ATLAS: Janko group J₄

The following information is available for J₄:

• Standard generators.
• Black box algorithms.
• Presentation.
• Representations.
• Maximal subgroups.
• Conjugacy classes.
• Additional information.
• Checks applied.

Standard generators

Type II standard generators of the Janko group J₄ are x, y and t where x has order 2 (necessarily class 2B), y has order 3, t has order 2 (necessarily class 2A), (x, y) is a pair of [Type I] standard generators of M₂₄, [t, x] has order 1 and [t, yxyxy²xy²xyxyxy] has order 1.
(Alternatively, x, y and t are non-trivial elements of J₄ satisfying the presentation given below.)

We convert from Type I to Type II standard generators by applying this program, which also has a Magma version.

Black box algorithms

Finding generators

To find Type I standard generators for J₄:

• Find any element of order 20, 40 or 44. It powers up to a 2A-element x and a 4A-element y.
  [The probability of success at each attempt is 15 in 176 (about 1 in 12).]
• Find a conjugate a of x and a conjugate b of y such that ab has order 37 and ababb has order 10.
  [The probability of success at each attempt is 491520 in 361868177 (about 1 in 736).]
• Now a and b are standard generators of J₄.

To find Type II standard generators for J₄, first find Type I standard generators, and then apply the given word program.

Checking generators

To check that elements x and y of J₄ are Type I standard generators:

• Check o(x) = 2
• Check o(y) = 4
• Check o(xy) = 37
• Check o(xyxyy) = 10
• Let z = xyxyxyy
• Check o(z) = 24
• Check o(x(z¹²)^xyyyxyyy) = 11
• Let r = xyxy³xyxy
• Let s = xy²xy³xy²xy
• Let t = r(yy(yy)^r)⁵
• Let u = s(yy(yy)^s)
• Let v = (tut)³(ut)⁴u
• Check o(v) = 20
• Check o([v,y]) = 1

Presentation

< x, y, t | x² = y³ = (xy)²³ = [x, y]¹² = [x, yxy]⁵ = (xyxyxy^-1)³(xyxy^-1xy^-1)³ = (xy(xyxy^-1)³)⁴ = t² = [t, x] = [t, yxy(xy^-1)²(xy)³] = (yt^yxy-1xyxy-1x)³ = ((yxyxyxy)³tt^(xy)3y(xy)6y)² = 1 >.

This presentation is available in Magma format as follows: J₄ on x, y and t.

Representations

• Dimension 112 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 173067389 images of the vector: v1 (Meataxe), v1 (Meataxe binary), v1 (GAP).
• Permutations on 8474719242 images of the vector: v3 (Meataxe), v3 (Meataxe binary), v3 (GAP).
• The 112­dimensional representation and the two vectors are also available as a, b, v1 and v3 (Magma).
• Dimension 1220 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
• Dimension 1333 over GF(11) - kindly provided by Wolfgang Lempken: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).

Maximal subgroups

• 2¹¹:M₂₄, with (non­standard) generators a^b, (abbb)^-1(abababb)⁸abbb.
• 2¹⁺¹².3.M₂₂:2, with (non­standard) generators (ab)^-9(ababb)⁵(ab)⁹, (abbb)¹⁶(abababb)⁸(abbb)²¹.
• 2¹⁰:L₅(2), with (non­standard) generators (abbb)^-4a(abbb)⁴, (ab)^-8b(ab)⁸.
• 2³⁺¹².(S₅ × L₃(2)), with generators (ba)^-2(ababab^-1)⁶(ba)², (ab)^-13(ba)^-22(ababab^-1)³(ba)²²(ab)¹³.
• U₃(11):2, with (non­standard) generators (ab)^-5(ababb)⁵(ab)⁵, (abbb)^-4(abababb)⁸(abbb)⁴.
• M₂₂:2, with standard generators a^b, ((abababb)⁶)^((ab)³(abababbabb)⁴²).
• 11¹⁺²:(5 × 2S₄), with generators here.
• L₂(32):5, with standard generators (babb)^-9(abababbb)⁶(babb)⁹, (abababbabb)¹⁷(abbabababb)^-27(ababb)³(abbabababb)²⁷(abababbabb)^-17.
• L₂(23):2, with (non­standard) generators (ab)^-5(ababb)⁵(ab)⁵, (abbb)¹⁷(abababb)⁸(abbb)²⁰.
• U₃(3), with standard generators (ab)^-18(ba)^-8b^-1ab(ba)⁸(ab)¹⁸, (ba)^-24(ab)¹⁶(abababb)⁴(ab)^-16(ba)²⁴.
• 29:28 = F₈₁₂, with generators (abababbb)⁶, SomethingElse.
• 43:14 = F₆₀₂, with generators ((abababbb)⁶)^(abab(abababbabb)²⁶), ((ababbababbabbb)³)^((abababbabb)³³(ab)²²).
• 37:12 = F₄₄₄, with generators here.

Conjugacy class representatives

• 1A: aa
• 2A: a
• 2B: (abababbb)^6
• 3A: (abb)^4
• 4A: b
• 4B: (abababb)^6
• 4C: (abababbb)^3
• 5A: ababbababb
• 6A: (abababbabb)^11
• 6B: (abababb)^4
• 6C: (abababbb)^2
• 7A:
• 7B:
• 8A:
• 8B: (abababb)^3
• 8C:
• 10A: ababbb
• 10B: ababb
• 11A: (abababbabb)^6
• 11B: (abababababb)^2
• 12A:
• 12B: abababbabababb
• 12C: abababbb
• 14A:
• 14B:
• 14C:
• 14D:
• 15A:
• 16A:
• 20A: ababbababbb
• 20B:
• 21A:
• 21B:
• 22A: abababbababb
• 22B: abababababb
• 23A:
• 24A: abababb
• 24B:
• 28A:
• 28B:
• 29A:
• 30A:
• 31A:
• 31B:
• 31C:
• 33A:
• 33B:
• 35A:
• 35B:
• 37A: ab
• 37B: abab
• 37C: abababab
• 40A:
• 40B:
• 42A:
• 42B:
• 43A:
• 43B:
• 43C:
• 44A:
• 66A: abababbabb
• 66B:

A set of generators for the maximal cyclic subgroups can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Problems of algebraic conjugacy are not dealt with.

Additional information

A subgroup U₃(11) has standard generators a, (abababbabb)^-8(abb)⁴(abababbabb)⁸.
A subgroup L₅(2) is generated by b^-1ab, (abbb)^-2(abababb)⁸(abbb)².
A subgroup L₂(32) is generated by (ab)^-9(ababb)⁵(ab)⁹, (abbb)⁸(abababb)⁸(abbb)^-8.

Checks applied

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