# ATLAS: Janko group J1

Order = 175560 = 23.3.5.7.11.19.
Mult = 1.
Out = 1.

The following information is available for J1:

### Standard generators

Standard generators of the Janko group J1 are a and b where a has order 2, b has order 3, ab has order 7 and ababb has order 19.

### Black box algorithms

To find standard generators for J1:
• Find an element order 2, 6 or 10. This powers up to x of order 2.
[The probability of success at each attempt is 3 in 8 (about 1 in 3).]
• Find an element order 3, 6 or 15. This powers up to y of order 3.
[The probability of success at each attempt is 1 in 3.]
• Find a conjugate a of x and a conjugate b of y such that ab has order 7 and ababb has order 19.
[The probability of success at each attempt is 30 in 1463 (about 1 in 49).]
Alternatively, we can use:
• Find an element order 6. This cubes to x of order 2 and squares to y of order 3.
[The probability of success at each attempt is 1 in 6.]
• Find a conjugate a of x and a conjugate b of y such that ab has order 7 and ababb has order 19.
[The probability of success at each attempt is 30 in 1463 (about 1 in 49).]

### Presentation

A presentation for J1 in terms of its standard generators is given below.

< a, b | a2 = b3 = (ab)7 = (ab(abab-1)3)5 = (ab(abab-1)6abab(ab-1)2)2 = 1 >.

### Representations

The representations of J1 available are as follows. Choose from
permutation representations and matrix representations in characteristic 2, 3, 5, 7, 11 or 19. They should be in the Atlas order, defined by the conjugacy class representatives: ababb in 19A, and either abababbababb in 15B, or ababababbababbabb in 10B (these last two statements are equivalent).
• All primitive permutation representations.
• a and b as permutations on 266 points.
• a and b as permutations on 1045 points.
• a and b as permutations on 1463 points.
• a and b as permutations on 1540 points.
• a and b as permutations on 1596 points.
• a and b as permutations on 2926 points.
• a and b as permutations on 4180 points.
• All faithful irreducibles in characteristic 2.
• a and b as 20 × 20 matrices over GF(2).
• a and b as 56 × 56 matrices over GF(4) - a symplectic representation.
• a and b as 56 × 56 matrices over GF(4) - the Frobenius automorph of the above.
• a and b as 56 × 56 matrices over GF(4) - an orthogonal representation.
• a and b as 56 × 56 matrices over GF(4) - the Frobenius automorph of the above.
• a and b as 76 × 76 matrices over GF(2).
• a and b as 76 × 76 matrices over GF(2).
• a and b as 120 × 120 matrices over GF(8).
• a and b as 120 × 120 matrices over GF(8) - the image of the above under *4.
• a and b as 120 × 120 matrices over GF(8) - the image of the first such representation under *2.
• All faithful irreducibles in characteristic 3.
• a and b as 56 × 56 matrices over GF(9).
• a and b as 56 × 56 matrices over GF(9) - the Frobenius automorph of the above.
• a and b as 76 × 76 matrices over GF(3).
• a and b as 76 × 76 matrices over GF(3).
• a and b as 77 × 77 matrices over GF(9).
• a and b as 77 × 77 matrices over GF(9) - the Frobenius automorph of the above.
• a and b as 120 × 120 matrices over GF(27).
• a and b as 120 × 120 matrices over GF(27).
• a and b as 120 × 120 matrices over GF(27).
• a and b as 133 × 133 matrices over GF(3).
• All faithful irreducibles in characteristic 5.
• a and b as 56 × 56 matrices over GF(5).
• a and b as 76 × 76 matrices over GF(5) - phi3 in modular atlas.
• a and b as 76 × 76 matrices over GF(5) - phi4 in modular atlas.
• a and b as 77 × 77 matrices over GF(5).
• a and b as 120 × 120 matrices over GF(125).
• a and b as 120 × 120 matrices over GF(125).
• a and b as 120 × 120 matrices over GF(125).
• a and b as 133 × 133 matrices over GF(5).
• All faithful irreducibles in characteristic 7.
• a and b as 31 × 31 matrices over GF(7).
• a and b as 45 × 45 matrices over GF(7).
• a and b as 56 × 56 matrices over GF(49).
• a and b as 56 × 56 matrices over GF(49).
• a and b as 75 × 75 matrices over GF(7).
• a and b as 77 × 77 matrices over GF(7) - with rational character.
• a and b as 77 × 77 matrices over GF(49) - phi8 in the modular atlas.
• a and b as 77 × 77 matrices over GF(49) - phi9 in the modular atlas.
• a and b as 89 × 89 matrices over GF(7).
• a and b as 120 × 120 matrices over GF(7).
• a and b as 133 × 133 matrices over GF(7).
• a and b as 133 × 133 matrices over GF(49) - phi13 in the modular atlas.
• a and b as 133 × 133 matrices over GF(49) - phi14 in the modular atlas.
• All faithful irreducibles in characteristic 11.
• a and b as 7 × 7 matrices over GF(11).
• a and b as 14 × 14 matrices over GF(11).
• a and b as 27 × 27 matrices over GF(11).
• a and b as 49 × 49 matrices over GF(11).
• a and b as 56 × 56 matrices over GF(11).
• a and b as 64 × 64 matrices over GF(11).
• a and b as 69 × 69 matrices over GF(11).
• a and b as 77 × 77 matrices over GF(11) - the rational representation.
• a and b as 77 × 77 matrices over GF(11) - phi10 in the modular atlas.
• a and b as 77 × 77 matrices over GF(11) - phi11 in the Modular Atlas.
• a and b as 106 × 106 matrices over GF(11).
• a and b as 119 × 119 matrices over GF(11).
• a and b as 209 × 209 matrices over GF(11).
• All faithful irreducibles in characteristic 19.
• a and b as 22 × 22 matrices over GF(19).
• a and b as 34 × 34 matrices over GF(19).
• a and b as 43 × 43 matrices over GF(19).
• a and b as 55 × 55 matrices over GF(19).
• a and b as 76 × 76 matrices over GF(19).
• a and b as 76 × 76 matrices over GF(19).
• a and b as 77 × 77 matrices over GF(19).
• a and b as 133 × 133 matrices over GF(19).
• a and b as 133 × 133 matrices over GF(19) - phi9 in the modular atlas.
• a and b as 133 × 133 matrices over GF(19) - phi10 in the modular atlas.
• a and b as 209 × 209 matrices over GF(19) - phi11 in the modular atlas.
Source: Janko's original 7 × 7 matrices over GF(11), from which most of the given representations can be derived with the Meat-axe. Some of this work has been done by Peter Walsh in his Ph.D. thesis (Birmingham, 1996), with details given in his M.Phil. thesis (Birmingham, 1994). The matrix representations were mostly made by uncondensing them out of condensed permutation representations.

### Maximal subgroups

The maximal subgroups of J1 are as follows. Some words provided by Peter Walsh.

### 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. Notation for algebraically conjugate elements is consistent with the Atlas of Brauer Characters.