# ATLAS: Janko group J4

Order = 86775571046077562880 = 221.33.5.7.113.23.29.31.37.43.
Mult = 1.
Out = 1.

The following information is available for J4:

### Standard generators

Type I standard generators of the Janko group J4 are a and b where a is in class 2A, b is in class 4A, ab has order 37 and ababb has order 10.

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.
This algorithm is available in computer readable format: finder for 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
This algorithm is available in computer readable format: checker for J4.

### Presentation

A presentation of J4 on its Type II standard generators is given below:

< 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] = (ytyxy-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

The representations of J4 available are:
• 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

The maximal subgroups of J4 are:

### 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.

Here we may add additional information which does not necessarily fit into the above categories. It may not have been checked to the same standard as other 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

CheckDateBy whomRemarks
Links work (except representations)27.02.01JNB Except to L2(23) and L2(32)
— but they now (21/5/03) do work.
Links to (meataxe) representations work and have right degree and field24.01.01RAW
All info from v1 is included24.01.01RAW
HTML page standard
Word program syntax24.01.01RAW
Word programs applied
All necessary standard generators are defined24.01.01RAW
All representations are in standard generators Go to main ATLAS (version 2.0) page. Go to sporadic groups page. Go to old J4 page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 14th April 1999.
Last updated 17.05.06 by JNB.
Information checked to Level 1 on 22.04.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.