ATLAS: Hall–Janko group HJ = J_{2}
Order = 604800 = 2^{7}.3^{3}.5^{2}.7.
Mult = 2.
Out = 2.
The following information is available for J_{2}:
Standard generators of the Janko group J_{2} are a and b
where a is in class 2B, b is in class 3B, ab has order 7
and ababb has order 12.
Standard generators of the double cover 2.J_{2} are preimages A
and B where B has order 3, and AB has order 7.
Standard generators of the automorphism group J_{2}:2 are c
and d where c is in class 2C, d is in class 5AB and
cd has order 14.
Standard generators of either group 2.J_{2}.2 are preimages C
and D where D has order 5.
A pair of generators conjugate to A, B can be obtained as
A' = (CDCDCDD)^{18}, B' = (CDD)^{−3}(CDCDCDD)^{16}(CDD)^{3}.
An outer automorphism of J_{2} maps (a, b) to (a, bb) = (a, b^{−1});
an outer automorphism of 2.J_{2} maps (A, B) to
(A^{−1}, BB) = (A^{−1}, B^{−1}).
This automorphism resides in class 2C.
Finding generators
To find standard generators for J_{2}:
 Find any elements x of order 2 and y of order 3.
 Try to find a conjugate a of x and a conjugate b of y, whose product has order 7.
 If you fail, then y is in the wrong conjugacy class.
 If you succeed, but ababb has order 4, then x is in the wrong conjugacy class.
 Otherwise, x and y are in the right classes, so find a conjugate a of x
and a conjugate b of y, such that ab has order 7 and ababb has order 12.
This algorithm is available in computer readable format:
finder for J_{2}.
To find standard generators for J_{2}.2:
 Find any element of order 14. Its seventh power is a 2Celement, x say.
 Find any element of order 15. Its cube is a 5ABelement, y say.
 Find a conjugate c of x and a conjugate d of y, whose product has order 14.
This algorithm is available in computer readable format:
finder for J_{2}.2.
Checking generators
To check that elements x and y of J_{2}
are standard generators:
 Check o(x) = 2.
 Check o(y) = 3.
 Check o(xy) = 7.
 Check o(xyxyy) = 12.
This algorithm is available in computer readable format:
checker for J_{2}.
To check that elements x and y of J_{2}.2
are standard generators:
 Check o(x) = 2.
 Check o(y) = 5.
 Check o(xy) = 14.
 Check o(xyy) = 24.
This algorithm is available in computer readable format:
checker for J_{2}.2.
Presentations of J_{2} and J_{2}:2 in terms of their standard
generators are given below. [The second J_{2} presentation is shorter,
and the former is better for coset enumeration.]
< a, b  a^{2} = b^{3} =
(ab)^{7} = [a, b]^{12} =
(ababab^{−1}abab^{−1}ab^{−1}ababab^{−1}ab^{−1}abab^{−1})^{3}
= 1 >.
< a, b  a^{2} = b^{3} =
(ab)^{7} = [a, b]^{12} =
(ababab^{−1}abab^{−1})^{6}
= 1 >.
< c, d  c^{2} = d^{5} =
(cd)^{14} = [c, d]^{7} =
(cdcdcd^{−2}cd^{−2})^{3} =
[c, dcd]^{3} =
(cdcdcd^{2})^{3}cd^{−1}cdcdcd^{−1}cd^{2}
= 1 >.
The relation [c, dcd]^{3} = 1 is redundant. These
presentations, and those of the covering groups, are available in Magma
format as follows:
J_{2} on a and b [v1],
J_{2} on a and b [v2],
2.J_{2} on A and B [v1],
2.J_{2} on A and B [v2],
J_{2}:2 on c and d,
2.J_{2}.2 (+) on C and D and
2.J_{2}:2 (−) on C and D.
The representations of J_{2} available are:
 Primitive permutation representations.

Permutations on 100 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Permutations on 280 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Permutations on 315 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Permutations on 525 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Permutations on 840 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Permutations on 1008 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Permutations on 1800 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 The faithful irreducibles in characteristic 2 (up to Frobenius automorphisms).

Dimension 6 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 14 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 36 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 64 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 84 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 160 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 The faithful irreducibles in characteristic 3 (up to Frobenius automorphisms).

Dimension 13 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 21 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 36 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 57 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 63 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 90 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

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

Dimension 189 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 225 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 All faithful irreducibles in characteristic 5.

Dimension 14 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 21 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 41 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 70 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

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

Dimension 90 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 175 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 189 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 225 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 300 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 All faithful irreducibles in characteristic 7 (up to Frobenius automorphisms).

Dimension 14 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 21 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 36 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 63 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 70 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

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

Dimension 101 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 124 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 126 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 175 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 189 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 199 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 224 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 336 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 a and
b as 36 × 36 matrices over Z  not there, see version 1.
The representations of 2.J_{2} available are:

Permutations on 200 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Permutations on 1120 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).
 The faithful irreducibles in characteristic 3 (up to Frobenius automorphisms).

Dimension 6 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 14 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 36 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 50 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 126 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 216 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 236 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).
 Faithful irreducibles in characteristic 5.

Dimension 6 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 14 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 50 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 50 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 56 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 64 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).
 Faithful irreducibles in characteristic 7.

Dimension 6a over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 14 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 50 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 58 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).
The representations of J_{2}:2 available are:

Permutations on 100 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 The faithful irreducibles in characteristic 2.

Dimension 12 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 28 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 36 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 84 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 128 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 160 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 The faithful irreducibles in characteristic 3 (up to tensoring with the sign character):

Dimension 26 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

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

Dimension 42 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 63 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 90 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 114 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 133 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 225 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 378 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 The faithful irreducibles in characteristic 5 (up to tensoring with the sign character):

Dimension 14 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 21 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 41 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 70 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 85 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 90 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 175 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 189 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 225 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 300 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 The faithful irreducibles in characteristic 7 (up to tensoring with the sign character):

Dimension 28 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 36 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 42 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 63 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 89 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 101 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 124 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 126 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 140 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 175 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 199 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 336 over GF(49):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 378 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 448 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
The representations of 2.J_{2}.2 available are:
 The faithful irreducibles in characteristic 3 (up to tensoring with the sign character):

Dimension 12 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 14 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 72 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 100 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 216 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 236 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 252 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).
 The faithful irreducibles in characteristic 5 (up to tensoring with the sign character):

Dimension 6 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 14 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 100 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 56 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 64 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 190 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 202 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 350 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).
 The faithful irreducibles in characteristic 7 (up to tensoring with the sign character):

Dimension 12 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 14 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 84 over GF(49):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 100 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 112 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 116 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 252 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 252 over GF(49):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 336 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 350 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Dimension 448 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).
The maximal subgroups of J_{2} are as follows. Words provided by Peter
Walsh, implemented and checked by Ibrahim Suleiman.
The maximal subgroups of J_{2}:2 are:
A set of generators for the maximal cyclic subgroups of J_{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.
A set of generators for the maximal cyclic subgroups of J_{2}: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 J2 page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 20th June 2000.
Last updated 02.08.06 by JNB.
Information checked to
Level 0 on 20.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.