ATLAS: Janko group J_{4}
Order = 86775571046077562880 = 2^{21}.3^{3}.5.7.11^{3}.23.29.31.37.43.
Mult = 1.
Out = 1.
The following information is available for J_{4}:
Type I standard generators of the Janko group J_{4} are a
and b where a is in class 2A, b is in class 4A,
ab has order 37 and ababb has order 10.
Type II standard generators of the Janko group J_{4} are x,
y and t where x has order 2 (necessarily class 2B),
y has order 3, t has order 2 (necessarily class 2A),
(x, y) is a pair of [Type I] standard generators of
M_{24}, [t, x] has order 1 and
[t, yxyxy^{2}xy^{2}xyxyxy] has order 1.
(Alternatively, x, y and t are nontrivial elements of
J_{4} satisfying the presentation given below.)
We convert from Type I to Type II standard generators by applying
this program, which also has a
Magma version.
Finding generators
To find Type I standard generators for J_{4}:

Find any element of order 20, 40 or 44. It powers up to a 2Aelement
x and a 4Aelement y.
[The probability of success at each attempt is 15 in 176 (about 1 in 12).]

Find a conjugate a of x and a conjugate b of y
such that ab has order 37 and ababb has order 10.
[The probability of success at each attempt is 491520 in 361868177 (about 1 in 736).]

Now a and b are standard generators of J_{4}.
This algorithm is available in computer readable format:
finder for J_{4}.
To find Type II standard generators for J_{4}, first find Type I
standard generators, and then apply
the given word program.
Checking generators
To check that elements x and y of J_{4}
are Type I standard generators:
 Check o(x) = 2
 Check o(y) = 4
 Check o(xy) = 37
 Check o(xyxyy) = 10
 Let z = xyxyxyy
 Check o(z) = 24
 Check o(x(z^{12})^{xyyyxyyy}) = 11
 Let r = xyxy^{3}xyxy
 Let s = xy^{2}xy^{3}xy^{2}xy
 Let t = r(yy(yy)^{r})^{5}
 Let u = s(yy(yy)^{s})
 Let v = (tut)^{3}(ut)^{4}u
 Check o(v) = 20
 Check o([v,y]) = 1
This algorithm is available in computer readable format:
checker for J_{4}.
A presentation of J_{4} on its Type II standard generators is
given below:
< x, y, t  x^{2} =
y^{3} = (xy)^{23} =
[x, y]^{12} = [x, yxy]^{5} =
(xyxyxy^{1})^{3}(xyxy^{1}xy^{1})^{3} =
(xy(xyxy^{1})^{3})^{4} =
t^{2} = [t, x] =
[t, yxy(xy^{1})^{2}(xy)^{3}] =
(yt^{yxy1xyxy1x})^{3} =
((yxyxyxy)^{3}tt^{(xy)3y(xy)6y})^{2}
= 1 >.
This presentation is available in Magma format as follows:
J_{4} on x, y and t.
The representations of J_{4} available are:

Dimension 112 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 173067389 images of the vector:
v1 (Meataxe),
v1 (Meataxe binary),
v1 (GAP).

Permutations on 8474719242 images of the vector:
v3 (Meataxe),
v3 (Meataxe binary),
v3 (GAP).

The 112dimensional representation and the two vectors are also available as
a, b, v1 and v3
(Magma).

Dimension 1220 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 1333 over GF(11)  kindly provided by Wolfgang Lempken:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
The maximal subgroups of J_{4} are:

2^{11}:M_{24}, with (nonstandard) generators
a^{b},
(abbb)^{1}(abababb)^{8}abbb.

2^{1+12}.3.M_{22}:2, with (nonstandard) generators
(ab)^{9}(ababb)^{5}(ab)^{9},
(abbb)^{16}(abababb)^{8}(abbb)^{21}.

2^{10}:L_{5}(2), with (nonstandard) generators
(abbb)^{4}a(abbb)^{4},
(ab)^{8}b(ab)^{8}.

