y₂⁵
w₃.v = x₁.u₁ + y₃.t + y₃.w₂.w₃ + y₂.w₂.w₃ + y₂.x₁³ + y₂².u₁ + y₂².x₁.w₂ + y₂².y₃.v + y₂⁴.w₁ + y₂⁵.x₁ + y₂⁷ + y₁.y₂³.w₁ + y₁.y₂⁶
x₂.u₂ = x₂.u₁ + y₂².u₂ + y₂².x₁.w₂ + y₂².y₃.v + y₂⁵.x₁ + y₂⁷ + y₁.y₂.x₂.w₁ + y₁.y₂³.w₁
x₁.u₂ = x₁².w₂ + y₃.x₁.v + y₂².u₂ + y₂².x₁.w₂ + y₂².y₃.v + y₂³.x₁² + y₂⁷ + y₁.y₂³.w₁ + y₁.y₂⁶
y₂.y₃.u₁ = y₂².u₁ + y₂⁴.w₁ + y₂⁶.y₃ + y₂⁷ + y₁.y₂⁶
y₁.t = y₁.y₂³.w₁
w₃.u₂ = x₁.w₂.w₃ + y₃.x₁.u₁ + y₃².t + y₃².w₂.w₃ + y₂².w₂.w₃ + y₂².x₁³ + y₂³.u₁ + y₂³.x₁.w₃ + y₂³.x₁.w₂ + y₂³.y₃.v + y₂⁵.w₃ + y₂⁶.x₁ + y₂⁷.y₃ + y₁.y₂⁴.w₁ + y₁.y₂⁷
w₃.u₁ = x₁².v + y₃.s + y₃.w₂.v + y₃².t + y₃².w₂.w₃ + y₃².x₁.v + y₃³.x₁.w₂ + y₃⁴.v + y₂.x₁².w₃ + y₂.x₁².w₂ + y₂.y₃.t + y₂³.x₁.w₂ + y₂⁴.v + y₂⁴.x₁² + y₂⁵.w₂ + y₂⁵.w₁
w₂.u₂ = x₁².v + y₃.w₂.v + y₃.x₁.u₁ + y₃².x₁.v + y₃³.x₁.w₂ + y₂.x₁².w₃ + y₂.x₁².w₂ + y₂.y₃.t + y₂².t + y₂³.u₂ + y₂³.x₁.w₃ + y₂³.x₁.w₂ + y₂³.y₃.v + y₂⁴.x₁² + y₂⁵.w₂ + y₂⁵.w₁ + y₂⁶.x₁ + y₂⁷.y₃ + y₁.y₂⁷
w₁.u₂ = x₂.t + y₂.x₂.u₁ + y₂.x₂².w₁ + y₂³.y₃.v + y₂⁴.v + y₁.x₂².w₁ + y₁.y₂⁴.w₁ + y₁.y₂⁷
w₁.u₁ = x₂.t + y₂.x₂.u₁ + y₂.x₂².w₁ + y₂.y₃.t + y₂².t + y₂³.y₃.v + y₂⁴.v + y₂⁵.w₁ + y₂⁷.y₃ + y₂⁸ + y₁.x₂².w₁ + y₁.y₂⁴.w₁ + y₁.y₂⁷
y₁.s = y₁.y₂⁷
v.u₂ = w₁.t + x₁.w₂.v + y₃.v² + y₂.x₂².v + y₂³.x₁.v + y₂⁵.v + y₂⁶.w₁ + y₁.y₂.x₂².w₁ + y₁.y₂⁵.w₁
w₃.t = x₁.s + x₁.w₂.v + x₁².u₁ + x₁³.w₂ + y₃.v² + y₃.x₁.w₂.w₃ + y₃.x₁².v + y₃².x₁².w₂ + y₃⁴.u₁ + y₃⁵.v + y₃⁶.w₂ + y₂.x₁.t + y₂.x₁.w₂.w₃ + y₂.x₁².v + y₂.x₁⁴ + y₂².s + y₂².w₂.v + y₂².x₁.u₁ + y₂².x₁².w₃ + y₂².x₁².w₂ + y₂².y₃.t + y₂³.t + y₂³.w₂.w₃ + y₂³.x₁³ + y₂⁴.u₁ + y₂⁴.x₁.w₃ + y₂⁴.x₁.w₂ + y₂⁴.y₃.v + y₂⁵.v + y₂⁶.w₃ + y₂⁷.x₁ + y₂⁸.y₃ + y₂⁹
x₂.s = x₂.w₁.v + x₂².u₁ + x₂³.w₁ + y₂.x₂.t + y₂.x₂².v + y₂⁴.y₃.v + y₂⁵.v + y₁.y₂.x₂².w₁
y₂.y₃.s = y₂².s + y₂².w₁.v + y₂².y₃.t + y₂³.t + y₂⁸.y₃ + y₂⁹ + y₁.y₂⁵.w₁
