ATLAS: Hall-Janko group HJ = J2

Order = 604800 =
Mult = 2.
Out = 2.

Standard generators

Standard generators of the Janko group J2 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.J2 are preimages A and B where B has order 3, and AB has order 7.

Standard generators of the automorphism group J2:2 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.J2.2 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.

Black box algorithms

To find standard generators for J2: To find standard generators for J2.2:


The representations of J2 available are The representations of 2.J2 available are The representations of J2:2 available are The representations of 2.J2.2 available are

Maximal subgroups

The maximal subgroups of J2 are as follows. Words provided by Peter Walsh, implemented and checked by Ibrahim Suleiman. The maximal subgroups of J2:2 are

Conjugacy classes

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.
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