2^{3+12}.(S_{5} ×
L_{3}(2)), with generators
(ba)^{2}(ababab^{1})^{6}(ba)^{2},
(ab)^{13}(ba)^{22}(ababab^{1})^{3}(ba)^{22}(ab)^{13}.

U_{3}(11):2, with (nonstandard) generators
(ab)^{5}(ababb)^{5}(ab)^{5},
(abbb)^{4}(abababb)^{8}(abbb)^{4}.

M_{22}:2, with standard generators
a^{b}, ((abababb)^{6})^((ab)^{3}(abababbabb)^{42}).

11^{1+2}:(5 × 2S_{4}), with generators
here.

L_{2}(32):5, with standard generators
(babb)^{9}(abababbb)^{6}(babb)^{9},
(abababbabb)^{17}(abbabababb)^{27}(ababb)^{3}(abbabababb)^{27}(abababbabb)^{17}.

L_{2}(23):2, with (nonstandard) generators
(ab)^{5}(ababb)^{5}(ab)^{5},
(abbb)^{17}(abababb)^{8}(abbb)^{20}.

U_{3}(3), with standard generators
(ab)^{18}(ba)^{8}b^{1}ab(ba)^{8}(ab)^{18},
(ba)^{24}(ab)^{16}(abababb)^{4}(ab)^{16}(ba)^{24}.

29:28 = F_{812}, with generators
(abababbb)^{6}, SomethingElse.

43:14 = F_{602}, with generators
((abababbb)^{6})^(abab(abababbabb)^{26}),
((ababbababbabbb)^{3})^((abababbabb)^{33}(ab)^{22}).

37:12 = F_{444}, with generators
here.
 1A: aa
 2A: a
 2B: (abababbb)^6
 3A: (abb)^4
 4A: b
 4B: (abababb)^6
 4C: (abababbb)^3
 5A: ababbababb
 6A: (abababbabb)^11
 6B: (abababb)^4
 6C: (abababbb)^2
 7A:
 7B:
 8A:
 8B: (abababb)^3
 8C:
 10A: ababbb
 10B: ababb
 11A: (abababbabb)^6
 11B: (abababababb)^2
 12A:
 12B: abababbabababb
 12C: abababbb
 14A:
 14B:
 14C:
 14D:
 15A:
 16A:
 20A: ababbababbb
 20B:
 21A:
 21B:
 22A: abababbababb
 22B: abababababb
 23A:
 24A: abababb
 24B:
 28A:
 28B:
 29A:
 30A:
 31A:
 31B:
 31C:
 33A:
 33B:
 35A:
 35B:
 37A: ab
 37B: abab
 37C: abababab
 40A:
 40B:
 42A:
 42B:
 43A:
 43B:
 43C:
 44A:
 66A: abababbabb
 66B:
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.
Problems of algebraic conjugacy are not dealt with.
Here we may add additional information which does not necessarily fit into
the above categories. It may not have been checked to the same standard as
other information.
A subgroup U_{3}(11) has standard generators
a,
(abababbabb)^{8}(abb)^{4}(abababbabb)^{8}.
A subgroup L_{5}(2) is
generated by
b^{1}ab,
(abbb)^{2}(abababb)^{8}(abbb)^{2}.
A subgroup L_{2}(32) is
generated by
(ab)^{9}(ababb)^{5}(ab)^{9},
(abbb)^{8}(abababb)^{8}(abbb)^{8}.
Check  Date  By whom  Remarks 
Links work (except representations)  27.02.01  JNB 
Except to L2(23) and L2(32)
— but they now (21/5/03) do work. 
Links to (meataxe) representations work and have right degree and field  24.01.01  RAW 
All info from v1 is included  24.01.01  RAW 
HTML page standard   
Word program syntax  24.01.01  RAW 
Word programs applied   
All necessary standard generators are defined  24.01.01  RAW 
All representations are in standard generators  
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old J4 page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 14th April 1999.
Last updated 17.05.06 by JNB.
Information checked to
Level 1 on 22.04.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.