# ATLAS: Hall–Janko group HJ = J2

Order = 604800 = 27.33.52.7.
Mult = 2.
Out = 2.

The following information is available for J2:

### Standard generators

Standard generators of the Janko group J2 are a and b where a is in class 2B, b is in class 3B, ab has order 7 and ababb has order 12.
Standard generators of the double cover 2.J2 are preimages A and B where B has order 3, and AB has order 7.

Standard generators of the automorphism group J2:2 are c and d where c is in class 2C, d is in class 5AB and cd has order 14.
Standard generators of either group 2.J2.2 are preimages C and D where D has order 5.

A pair of generators conjugate to A, B can be obtained as
A' = (CDCDCDD)18, B' = (CDD)−3(CDCDCDD)16(CDD)3

### Automorphisms

An outer automorphism of J2 maps (a, b) to (a, bb) = (a, b−1); an outer automorphism of 2.J2 maps (A, B) to (A−1, BB) = (A−1, B−1). This automorphism resides in class 2C.

### Black box algorithms

#### Finding generators

To find standard generators for J2:

• Find any elements x of order 2 and y of order 3.
• Try to find a conjugate a of x and a conjugate b of y, whose product has order 7.
• If you fail, then y is in the wrong conjugacy class.
• If you succeed, but ababb has order 4, then x is in the wrong conjugacy class.
• Otherwise, x and y are in the right classes, so find a conjugate a of x and a conjugate b of y, such that ab has order 7 and ababb has order 12.
This algorithm is available in computer readable format: finder for J2.

To find standard generators for J2.2:

• Find any element of order 14. Its seventh power is a 2C-element, x say.
• Find any element of order 15. Its cube is a 5AB-element, y say.
• Find a conjugate c of x and a conjugate d of y, whose product has order 14.
This algorithm is available in computer readable format: finder for J2.2.

#### Checking generators

To check that elements x and y of J2 are standard generators:

• Check o(x) = 2.
• Check o(y) = 3.
• Check o(xy) = 7.
• Check o(xyxyy) = 12.
This algorithm is available in computer readable format: checker for J2.

To check that elements x and y of J2.2 are standard generators:

• Check o(x) = 2.
• Check o(y) = 5.
• Check o(xy) = 14.
• Check o(xyy) = 24.
This algorithm is available in computer readable format: checker for J2.2.

### Presentations

Presentations of J2 and J2:2 in terms of their standard generators are given below. [The second J2 presentation is shorter, and the former is better for coset enumeration.]

< a, b | a2 = b3 = (ab)7 = [a, b]12 = (ababab−1abab−1ab−1ababab−1ab−1abab−1)3 = 1 >.

< a, b | a2 = b3 = (ab)7 = [a, b]12 = (ababab−1abab−1)6 = 1 >.

< c, d | c2 = d5 = (cd)14 = [c, d]7 = (cdcdcd−2cd−2)3 = [c, dcd]3 = (cdcdcd2)3cd−1cdcdcd−1cd2 = 1 >.

The relation [c, dcd]3 = 1 is redundant. These presentations, and those of the covering groups, are available in Magma format as follows:
J2 on a and b [v1], J2 on a and b [v2], 2.J2 on A and B [v1], 2.J2 on A and B [v2], J2:2 on c and d, 2.J2.2 (+) on C and D and 2.J2:2 (−) on C and D.

### Representations

The representations of J2 available are:
• Primitive permutation representations.
• The faithful irreducibles in characteristic 2 (up to Frobenius automorphisms).
• The faithful irreducibles in characteristic 3 (up to Frobenius automorphisms).
• All faithful irreducibles in characteristic 5.
• All faithful irreducibles in characteristic 7 (up to Frobenius automorphisms).
• a and b as 36 × 36 matrices over Z - not there, see version 1.
The representations of 2.J2 available are:
• Permutations on 200 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Permutations on 1120 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• The faithful irreducibles in characteristic 3 (up to Frobenius automorphisms).
• Faithful irreducibles in characteristic 5.
• Faithful irreducibles in characteristic 7.
The representations of J2:2 available are:
• Permutations on 100 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• The faithful irreducibles in characteristic 2.
• The faithful irreducibles in characteristic 3 (up to tensoring with the sign character):
• The faithful irreducibles in characteristic 5 (up to tensoring with the sign character):
• The faithful irreducibles in characteristic 7 (up to tensoring with the sign character):
The representations of 2.J2.2 available are:
• The faithful irreducibles in characteristic 3 (up to tensoring with the sign character):
• The faithful irreducibles in characteristic 5 (up to tensoring with the sign character):
• The faithful irreducibles in characteristic 7 (up to tensoring with the sign character):

### Maximal subgroups

The maximal subgroups of J2 are as follows. Words provided by Peter Walsh, implemented and checked by Ibrahim Suleiman.
• U3(3), with generators here.
• 3.A6.22 = 3.PGL2(9), with generators here.
• 21+4.A5, with generators here.
• 22+4:(3 × S3), with generators here.
• A4 × A5, with generators here.
• A5 × D10, with generators here.
• L3(2):2, with generators here.
• 52:D12, with generators here.
• A5, with generators here.
The maximal subgroups of J2:2 are:

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of J2 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

A set of generators for the maximal cyclic subgroups of J2:2 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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