Janko group J3

In the area of modern algebra known as group theory, the Janko group J₃ or the Higman-Janko-McKay group HJM is a sporadic simple group of order

J₃ is one of the 26 Sporadic groups and was predicted by Zvonimir Janko in 1969 as one of two new simple groups having 2¹⁺⁴:A₅ as a centralizer of an involution (the other is the Janko group J₂). J₃ was shown to exist by Graham Higman and John McKay (1969).

In 1982 R. L. Griess showed that J₃ cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.

J₃ has an outer automorphism group of order 2 and a Schur multiplier of order 3, and its triple cover has a unitary 9-dimensional representation over the finite field with 4 elements. Weiss (1982) constructed it via an underlying geometry. It has a modular representation of dimension eighteen over the finite field with 9 elements. It has a complex projective representation of dimension eighteen.

In terms of generators a, b, c, and d its automorphism group J₃:2 can be presented as ${\displaystyle a^{17}=b^{8}=a^{b}a^{-2}=c^{2}=b^{c}b^{3}=(abc)^{4}=(ac)^{17}=d^{2}=[d,a]=[d,b]=(a^{3}b^{-3}cd)^{5}=1.}$

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