ATLAS: Janko group J_{1}
Order = 175560 = 2^{3}.3.5.7.11.19.
Mult = 1.
Out = 1.
The following information is available for J_{1}:
Standard generators of the Janko group J_{1} are a and b
where a has order 2, b has order 3, ab has order 7
and ababb has order 19.
Finding generators
To find standard generators for J_{1}:

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 J_{1}.
This algorithm is available in computer readable format:
finder for J_{1}.
Checking generators
To check that elements x and y of J_{1}
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 J_{1}.
A presentation for J_{1} in terms of its standard generators is given below.
< a, b  a^{2} = b^{3} = (ab)^{7} = (ab(abab^{1})^{3})^{5} = (ab(abab^{1})^{6}abab(ab^{1})^{2})^{2} = 1 >.
This presentation is available in Magma format as follows:
J1 on a and b.
The representations of J_{1} 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.

Permutations on 266 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1045 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1463 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1540 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1596 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 2926 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 4180 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 2.

Dimension 20 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 56 over GF(4)  a nonorthogonal representation:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 56 over GF(4)  the Frobenius automorph of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 56 over GF(4)  an orthogonal representation:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 56 over GF(4)  the Frobenius automorph of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 76 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 76 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 112 over GF(2)  reducible over GF(4) [rep 56a + 56b]:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 112 over GF(2)  reducible over GF(4) [rep 56c + 56d]:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 120 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 120 over GF(8)  the image of the above under *4:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 120 over GF(8)  the image of the first such under *2:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 360 over GF(2)  reducible over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 3.

Dimension 56 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 56 over GF(9)  the Frobenius automorph of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 76 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 76 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(9)  the Frobenius automorph of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 112 over GF(3)  reducible over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 120 over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 120 over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 120 over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(3)  reducible over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 360 over GF(3)  reducible over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

All faithful irreducibles in characteristic 5.

Dimension 56 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 76 over GF(5)  phi3 in the modular atlas:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 76 over GF(5)  phi4 in the modular atlas:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 120 over GF(125):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 120 over GF(125):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 120 over GF(125):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 360 over GF(5)  reducible over GF(125):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 7.

Dimension 31 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 56 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 56 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 75 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(7)  with rational character:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(49)  phi8 in the modular atlas:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(49)  phi9 in the modular atlas:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 89 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 112 over GF(7)  reducible over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 120 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133 over GF(49)  phi13 in the modular atlas:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133 over GF(49)  phi14 in the modular atlas:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(7)  reducible over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 266 over GF(7)  reducible over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 11.

Dimension 7 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 14 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 49 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 56 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 64 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 69 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(11)  with rational character:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(11)  phi10 in the modular atlas:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(11)  phi11 in the modular atlas:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 106 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 119 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 209 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 19.

Dimension 22 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 34 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 43 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 55 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 76 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 76 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133 over GF(19)  phi9 in the modular atlas:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133 over GF(19)  phi10 in the modular atlas:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133 over GF(19)  phi11 in the modular atlas:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 209 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
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.
The maximal subgroups of J_{1} are as follows. Some words provided by Peter Walsh.

L_{2}(11), with standard generators
(ab)^2bab, (ababb)^4b(ababb)^4.
To change the previous generators to standard generators of L_{2}(11), use the program
here.

2^{3}:7:3 = F_{168}, with generators
(ab)^3bab, (ababb)^5(ab)^1bab(ababb)^5.

2 × A_{5}, with generators
a, (ababb)^4(ab)^1(ab(ababb)^2)^3ab(ababb)^4.

19:6 = F_{114}, with generators
(ab)^2bab, (abb)^3b(abb)^3.

11:10 = F_{110}, with generators
(ab)^2bab, (abb)^2(abababbababb)^3(abb)^2.

D_{6} × D_{10}, with generators
a, (ababb)^2(ab)^2(a(abab(ababb)^3)^1babab(ababb)^3)abab(ababb)^2.

7:6 = F_{42}, with generators
(ab)^3b(ab)^2, (abb)^2b(abb)^2.
Representatives of the 15 conjugacy classes J_{1} are given below.
 1A: identity [or a^{2}].
 2A: a.^{ }
 3A: b.^{ }
 5A: [a, bab]^{3} or ab(abab^{2})^{3}.
 5B: [a, bab]^{6}.
 6A: ab(abab^{2})^{3}ab^{2}.
 7A: ab.^{ }
 10A: (ababab^{2})^{2}ab^{2}.
 10B: ababab^{2}abab^{2}ab^{2} or [a, babab^{2}].
 11A: ab(abab^{2})^{4}.
 15A: [a, bab]^{2}.
 15B: ababab^{2}ab^{2} or [a, bab].
 19A: abab^{2} 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.
Go to sporadic groups page.
Go to old J1 page  ATLAS version 1.
Anonymous ftp access is also available on
for.mat.bham.ac.uk, user atlasftp, password atlasftp.
Files can be found in directory v2.0 and subdirectories.
Version 2.0 created on 24th May 1999.
Last updated 7.1.05 by SJN.
Information checked to
Level 1 on 04.12.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.