u₂² = x₂.v² + x₂³.v + x₁³.v + y₃.x₁².u₁ + y₃².v² + y₃².x₁².v + y₃³.x₁².w₂ + y₂.x₂³.w₁ + y₂.x₁³.w₃ + y₂.x₁³.w₂ + y₂³.x₁².w₃ + y₂³.x₁².w₂ + y₂⁴.x₁³ + y₂⁶.x₁² + y₂⁷.w₁ + y₂⁹.y₃ + y₁.x₂³.w₁ + y₁.y₂⁶.w₁ + y₁.y₂⁹
u₁.u₂ = x₂.v² + x₂³.v + x₁.w₂.u₁ + y₃.v.u₁ + y₂.x₂³.w₁ + y₂.y₃.v² + y₂².v² + y₂³.x₁.u₁ + y₂³.y₃.t + y₂⁴.t + y₂⁵.u₂ + y₂⁵.u₁ + y₂⁵.x₁.w₂ + y₂⁶.v + y₂⁸.x₁ + y₂⁹.y₃ + y₁.x₂³.w₁ + y₁.y₂⁹
u₁² = x₂.v² + x₂³.v + x₁.v² + y₃.v.u₁ + y₃².w₂.u₁ + y₃².x₁.t + y₃².x₁⁴ + y₃³.x₁.u₁ + y₃³.x₁².w₃ + y₃⁴.t + y₃⁴.x₁.v + y₃⁶.v + y₂.x₂³.w₁ + y₂².w₂.u₁ + y₂².x₁⁴ + y₂³.w₂.v + y₂³.w₁.v + y₂³.x₁².w₃ + y₂³.x₁².w₂ + y₂³.y₃.t + y₂⁴.w₂.w₃ + y₂⁵.u₁ + y₂⁵.y₃.v + y₂⁶.x₁² + y₂⁷.w₃ + y₂⁷.w₂ + y₂⁸.x₁ + y₁.x₂³.w₁ + y₃².r
w₃.s = x₁.w₂.u₁ + x₁².t + x₁².w₂.w₃ + x₁³.v + y₃.v.u₁ + y₃.w₂.t + y₃.x₁².u₁ + y₃².v² + y₃².w₂.u₁ + y₃².x₁.t + y₃².x₁².v + y₃³.s + y₃⁴.t + y₃⁴.x₁.v + y₃⁵.x₁.w₂ + y₃⁶.v + y₂.x₁.s + y₂.x₁.w₂.v + y₂.x₁².u₁ + y₂.y₃.v² + y₂².x₁⁴ + y₂³.w₂.v + y₂³.w₁.v + y₂³.x₁².w₂ + y₂³.y₃.t + y₂⁴.w₂.w₃ + y₂⁵.u₁ + y₂⁵.x₁.w₃ + y₂⁵.x₁.w₂ + y₂⁵.y₃.v + y₂⁶.x₁² + y₂⁷.w₂ + y₂⁷.w₁ + y₂⁸.x₁ + y₁.y₂⁶.w₁ + y₁.y₂⁹
w₁.s = x₂.v² + x₂².t + x₂³.v + y₂.w₁.t + y₂.x₂².u₁ + y₂³.y₃.t + y₂⁴.t + y₂⁵.y₃.v + y₂⁶.v + y₂⁹.y₃ + y₂¹⁰ + y₁.y₂⁹
u₂.t = w₁.v² + x₂².w₁.v + x₁.w₂.t + y₃.v.t + y₂.x₂.v² + y₂.x₂².t + y₂².y₃.v² + y₂³.v² + y₂³.x₁.t + y₂⁵.t + y₂⁶.u₂ + y₂⁶.x₁.w₂ + y₂⁶.y₃.v + y₂⁹.x₁ + y₂¹⁰.y₃ + y₁.y₂.x₂³.w₁ + y₁.y₂¹⁰
u₁.t = v.s + w₂.v² + x₂.w₁.t + x₁.v.u₁ + x₁².w₂.v + y₃.v.t + y₃.w₂.s + y₃.x₁².t + y₃.x₁⁵ + y₃².v.u₁ + y₃².x₁².u₁ + y₃².x₁³.w₃ + y₃³.x₁.t + y₃⁴.x₁.u₁ + y₃⁶.x₁.w₂ + y₂.v.t + y₂.w₂.s + y₂.x₂².t + y₂.x₂³.v + y₂.x₁².t + y₂².v.u₁ + y₂².w₂.t + y₂².x₁.w₂.v + y₂².x₁².u₁ + y₂².x₁³.w₃ + y₂².x₁³.w₂ + y₂³.w₂.u₁ + y₂³.x₁.w₂.w₃ + y₂⁴.w₂.v + y₂⁴.x₁.u₁ + y₂⁴.x₁².w₃ + y₂⁴.y₃.t + y₂⁵.t + y₂⁵.w₂.w₃ + y₂⁶.x₁.w₂ + y₂⁸.w₃ + y₂⁸.w₁ + y₂⁹.x₁ + y₂¹⁰.y₃ + y₂¹¹ + y₃.x₁.r + y₂².y₃.r
