# ATLAS: Janko group J3

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

### 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 3J3 are pre-images A and B where A has order 2, and B is in class +3A.
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 3J3:2 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

To find standard generators for J3:
• Find any element x of order 2.
• 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.
To find standard generators for J3.2:
• Find any element of order 18 or 34. It powers up to a 2B-element.
• Find any element of order 15 or 24. It powers up to a 3A-element.
• Find a conjugate a of x and a conjugate b of y, whose product has order 24, and commutator has order 9.

### Representations

NB: As usual, the ordering of the representations [and their labellings] is NOT guaranteed to remain constant unless explicitly indicated to the contrary.

The representations of J3 available are

• Some representations in characteristic 2.
• a and b as 78 x 78 matrices over GF(4).
• a and b as 80 x 80 matrices over GF(2).
• a and b as 84 x 84 matrices over GF(4).
• a and b as 248 x 248 matrices over GF(2) which exhibit the inclusion in E8(2), written with respect to a Chevalley basis.
• Some representations in characteristic 3.
• a and b as 18 x 18 matrices over GF(9).
• a and b as 84 x 84 matrices over GF(9).
• a and b as 153 x 153 matrices over GF(9).
• a and b as 324 x 324 matrices over GF(3).
• a and b as 85 x 85 matrices over GF(5).
• a and b as 323 x 323 matrices over GF(5).
• a and b as 85 x 85 matrices over GF(17).
• a and b as 85 x 85 matrices over GF(19).
• a and b as 110 x 110 matrices over GF(19).
• a and b as 214 x 214 matrices over GF(19).
• All primitive permutation representations.
• a and b as permutations on 6156 points.
• a and b as permutations on 14688 points.
• a and b as permutations on 14688 points.
• a and b as permutations on 17442 points.
• a and b as permutations on 20520 points.
• a and b as permutations on 23256 points.
• a and b as permutations on 25840 points.
• a and b as permutations on 26163 points.
• a and b as permutations on 43605 points.
The representations of 3J3 available are
• A and B as permutations on 18468 points.
• A and B as 9 x 9 matrices over GF(4).
• A and B as 18 x 18 matrices over GF(4).
• A and B as 18 x 18 matrices over GF(25).
• A and B as 36 x 36 matrices over GF(5) - reducible over GF(25).
• A and B as 36 x 36 matrices over GF(17) - reducible over GF(289).
• A and B as 36 x 36 matrices over GF(17) - reducible over GF(289).
• A and B as 18 x 18 matrices over GF(19).
• A and B as 18 x 18 matrices over GF(19).
The representations of J3:2 available are
• c and d as 36 x 36 matrices over GF(3).
• c and d as 323 x 323 matrices over GF(5).
• c and d as 214 x 214 matrices over GF(19).
• c and d as permutations on 6156 points.
The representations of 3J3:2 available are
• C and D as 18 x 18 matrices over GF(2).

### 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.

### 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.