---
tag: PSEC8D7CX0
kind: object
name: Janko groups
msc: [20D08]
uses: [PZ49Z4CBFZ, 0YMYMNQ0QZ, 0B85S2V2DR, ZYZ5FYG7QC, C9VY20TAPT]
sources:
  - url: https://en.wikipedia.org/wiki/Janko_group
same_as:
  wikidata: Q1099545
  wikipedia: Janko group
---

## Description

The Janko groups $J_1$, $J_2$, $J_3$ and $J_4$ are four [[sporadic groups|PZ49Z4CBFZ]], found by Zvonimir Janko.
Their [[orders|0B85S2V2DR]] are:

1. $\abs{J_1} = 175560 = 2^3 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19$;
2. $\abs{J_2} = 604800 = 2^7 \cdot 3^3 \cdot 5^2 \cdot 7$;
3. $\abs{J_3} = 50232960 = 2^7 \cdot 3^5 \cdot 5 \cdot 17 \cdot 19$;
4. $\abs{J_4} = 86775571046077562880 = 2^{21} \cdot 3^3 \cdot 5 \cdot 7 \cdot 11^3 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 43$.

Each of them is a finite [[simple group|0YMYMNQ0QZ]].

## Remarks

$J_1$ was the first sporadic group found after the [[Mathieu groups|T1KR4KPX8P]], about a century later.
Janko characterised it by the [[centraliser|02YYV36BFM]] of an involution, which is $C_2 \times A_5$.
$J_1$ is a subgroup of the Chevalley group $G_2(\F_{11})$.
$J_2$ is also called the Hall–Janko group.
$J_1$, $J_3$ and $J_4$ are [[pariah groups|ZYZ5FYG7QC]], since they are not subquotients of the [[Monster group|C9VY20TAPT]], while $J_2$ is a subquotient of the Monster.

## Explanations

### L1 human

The Janko groups are four of the exceptional finite simple groups.
The first of them showed, after a century without new examples, that the Mathieu groups were not the only exceptions.