# 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

#### Finding generators

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).]
• Now a and b are standard generators of J1.
This algorithm is available in computer readable format: finder for J1.

#### Checking generators

To check that elements x and y of J1 are standard generators:
• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 7
• Check o(xyxyy) = 19
This algorithm is available in computer readable format: checker for J1.

### 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 >.

This presentation is available in Magma format as follows: J1 on a and b.

### 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).
Source: Janko's original 7 × 7 matrices over GF(11), from which most of the given representations can be derived with the Meataxe. 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

Representatives of the 15 conjugacy classes J1 are given below.
• 1A: identity [or a2].
• 2A: a.
• 3A: b.
• 5A: [a, bab]3 or ab(abab2)3.
• 5B: [a, bab]6.
• 6A: ab(abab2)3ab2.
• 7A: ab.
• 10A: (ababab2)2ab2.
• 10B: ababab2abab2ab2 or [a, babab2].
• 11A: ab(abab2)4.
• 15A: [a, bab]2.
• 15B: ababab2ab2 or [a, bab].
• 19A: abab2 or [a, b].
• 19B: [a, b]2.
• 19C: [a, b]4.
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.
Go to main ATLAS (version 2.0) page.