t² = v³ + x₂².v² + x₁.v.t + x₁³.t + x₁⁴.v + x₁⁶ + y₃.w₂.v² + y₃.x₁.w₂.t + y₃.x₁⁴.w₃ + y₃².x₁.v² + y₃².x₁².t + y₃³.v.u₁ + y₃³.x₁.w₂.v + y₃³.x₁³.w₂ + y₃⁴.v² + y₃⁴.w₂.u₁ + y₃⁴.x₁².v + y₃⁶.x₁.v + y₃⁷.u₁ + y₃⁸.v + y₃⁹.w₂ + y₂.w₂.v² + y₂.w₁.v² + y₂.x₁.w₂.t + y₂.x₁³.u₁ + y₂.x₁⁴.w₃ + y₂.x₁⁴.w₂ + y₂².v.t + y₂².x₁.v² + y₂².x₁².t + y₂².x₁².w₂.w₃ + y₂².x₁³.v + y₂³.v.u₁ + y₂³.w₁.t + y₂³.x₁³.w₃ + y₂⁴.w₂.u₁ + y₂⁴.x₁.t + y₂⁴.x₁².v + y₂⁴.x₁⁴ + y₂⁵.w₂.v + y₂⁵.x₁.u₁ + y₂⁶.w₂.w₃ + y₂⁹.w₃ + y₂⁹.w₁ + y₂¹⁰.x₁ + y₂¹¹.y₃ + y₁.y₂⁸.w₁ + y₁.y₂¹¹ + x₁².r + y₂⁴.r
u₂.s = x₂.v.t + x₂².v² + x₂³.t + x₂⁴.v + x₁.w₂.s + y₃.v.s + y₂.w₁.v² + y₂.x₂³.u₁ + y₂³.x₁.s + y₂³.y₃.v² + y₂⁴.v² + y₂⁵.s + y₂⁹.w₁ + y₁.y₂⁸.w₁
u₁.s = w₂.v.u₁ + x₂.v.t + x₂².v² + x₂³.t + x₂⁴.v + x₁.v.t + x₁².v² + x₁².w₂.u₁ + y₃.x₁.w₂.t + y₃.x₁².s + y₃.x₁⁴.w₃ + y₃.x₁⁴.w₂ + y₃².w₂.s + y₃².x₁³.v + y₃³.w₂.t + y₃³.x₁.s + y₃³.x₁².u₁ + y₃⁴.x₁.w₂.w₃ + y₃⁴.x₁².v + y₃⁵.s + y₃⁵.w₂.v + y₃⁶.t + y₃⁷.u₁ + y₃⁷.x₁.w₂ + y₃⁹.w₂ + y₂.v.s + y₂.w₂.v² + y₂.x₂.w₁.t + y₂.x₂².w₁.v + y₂.x₂³.u₁ + y₂.x₁.w₂.t + y₂.x₁².s + y₂.x₁².w₂.v + y₂.x₁⁴.w₂ + y₂.y₃.v.t + y₂².v.t + y₂².x₁².w₂.w₃ + y₂³.v.u₁ + y₂³.w₂.t + y₂³.x₁.s + y₂³.x₁³.w₃ + y₂⁴.v² + y₂⁴.w₂.u₁ + y₂⁴.x₁.t + y₂⁴.x₁.w₂.w₃ + y₂⁵.s + y₂⁵.w₂.v + y₂⁵.x₁.u₁ + y₂⁵.y₃.t + y₂⁶.x₁³ + y₂⁷.x₁.w₃ + y₂⁷.y₃.v + y₂⁸.x₁² + y₂⁹.w₃ + y₂¹⁰.x₁ + y₁.y₂⁸.w₁ + y₃.w₃.r + y₃².x₁.r + y₂².x₁.r + y₂³.y₃.r + y₂⁴.r + y₁.w₂.r + y₁.y₂³.r
