ATLAS: Janko group J4

Order = 86775571046077562880 = 221.33.5.7.113.23.29.31.37.43.
Mult = 1.
Out = 1.

Standard generators

Standard generators of the Janko group J4 are a and b where a is in class 2A, b is in class 4A, ab has order 37 and ababb has order 10.

Black box algorithms

To find standard generators for J4:
• Find any element of order 20, 40 or 44. It powers up to a 2A-element x and a 4A-element y.
• Find a conjugate a of x and a conjugate b of y such that ab has order 37 and ababb has order 10.
• Now a and b are standard generators of J4.

Representations

The representations of J4 available are:
• a and b as 112 × 112 matrices over GF(2).
• A vector which has 173067389 images in the above representation.
• Another vector which has 8474719242 images in the above representation.
• a and b as 1333 × 1333 matrices over GF(11) - kindly provided by Wolfgang Lempken.

Maximal subgroups

The maximal subgroups of J4 are

Conjugacy class representatives

• 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.