First Janko group J1

The first Janko group $J_1$ is the group with the following complete presentation. With $c=abab^{-1}$ as an abbreviation for a word, set

J_1=\left\langle a,b\;\middle|\; \begin{aligned} a^2&=b^3=(ab)^7=1,\\ (abc^3)^5&=1,\\ \bigl(abc^6abab(ab^{-1})^2\bigr)^2&=1 \end{aligned}\right\rangle.

Here $c$ is not an additional generator. Every displayed relation is part of the definition.

Order and structure

This is a nonabelian finite simple group of order

|J_1|=175560=2^3\cdot3\cdot5\cdot7\cdot11\cdot19.

It is one of the six sporadic groups in the pariah display block. The labels $J_1$ , $J_3$ , and $J_4$ name distinct sporadic groups, rather than the first members of a family defined for every integer subscript.

A relationship to the O'Nan group

The O'Nan group contains a maximal subgroup isomorphic to $J_1$ , of index $2624832$ . This gives an injective group homomorphism $J_1\hookrightarrow\mathrm{O'N}$ ; its image is proper.

References

ATLAS of Finite Group Representations, J1 presentation J1G1-P1, “About this presentation”; complete Magma relator file.
ATLAS, Janko group J1, order heading and “Presentations.”
ATLAS, O'Nan group, “Maximal subgroups of O'N,” row J1.
ATLAS, sporadic groups, “Pariahs.”