t.s = v².u₁ + w₂.v.t + x₂.w₁.v² + x₂².w₁.t + x₂³.w₁.v + x₁².w₂.t + x₁³.s + x₁³.w₂.v + x₁⁵.w₃ + x₁⁵.w₂ + y₃.v³ + y₃.x₁.w₂.s + y₃.x₁².w₂.u₁ + y₃.x₁⁴.v + y₃².x₁².s + y₃².x₁³.u₁ + y₃³.v.t + y₃³.x₁².w₂.w₃ + y₃³.x₁³.v + y₃⁴.v.u₁ + y₃⁴.x₁.s + y₃⁴.x₁.w₂.v + y₃⁵.v² + y₃⁵.x₁⁴ + y₃⁶.w₂.v + y₃⁶.x₁².w₃ + y₃⁶.x₁².w₂ + y₃⁷.t + y₃⁷.x₁.v + y₃⁸.x₁.w₂ + y₃⁹.v + y₂.v³ + y₂.x₂.v.t + y₂.x₂².v² + y₂.x₂³.t + y₂.x₁.w₂.s + y₂.x₁².w₂.u₁ + y₂.x₁³.w₂.w₃ + y₂².v.s + y₂².w₁.v² + y₂².x₁.v.u₁ + y₂².x₁².s + y₂².x₁².w₂.v + y₂².x₁³.u₁ + y₂².x₁⁴.w₃ + y₂².x₁⁴.w₂ + y₂³.v.t + y₂³.w₂.s + y₂³.x₁².t + y₂⁴.v.u₁ + y₂⁴.w₂.t + y₂⁴.w₁.t + y₂⁴.x₁.w₂.v + y₂⁴.x₁².u₁ + y₂⁴.y₃.v² + y₂⁵.v² + y₂⁵.x₁².v + y₂⁶.s + y₂⁶.w₂.v + y₂⁶.x₁².w₃ + y₂⁶.x₁².w₂ + y₂⁶.y₃.t + y₂⁷.x₁.v + y₂⁸.u₁ + y₂⁸.x₁.w₃ + y₂¹⁰.w₃ + y₂¹².y₃ + y₂¹³ + y₁.y₂.x₂⁴.w₁ + y₁.y₂¹² + x₁.w₃.r + y₃.x₁².r + y₃⁵.r + y₂.x₁².r + y₂².w₃.r + y₂⁵.r + y₁.y₂⁴.r
s² = x₂.v³ + x₂³.v² + x₁².v.t + x₁³.v² + x₁⁴.t + x₁⁷ + y₃.x₁.v.s + y₃.x₁².w₂.t + y₃.x₁³.s + y₃.x₁⁴.u₁ + y₃.x₁⁵.w₂ + y₃².x₁.v.t + y₃².x₁.w₂.s + y₃².x₁².w₂.u₁ + y₃².x₁³.t + y₃².x₁⁴.v + y₃³.x₁.v.u₁ + y₃³.x₁².s + y₃³.x₁².w₂.v + y₃³.x₁⁴.w₂ + y₃⁴.v.t + y₃⁴.x₁.v² + y₃⁴.x₁.w₂.u₁ + y₃⁴.x₁².t + y₃⁴.x₁².w₂.w₃ + y₃⁴.x₁³.v + y₃⁵.w₂.t + y₃⁵.x₁.s + y₃⁵.x₁².u₁ + y₃⁶.x₁².v + y₃⁶.x₁⁴ + y₃⁷.w₂.v + y₃⁷.x₁².w₃ + y₃⁸.t + y₃⁸.x₁.v + y₃⁹.x₁.w₂ + y₃¹⁰.v + y₂.x₂.w₁.v² + y₂.x₁².v.u₁ + y₂.x₁².w₂.t + y₂.x₁⁴.u₁ + y₂².v³ + y₂².x₁.w₂.s + y₂².x₁².v² + y₂².x₁².w₂.u₁ + y₂².x₁³.t + y₂³.v.s + y₂³.x₁.v.u₁ + y₂³.x₁⁴.w₃ + y₂⁴.x₁.v² + y₂⁴.x₁².t + y₂⁵.x₁.s + y₂⁵.x₁².u₁ + y₂⁵.x₁³.w₃ + y₂⁶.w₂.u₁ + y₂⁶.x₁.t + y₂⁶.x₁².v + y₂⁶.x₁⁴ + y₂⁷.x₁.u₁ + y₂⁷.x₁².w₃ + y₂⁷.x₁².w₂ + y₂⁷.y₃.t + y₂⁸.t + y₂⁸.w₂.w₃ + y₂⁸.x₁³ + y₂⁹.x₁.w₃ + y₂⁹.y₃.v + y₂¹⁰.x₁² + y₂¹¹.w₃ + y₂¹¹.w₂ + y₂¹².x₁ + y₂¹³.y₃ + x₁³.r + y₃.x₁.w₃.r + y₃².v.r + y₃².x₁².r + y₃³.w₂.r + y₃⁶.r + y₂².x₁².r + y₂³.w₃.r + y₂³.w₂.r + y₂³.w₁.r + y₂⁵.y₃.r + y₂⁶.r + y₁.y₂⁵.r

