# ATLAS: Janko group J3

Order = 50232960 = 27.35.5.17.19.
Mult = 3.
Out = 2.

The following information is available for J3:

### Standard generators

Standard generators of the Janko group J3 are a and b where a has order 2, b is in class 3A, ab has order 19 and ababb has order 9.
Standard generators of the triple cover 3.J3 are preimages A and B where A has order 2 and B is in class +3A. The condition that B is in class +3A is equivalent to the condition that ABABABB has order 17.

Standard generators of the automorphism group J3:2 are c and d where c is in class 2B, d is in class 3A, cd has order 24 and cdcdd has order 9.
Standard generators of 3.J3:2 are preimages C and D where D is in class +3A.

A pair of generators conjugate to a, b can be obtained as
a' = (cd)^{12}, b' = (cdcdd)^{-1}dcdcdd.

### Black box algorithms

#### Finding generators

To find standard generators for J3:

• Find any element x of order 2 [as a suitable power of any element of even order].
• Find any element of order 6, 12 or 15. This powers up to a 3A-element, y say.
• Find a conjugate a of x and a conjugate b of y, whose product has order 19, such that ababb has order 9.
This algorithm is available in computer readable format: finder for J3.

To find standard generators for J3.2:

• Find any element of order 18 or 34. It powers up to a 2B-element, x say.
• Find any element of order 15 or 24. It powers up to a 3A-element, y say.
• Find a conjugate c of x and a conjugate d of y, whose product has order 24, and commutator has order 9.
This algorithm is available in computer readable format: finder for J3.2.

#### Checking generators

To check that elements x and y of J3 are standard generators:

• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 19
• Check o(xyxyxyy) = 17
This algorithm is available in computer readable format: checker for J3.

To check that elements x and y of J3.2 are standard generators:

• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 24
• Check o(xyxyxyxyy) = 9
This algorithm is available in computer readable format: checker for J3.2.

### Presentations

Presentations of J3 and J3:2 in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)19 = [a, b]9 = ((ab)6(ab-1)5)2 = ((ababab-1)2abab-1ab-1abab-1)2 = abab(abab-1)3abab(abab-1)4ab-1(abab-1)3 = (ababababab-1abab-1)4 = 1 >.

< c, d | c2 = d3 = (cd)24 = [c, d]9 = (cd(cdcd-1)2)4 = (cdcdcd-1(cdcdcd-1cd-1)2)2 = [c, (dc)4(d-1c)2d]2 = [c, d(cd-1)2(cd)4]2 = 1 >.

These presentations are available in Magma format as follows:
J3 on a and b, 3.J3 on A and B [v1], 3.J3 on A and B [v2] and J3:2 on c and d.

### Representations

The representations of J3 available are:
• All primitive permutation representations.
• Permutations on 6156 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 14688 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 14688 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 17442 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 20520 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 23256 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 25840 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 26163 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 43605 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some representations in characteristic 2.
• Dimension 78 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 80 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 84 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 244 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 322 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 966 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 248 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
These matrices exhibit the inclusion in E8(2), written with respect to a Chevalley basis.
• Some representations in characteristic 3.
• Some representations in characteristic 5.
• Some representations in characteristic 17.
• Dimension 85 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 324 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 379 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 646 over GF(17) - reducible over GF(289): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 761 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 816 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 836 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 1292 over GF(17) - reducible over GF(289): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some representations in characteristic 19.
The representations of 3.J3 available are:
• Permutations on 18468 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Some representations in characteristic 2.
• Dimension 18 over GF(25): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 36 over GF(5) - reducible over GF(25): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 36 over GF(17) - reducible over GF(289): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 36 over GF(17) - reducible over GF(289): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
I don't know which is which of the above two representations.
• Dimension 342 over GF(17) - reducible over GF(289): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 648 over GF(17) - reducible over GF(289): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 18 over GF(19): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 18 over GF(19): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
The representations of J3:2 available are:
• Permutations on 6156 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Some representations in characteristic 2:
• Some representations in charactersitc 3:
• Some representations in characteristic 5:
• Some representations in characteristic 17:
• Some representations in characteristic 19:
The representations of 3.J3:2 available are:

### Maximal subgroups

The maximal subgroups of J3 are as follows. Words are given by Suleiman and Wilson in Experimental Math. 4 (1995), 11-18.
The maximal subgroups of J3:2 are as follows. Words are given by Suleiman and Wilson in Experimental Math. 4 (1995), 11-18. (Different generators given below.)

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of J3 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

The canonical central element of order 3 in 3.J3 is taken to be (AB)-19.

A set of generators for the maximal cyclic subgroups of J3.2 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

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