ATLAS: Janko group J3

Order = 50232960 = 27.35.5.17.19.
Mult = 3.
Out = 2.

The following information is available for J3:


Standard generators

Standard generators of the Janko group J3 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.J3 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 J3:2 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.J3:2 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.


Black box algorithms

Finding generators

To find standard generators for J3:

This algorithm is available in computer readable format: finder for J3.

To find standard generators for J3.2:

This algorithm is available in computer readable format: finder for J3.2.

Checking generators

To check that elements x and y of J3 are standard generators:

This algorithm is available in computer readable format: checker for J3.

To check that elements x and y of J3.2 are standard generators:

This algorithm is available in computer readable format: checker for J3.2.

Presentations

Presentations of J3 and J3:2 in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)19 = [a, b]9 = ((ab)6(ab-1)5)2 = ((ababab-1)2abab-1ab-1abab-1)2 = abab(abab-1)3abab(abab-1)4ab-1(abab-1)3 = (ababababab-1abab-1)4 = 1 >.

< c, d | c2 = d3 = (cd)24 = [c, d]9 = (cd(cdcd-1)2)4 = (cdcdcd-1(cdcdcd-1cd-1)2)2 = [c, (dc)4(d-1c)2d]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.


Representations

The representations of J3 available are: The representations of 3.J3 available are: The representations of J3:2 available are: The representations of 3.J3:2 available are:

Maximal subgroups

The maximal subgroups of J3 are as follows. Words are given by Suleiman and Wilson in Experimental Math. 4 (1995), 11-18. The maximal subgroups of J3:2 are as follows. Words are given by Suleiman and Wilson in Experimental Math. 4 (1995), 11-18. (Different generators given below.)

Conjugacy classes

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.J3 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.


Main ATLAS page Go to main ATLAS (version 2.0) page.
Sporadic groups page Go to sporadic groups page.
Old J3 page Go to old J3 page - ATLAS version 1.
ftp access 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.