A minimal Gröbner basis for the relations ideal consists of this minimal generating set, together with the following redundant relations:

y₁.y₂².w₂ = y₁.y₂².w₁ + y₁.y₂⁵
x₂.v.u₁ = x₂.w₁.t + y₂.x₂³.v + y₂².w₁.t + y₂⁴.w₁.v + y₁.y₂.x₂³.w₁ + y₁.y₂¹⁰

Completion information

This cohomology ring was obtained from a calculation out to degree 18. The cohomology ring approximation is stable from degree 14 onwards, and Benson's tests detect stability from degree 14 onwards.

This cohomology ring has dimension 4 and depth 2. Here is a homogeneous system of parameters:

h₁ = r in degree 8
h₂ = x₂ + y₃² in degree 2
h₃ = v in degree 4
h₄ = x₁ + y₂² in degree 2

The first 2 terms h₁, h₂ form a regular sequence of maximum length.

The first term h₁ forms a complete Duflot regular sequence. That is, its restriction to the greatest central elementary abelian subgroup forms a regular sequence of maximal length.

Data for Benson's test:

Raw filter degree type: -1, -1, 5, 10, 12.
Filter degree type: -1, -2, -3, -4, -4.
α = 0
The system of parameters is very strongly quasi-regular.
The regularity conjecture is satisfied.

Koszul information

A basis for R/(h₁, h₂, h₃, h₄) is as follows.

1 in degree 0
y₃ in degree 1
y₂ in degree 1
y₁ in degree 1
y₃² in degree 2
y₂.y₃ in degree 2
y₂² in degree 2
y₁.y₂ in degree 2
w₃ in degree 3
w₂ in degree 3
w₁ in degree 3
y₂².y₃ in degree 3
y₂³ in degree 3
y₁.y₂² in degree 3
y₃.w₃ in degree 4
y₃.w₂ in degree 4
y₂.w₃ in degree 4
y₂.w₂ in degree 4
y₂.w₁ in degree 4
y₁.y₂³ in degree 4
u₂ in degree 5
u₁ in degree 5
y₂².w₃ in degree 5
y₂².w₂ in degree 5
y₂².w₁ in degree 5
t in degree 6
w₂.w₃ in degree 6
y₃.u₁ in degree 6
y₂.u₂ in degree 6
y₂.u₁ in degree 6
y₂³.w₁ in degree 6
s in degree 7
y₃.w₂.w₃ in degree 7
y₂.t in degree 7
y₂.w₂.w₃ in degree 7
y₂².u₂ in degree 7
y₂².u₁ in degree 7
w₂.u₁ in degree 8
y₃.s in degree 8
y₂.s in degree 8
y₂².t in degree 8
y₂².w₂.w₃ in degree 8
y₂³.u₁ in degree 8
w₂.t in degree 9
y₃.w₂.u₁ in degree 9
y₂.w₂.u₁ in degree 9
y₂².s in degree 9
y₂³.t in degree 9
w₂.s in degree 10
y₂.w₂.t in degree 10
y₂².w₂.u₁ in degree 10
y₃.w₂.s in degree 11
y₂.w₂.s in degree 11
y₂².w₂.s in degree 12

A basis for Ann_{R/(h₁, h₂, h₃)}(h₄) is as follows.

