ATLAS: Janko group J_{3}
Order = 50232960 = 2^{7}.3^{5}.5.17.19.
Mult = 3.
Out = 2.
The following information is available for J_{3}:
Standard generators of the Janko group J_{3} 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 3.J_{3} are preimages A
and B where A has order 2 and B is in class +3A. The
condition that B is in class +3A is equivalent to the condition that
ABABABB has order 17.
Standard generators of the automorphism group J_{3}: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 3.J_{3}: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.
Finding generators
To find standard generators for J_{3}:
 Find any element x of order 2 [as a suitable power of any element
of even order].
 Find any element of order 6, 12 or 15. This powers up to a 3Aelement,
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.
This algorithm is available in computer readable format:
finder for J_{3}.
To find standard generators for J_{3}.2:
 Find any element of order 18 or 34. It powers up to a 2Belement,
x say.
 Find any element of order 15 or 24. It powers up to a 3Aelement,
y say.
 Find a conjugate c of x and a conjugate d of
y, whose product has order 24, and commutator has order 9.
This algorithm is available in computer readable format:
finder for J_{3}.2.
Checking generators
To check that elements x and y of J_{3}
are standard generators:
 Check o(x) = 2
 Check o(y) = 3
 Check o(xy) = 19
 Check o(xyxyxyy) = 17
This algorithm is available in computer readable format:
checker for J_{3}.
To check that elements x and y of J_{3}.2
are standard generators:
 Check o(x) = 2
 Check o(y) = 3
 Check o(xy) = 24
 Check o(xyxyxyxyy) = 9
This algorithm is available in computer readable format:
checker for J_{3}.2.
Presentations of J_{3} and J_{3}:2 in terms of their standard
generators are given below.
< a, b  a^{2} = b^{3} =
(ab)^{19} = [a, b]^{9} =
((ab)^{6}(ab^{1})^{5})^{2} =
((ababab^{1})^{2}abab^{1}ab^{1}abab^{1})^{2} =
abab(abab^{1})^{3}abab(abab^{1})^{4}ab^{1}(abab^{1})^{3}
= (ababababab^{1}abab^{1})^{4} = 1 >.
< c, d  c^{2} = d^{3} =
(cd)^{24} = [c, d]^{9} =
(cd(cdcd^{1})^{2})^{4} =
(cdcdcd^{1}(cdcdcd^{1}cd^{1})^{2})^{2} =
[c, (dc)^{4}(d^{1}c)^{2}d]^{2} =
[c, d(cd^{1})^{2}(cd)^{4}]^{2} = 1 >.
These presentations are available in Magma format as follows:
J3 on a and b,
3.J3 on A and B [v1],
3.J3 on A and B [v2] and
J3:2 on c and d.
The representations of J_{3} available are:
 All primitive permutation representations.

Permutations on 6156 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 14688 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 14688 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 17442 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 20520 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 23256 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 25840 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 26163 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 43605 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some representations in characteristic 2.

Dimension 78 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 80 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 84 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 244 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 322 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 966 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 248 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
These matrices exhibit the inclusion in
E8(2), written with respect to a Chevalley basis.
 Some representations in characteristic 3.

Dimension 18 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 84 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 153 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 324 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 934 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some representations in characteristic 5.

Dimension 85 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 323 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 646 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 816 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some representations in characteristic 17.

Dimension 85 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 324 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 379 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 646 over GF(17)  reducible over GF(289):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 761 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 816 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 836 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 1292 over GF(17)  reducible over GF(289):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some representations in characteristic 19.

Dimension 85 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 110 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 214 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 323 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 646 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 706 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 919 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 1001 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 3.J_{3} available are:

Permutations on 18468 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 Some representations in characteristic 2.

Dimension 9 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 18 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 18 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 126 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 153 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 153 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 324 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 720 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 18 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 36 over GF(5)  reducible over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 36 over GF(17)  reducible over GF(289):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 36 over GF(17)  reducible over GF(289):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
I don't know which is which of the above two representations.

Dimension 342 over GF(17)  reducible over GF(289):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 648 over GF(17)  reducible over GF(289):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 18 over GF(19):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 18 over GF(19):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of J_{3}:2 available are:

Permutations on 6156 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some representations in characteristic 2:

Dimension 80 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 156 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 168 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 244 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 644 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 966 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some representations in charactersitc 3:

Dimension 36 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 168 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 306 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 324 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 934 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some representations in characteristic 5:

Dimension 170 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 323 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 646 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 816 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some representations in characteristic 17:

Dimension 170 over GF(17):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 324 over GF(17):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 379 over GF(17):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 646 over GF(17):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 761 over GF(17):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 836 over GF(17):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some representations in characteristic 19:

Dimension 85 over GF(19):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 110 over GF(19):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 214 over GF(19):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 214 over GF(19):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 646 over GF(19):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 706 over GF(19):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 919 over GF(19):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 1001 over GF(19):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 1214 over GF(19):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 3.J_{3}:2 available are:

Dimension 18 over GF(2):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The maximal subgroups of J_{3} are as follows. Words are given by
Suleiman and Wilson in Experimental Math. 4 (1995), 1118.

L_{2}(16):2,
with generators
a, (ababab^{2}abab^{2})^{6}.

L_{2}(19),
with generators
b^{2}ab, ((abab^{2})^{3})^((ab^{2})^{4}).

L_{2}(19),
with generators
bab^{2}, ((ab^{2}ab)^{3})^((ab)^{4}).

2^{4}:(3 × A_{5}),
with generators
here.

L_{2}(17), with generators
here.

(3 × A_{6}):2_{2},
with generators
here.

3^{2+1+2}:8, with generators
here.

2^{1+4}:A_{5}, with generators
here.

2^{2+4}:(3 × S_{3}), with generators
here.
The maximal subgroups of J_{3}:2 are as follows. Words are given by
Suleiman and Wilson in Experimental Math. 4 (1995), 1118.
(Different generators given below.)
 J_{3}, with standard generators
(cd)^{12}, d^{cdcdd},
and generators
d, cdc [program gives (cdcd, d)].

L_{2}(16):4,
with generators
(cd)^{6}cd^{2}cdcd^{2}cd^{2}(cd)^{3}, d,
and here.

2^{4}:(3 × A_{5}).2,
with generators
(cdcdcd^{2})^{2}(cd^{2}cd)^{2}c ,d,
and here.

L_{2}(17) × 2,
with generators
c,
cd^{2}(cd)^{5}(cd^{2})^{4},
and here (mapping onto standard generators for L_{2}(17)),
and here.

(3 × M_{10}):2,
with generators
c,
cd(cd(cd^{2})^{3})^{2}cd^{2}cdcd,
and here.

3^{2+1+2}:8.2,
with generators
c,
cdcd^{2}cdcd(cd^{2})^{4}cd,
and here._{ }

2^{1+4}:S_{5},
with generators
c,
(cd)^{5}cd^{2}cdcd^{2}(cd)^{4},
and here.

2^{2+4}:(S_{3} × S_{3}),
with generators
c,
cdcd^{2}(cd)^{6}(cd^{2})^{2}cd,
and here.

19:18 = F_{342},
with generators
c,
(cdcd^{2})^{2}(cd)^{6}(cd^{2})^{2},
and here.
A set of generators for the maximal cyclic subgroups of J3 can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
The canonical central element of order 3 in 3.J_{3} is taken to be
(AB)^{19}.
A set of generators for the maximal cyclic subgroups of J3.2 can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old J3 page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 14th June 2000.
Last updated 7.1.05 by SJN.
Information checked to
Level 0 on 14.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.