Sporadic Hall-Janko group HJ=J₂

Order = 604 800 = 2⁷.3³.5².7.
Mult = 2.
Out = 2.

Robert Wilson's ATLAS page for the Janko group J₂ is available here.

J₂: Length 73, 2-generator, 4-relator.

< x, y | x² = y³ = (xy)⁷ = (xyxyxy^-1xyxyxy^-1xyxy^-1xy^-1)³ = 1 >

J₂: Length 82, 3-generator, 9-relator.

< x, y, t | x³ = y³ = (xy)⁴ = (xy^-1)⁵ = t² = (xt)³ = (x^yt)³ = [t, y^-1xyx] = (xyt)⁶ = 1 >

Remark: This is the progenitor (2²)^*6:3.A₆ factored out by the relation [(oo, 0, 1, 2)(3, 4)t₀]⁶ = 1. We have that N = < x, y > = 3.A₆ of order 1 080 and index 560 (in G = J₂) is the control subgroup and that H = < y, t > = 3.PGL₂(9) has order 2 160 and index 280 (in G = J₂) and that K = < x, t^yt^yx, t^xyt^xyx > = U₃(3) has order 6 048 and index 100 in G = J₂.

J₂:2: Length 117, 6-generator, 22-relator.

< a, b, c, d, e, f | a² = b² = c² = d² = e² = f² = 1 = (ab)³ = [a, c] = [a, d] = [a, e] = [a, f] = (bc)³ = [b, d] = [b, e] = [b, f] = a(cd)⁴ = [c, e] = [c, f] = (de)³ = [d, f] = (ef)³ = (bcde)⁸ >

Remark: LHS-presentation. We have that H = < a, b, c, d, e > = U₃(3):2 has index 100 in G = J₂:2.

J₂:2: Length 153, 2-generator, 7-relator.

< x, y | x² = y⁴ = (xy)¹⁴ = [x, y]⁴ = (xyxyxy²)⁷ = (xyxyxy^-1xy^-1)³ = (xyxy²xy²xy^-1)³ = 1 >

Last updated 16^th July, 1997

John N. Bray

Sporadic Hall-Janko group HJ=J2

J2: Length 73, 2-generator, 4-relator.

J2: Length 82, 3-generator, 9-relator.

J2:2: Length 117, 6-generator, 22-relator.

J2:2: Length 153, 2-generator, 7-relator.

Sporadic Hall-Janko group HJ=J₂

J₂: Length 73, 2-generator, 4-relator.

J₂: Length 82, 3-generator, 9-relator.

J₂:2: Length 117, 6-generator, 22-relator.

J₂:2: Length 153, 2-generator, 7-relator.