y₁ in degree 1
y₁.y₂ in degree 2
w₁ + y₂².y₃ in degree 3
y₂³ in degree 3
y₁.y₂² in degree 3
y₂.w₁ + y₂³.y₃ in degree 4
y₁.y₂³ in degree 4
y₃².h in degree 4
y₂.y₃.h + y₂².h in degree 4
u₂ + y₂².w₂ + w₂.h in degree 5
y₂².w₁ + y₂⁴.y₃ in degree 5
y₂².y₃.h + y₂³.h in degree 5
y₂.u₂ + y₂³.w₂ + y₂.w₂.h in degree 6
y₂³.w₁ + y₂⁵.y₃ in degree 6
y₂².u₂ + y₂⁴.w₂ + y₂².w₂.h in degree 7
y₂².t + y₂.u₁.h + y₂².h³ in degree 8
y₂².w₂.w₃ + y₃.u₁.h + y₂².h³ in degree 8
y₂³.u₁ in degree 8
y₃.w₂.u₁ + y₂².w₂.h² in degree 9
y₂³.t in degree 9
y₂².u₁.h in degree 9
y₂².w₂.u₁ in degree 10

A basis for Ann_{R/(h₁, h₂)}(h₃) is as follows.

y₁.y₂ in degree 2
y₁.y₂² in degree 3
y₁.w₂ in degree 4
y₁.w₁ in degree 4
y₁.y₂³ in degree 4
y₁.y₂.w₁ in degree 5
y₁.h in degree 5

Restriction information

Restriction to special subgroup number 1, which is 2gp1

y₁ restricts to 0
y₂ restricts to 0
y₃ restricts to 0
x₁ restricts to 0
x₂ restricts to 0
w₁ restricts to 0
w₂ restricts to 0
w₃ restricts to 0
v restricts to 0
u₁ restricts to 0
u₂ restricts to 0
t restricts to 0
s restricts to 0
r restricts to y⁸

Restriction to special subgroup number 2, which is 8gp5

y₁ restricts to 0
y₂ restricts to 0
y₃ restricts to 0
x₁ restricts to 0
x₂ restricts to y₂²
w₁ restricts to y₂.y₃² + y₂².y₃
w₂ restricts to 0
w₃ restricts to 0
v restricts to y₃⁴ + y₂².y₃²
u₁ restricts to y₂.y₃⁴ + y₂⁴.y₃
u₂ restricts to y₂.y₃⁴ + y₂⁴.y₃
t restricts to y₃⁶ + y₂.y₃⁵ + y₂³.y₃³ + y₂⁴.y₃²
s restricts to y₂.y₃⁶ + y₂².y₃⁵ + y₂⁴.y₃³ + y₂⁵.y₃²
r restricts to y₂².y₃⁶ + y₂³.y₃⁵ + y₂⁴.y₃⁴ + y₂⁵.y₃³ + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₃² + y₁⁴.y₃⁴ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 3, which is 8gp5

y₁ restricts to 0
y₂ restricts to y₃
y₃ restricts to 0
x₁ restricts to y₃²
x₂ restricts to y₃²
w₁ restricts to y₃³ + y₂.y₃² + y₂².y₃
w₂ restricts to y₂.y₃² + y₂².y₃
w₃ restricts to 0
v restricts to y₃⁴ + y₂².y₃² + y₂⁴
u₁ restricts to y₂.y₃⁴ + y₂².y₃³
u₂ restricts to y₃⁵ + y₂².y₃³ + y₂⁴.y₃
t restricts to y₃⁶ + y₂.y₃⁵ + y₂².y₃⁴ + y₂³.y₃³ + y₂⁴.y₃² + y₂⁵.y₃ + y₂⁶
s restricts to y₃⁷ + y₂².y₃⁵ + y₂⁴.y₃³
r restricts to y₃⁸ + y₂.y₃⁷ + y₂⁴.y₃⁴ + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₃² + y₁⁴.y₃⁴ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 4, which is 16gp14

