# ATLAS: Hall-Janko group HJ = J2

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

### 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.

### Black box algorithms

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.
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.

### Representations

The representations of J2 available are
• Primitive permutation representations.
• a and b as permutations on 100 points.
• a and b as permutations on 280 points.
• a and b as permutations on 315 points.
• a and b as permutations on 525 points.
• a and b as permutations on 840 points.
• a and b as permutations on 1008 points.
• a and b as permutations on 1800 points.
• The faithful irreducibles in characteristic 2 (up to Frobenius automorphisms)
• a and b as 6 × 6 matrices over GF(4).
• a and b as 14 × 14 matrices over GF(4).
• a and b as 36 × 36 matrices over GF(2).
• a and b as 64 × 64 matrices over GF(4).
• a and b as 84 × 84 matrices over GF(2).
• a and b as 160 × 160 matrices over GF(2).
• The faithful irreducibles in characteristic 3 (up to Frobenius automorphisms)
• a and b as 13 × 13 matrices over GF(9).
• a and b as 21 × 21 matrices over GF(9).
• a and b as 36 × 36 matrices over GF(3).
• a and b as 57 × 57 matrices over GF(9).
• a and b as 63 × 63 matrices over GF(3).
• a and b as 90 × 90 matrices over GF(3).
• a and b as 133 × 133 matrices over GF(3).
• a and b as 189 × 189 matrices over GF(9).
• a and b as 225 × 225 matrices over GF(3).
• All faithful irreducibles in characteristic 5.
• a and b as 14 × 14 matrices over GF(5).
• a and b as 21 × 21 matrices over GF(5).
• a and b as 41 × 41 matrices over GF(5).
• a and b as 70 × 70 matrices over GF(5).
• a and b as 85 × 85 matrices over GF(5).
• a and b as 90 × 90 matrices over GF(5).
• a and b as 175 × 175 matrices over GF(5).
• a and b as 189 × 189 matrices over GF(5).
• a and b as 225 × 225 matrices over GF(5).
• a and b as 300 × 300 matrices over GF(5).
• All faithful irreducibles in characteristic 7 (up to Frobenius automorphisms).
• a and b as 14 × 14 matrices over GF(49).
• a and b as 21 × 21 matrices over GF(49).
• a and b as 36 × 36 matrices over GF(7).
• a and b as 63 × 63 matrices over GF(7).
• a and b as 89 × 89 matrices over GF(7).
• a and b as 101 × 101 matrices over GF(7).
• a and b as 124 × 124 matrices over GF(7).
• a and b as 126 × 126 matrices over GF(7).
• a and b as 175 ×175 matrices over GF(7).
• a and b as 189 × 189 matrices over GF(49).
• a and b as 199 × 199 matrices over GF(7).
• a and b as 224 × 224 matrices over GF(49).
• a and b as 336 ×336 matrices over GF(7).
• a and b as 36 × 36 matrices over Z.
The representations of 2.J2 available are
• A and B as permutations on 200 points.
• A and B as permutations on 1120 points.
• The faithful irreducibles in characteristic 3 (up to Frobenius automorphisms)
• A and B as 6 × 6 matrices over GF(9).
• A and B as 14 × 14 matrices over GF(3).
• A and B as 36 × 36 matrices over GF(9).
• A and B as 50 × 50 matrices over GF(9).
• A and B as 126 × 126 matrices over GF(9).
• A and B as 216 × 216 matrices over GF(3).
• A and B as 236 × 236 matrices over GF(3).
• Faithful irreducibles in characteristic 5.
• A and B as 6 × 6 matrices over GF(5).
• A and B as 14 × 14 matrices over GF(5).
• A and B as 50 × 50 matrices over GF(5).
• A and B as 50 × 50 matrices over GF(5) - I may swap these two 50-dimensional representations at some point.
• A and B as 56 × 56 matrices over GF(5).
• A and B as 64 × 64 matrices over GF(5).
• Faithful irreducibles in characteristic 7.
• A and B as 6 × 6 matrices over GF(49).
• A and B as 14 × 14 matrices over GF(7).
• A and B as 50 × 50 matrices over GF(49).
• A and B as 58 × 58 matrices over GF(49).
The representations of J2:2 available are
• The faithful irreducibles in characteristic 2
• c and d as 12 × 12 matrices over GF(2).
• c and d as 28 × 28 matrices over GF(2).
• c and d as 36 × 36 matrices over GF(2).
• c and d as 84 × 84 matrices over GF(2).
• c and d as 128 × 128 matrices over GF(2).
• c and d as 160 × 160 matrices over GF(2).
• c and d as 26 × 26 matrices over GF(3).
• c and d as 14 × 14 matrices over GF(5).
• c and d as 28 × 28 matrices over GF(7).
• c and d as permutations on 100 points.
The representations of 2.J2.2 available are
• C and D as 12 × 12 matrices over GF(3).
• C and D as 6 × 6 matrices over GF(25).
• C and D as 12 × 12 matrices over GF(7).

### 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.A6.2, with generators here.
• 2^1+4.A5, with generators here.
• 2^2+4:(3 x S3), with generators here.
• A4 x A5, with generators here.
• A5 x D10, with generators here.
• L3(2):2, with generators here.
• 5^2: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 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.