y₁ restricts to 0
y₂ restricts to 0
y₃ restricts to y₂
x₁ restricts to y₃² + y₂.y₃
x₂ restricts to 0
w₁ restricts to 0
w₂ restricts to y₃.y₄² + y₃².y₄ + y₂.y₄² + y₂².y₄ + y₁.y₂² + y₁².y₂
w₃ restricts to y₃³ + y₂.y₄² + y₂².y₄ + y₂².y₃
v restricts to y₄⁴ + y₃².y₄² + y₂².y₃.y₄ + y₂³.y₄ + y₁.y₂³ + y₁².y₂²
u₁ restricts to y₃.y₄⁴ + y₃³.y₄² + y₂.y₃².y₄² + y₂.y₃³.y₄ + y₂².y₃.y₄² + y₂².y₃².y₄ + y₁.y₂².y₃² + y₁.y₂³.y₃ + y₁².y₂.y₃² + y₁².y₂².y₃ + y₁².y₂³ + y₁⁴.y₂
u₂ restricts to y₃³.y₄² + y₃⁴.y₄ + y₂.y₄⁴ + y₂.y₃².y₄² + y₂.y₃³.y₄ + y₂².y₃.y₄² + y₂².y₃².y₄ + y₂⁴.y₄ + y₁.y₂².y₃² + y₁.y₂³.y₃ + y₁.y₂⁴ + y₁².y₂.y₃² + y₁².y₂².y₃ + y₁².y₂³
t restricts to y₄⁶ + y₃⁴.y₄² + y₂.y₄⁵ + y₂.y₃⁴.y₄ + y₂².y₄⁴ + y₂².y₃².y₄² + y₂³.y₄³ + y₂³.y₃².y₄ + y₁.y₂.y₃⁴ + y₁.y₂³.y₃² + y₁².y₃⁴ + y₁².y₂³.y₃ + y₁⁴.y₃² + y₁⁴.y₂.y₃
s restricts to y₃².y₄⁵ + y₃⁴.y₄³ + y₃⁵.y₄² + y₃⁶.y₄ + y₂.y₃.y₄⁵ + y₂.y₃².y₄⁴ + y₂.y₃⁵.y₄ + y₂².y₃.y₄⁴ + y₂².y₃².y₄³ + y₂³.y₄⁴ + y₂³.y₃².y₄² + y₂⁴.y₃.y₄² + y₂⁴.y₃².y₄ + y₂⁶.y₄ + y₁.y₂.y₃⁵ + y₁.y₂².y₄⁴ + y₁.y₂².y₃⁴ + y₁.y₂⁴.y₃² + y₁.y₂⁵.y₄ + y₁.y₂⁵.y₃ + y₁.y₂⁶ + y₁².y₃⁵ + y₁².y₂.y₄⁴ + y₁².y₂.y₃⁴ + y₁².y₂².y₃³ + y₁².y₂³.y₄² + y₁².y₂⁴.y₃ + y₁⁴.y₃³ + y₁⁴.y₂.y₄² + y₁⁴.y₂.y₃² + y₁⁴.y₂².y₄ + y₁⁴.y₂³
r restricts to y₃².y₄⁶ + y₃⁴.y₄⁴ + y₃⁸ + y₂.y₃.y₄⁶ + y₂.y₃².y₄⁵ + y₂.y₃⁵.y₄² + y₂.y₃⁶.y₄ + y₂.y₃⁷ + y₂².y₄⁶ + y₂².y₃.y₄⁵ + y₂³.y₄⁵ + y₂³.y₃.y₄⁴ + y₂³.y₃².y₄³ + y₂³.y₃⁴.y₄ + y₂⁴.y₃.y₄³ + y₂⁴.y₃².y₄² + y₂⁴.y₃³.y₄ + y₂⁴.y₃⁴ + y₂⁵.y₄³ + y₂⁵.y₃.y₄² + y₂⁵.y₃³ + y₂⁷.y₄ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂.y₃⁶ + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃⁴.y₄ + y₁.y₂².y₃⁵ + y₁.y₂³.y₃⁴ + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁴.y₃².y₄ + y₁.y₂⁴.y₃³ + y₁.y₂⁵.y₃² + y₁.y₂⁶.y₃ + y₁.y₂⁷ + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₃⁶ + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃⁴.y₄ + y₁².y₂.y₃⁵ + y₁².y₂².y₄⁴ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃⁴ + y₁².y₂³.y₃³ + y₁².y₂⁴.y₄² + y₁².y₂⁴.y₃.y₄ + y₁².y₂⁶ + y₁⁴.y₄⁴ + y₁⁴.y₃².y₄² + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂².y₄² + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂³.y₃ + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 5, which is 16gp14

y₁ restricts to 0
y₂ restricts to y₄
y₃ restricts to y₄
x₁ restricts to y₂.y₄ + y₂²
x₂ restricts to 0
w₁ restricts to 0
w₂ restricts to y₃.y₄² + y₃².y₄ + y₂.y₄² + y₂.y₃² + y₂².y₄ + y₂².y₃ + y₁.y₄² + y₁².y₄
w₃ restricts to y₃.y₄² + y₃².y₄ + y₂.y₄² + y₂³
v restricts to y₄⁴ + y₃.y₄³ + y₃⁴ + y₂.y₄³ + y₂.y₃.y₄² + y₂.y₃².y₄ + y₂².y₃.y₄ + y₂².y₃² + y₂³.y₄
u₁ restricts to y₄⁵ + y₂.y₃².y₄² + y₂.y₃⁴ + y₂².y₃.y₄² + y₂².y₃².y₄ + y₂³.y₃.y₄ + y₂³.y₃² + y₁.y₂.y₄³ + y₁.y₂².y₄² + y₁².y₄³ + y₁².y₂.y₄² + y₁².y₂².y₄ + y₁⁴.y₄
u₂ restricts to y₃.y₄⁴ + y₃⁴.y₄ + y₂².y₃².y₄ + y₂³.y₄² + y₂³.y₃.y₄ + y₂³.y₃² + y₂⁴.y₄ + y₂⁴.y₃ + y₁.y₂.y₄³ + y₁.y₂².y₄² + y₁².y₂.y₄² + y₁².y₂².y₄
t restricts to y₄⁶ + y₃³.y₄³ + y₃⁴.y₄² + y₃⁵.y₄ + y₃⁶ + y₂.y₃².y₄³ + y₂.y₃⁴.y₄ + y₂².y₃².y₄² + y₂³.y₄³ + y₂⁴.y₃² + y₂⁵.y₄ + y₁.y₂².y₄³ + y₁.y₂⁴.y₄ + y₁².y₄⁴ + y₁².y₂.y₄³ + y₁².y₂⁴ + y₁⁴.y₄² + y₁⁴.y₂.y₄ + y₁⁴.y₂²
s restricts to y₃.y₄⁶ + y₃².y₄⁵ + y₃³.y₄⁴ + y₃⁴.y₄³ + y₃⁵.y₄² + y₃⁶.y₄ + y₂.y₃.y₄⁵ + y₂.y₃³.y₄³ + y₂.y₃⁴.y₄² + y₂.y₃⁵.y₄ + y₂².y₃⁴.y₄ + y₂².y₃⁵ + y₂³.y₄⁴ + y₂³.y₃².y₄² + y₂⁴.y₃³ + y₂⁵.y₃² + y₂⁶.y₄ + y₂⁶.y₃ + y₁.y₃².y₄⁴ + y₁.y₃⁴.y₄² + y₁.y₂³.y₄³ + y₁.y₂⁵.y₄ + y₁².y₃.y₄⁴ + y₁².y₃⁴.y₄ + y₁².y₂.y₄⁴ + y₁².y₂⁵ + y₁⁴.y₃.y₄² + y₁⁴.y₃².y₄ + y₁⁴.y₂.y₄² + y₁⁴.y₂³
r restricts to y₄⁸ + y₃².y₄⁶ + y₃³.y₄⁵ + y₃⁵.y₄³ + y₃⁶.y₄² + y₂.y₄⁷ + y₂.y₃⁴.y₄³ + y₂.y₃⁶.y₄ + y₂².y₄⁶ + y₂².y₃.y₄⁵ + y₂².y₃².y₄⁴ + y₂².y₃³.y₄³ + y₂².y₃⁶ + y₂³.y₄⁵ + y₂⁴.y₄⁴ + y₂⁴.y₃.y₄³ + y₂⁴.y₃².y₄² + y₂⁴.y₃³.y₄ + y₂⁴.y₃⁴ + y₂⁵.y₄³ + y₂⁸ + y₁.y₃².y₄⁵ + y₁.y₃⁴.y₄³ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₄⁵ + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃⁴.y₄ + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁴.y₃².y₄ + y₁.y₂⁶.y₄ + y₁².y₃.y₄⁵ + y₁².y₃².y₄⁴ + y₁².y₂.y₄⁵ + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃⁴.y₄ + y₁².y₂².y₄⁴ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃⁴ + y₁².y₂³.y₄³ + y₁².y₂⁴.y₃.y₄ + y₁².y₂⁴.y₃² + y₁².y₂⁶ + y₁⁴.y₄⁴ + y₁⁴.y₃.y₄³ + y₁⁴.y₃⁴ + y₁⁴.y₂.y₄³ + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂³.y₄ + y₁⁸

Poincaré series

(1 + 3t + 4t² + 5t³ + 5t⁴ + 2t⁵ + t⁶ + 2t⁷ + 2t⁸ + 3t⁹ + 3t¹⁰ + t¹¹) / (1 - t²)² (1 - t⁴) (1 - t⁸)

Back to the groups of order 128