Sporadic groups

In mathematics, specifically group theory, the sporadic groups are a type of group which cannot be classified into any of the infinite families of simple groups.

There are 26 sporadic groups, the biggest being the Monster group (or Fischer-Griess monster), with a cardinality of approx. $8.1\times 10^{53}$. The sporadic groups go as follows:

• 1. Mathieu groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄
• 2. Janko groups J₁, J₂ or HJ, J₃ or HJM, J₄
• 3. Conway groups Co₁, Co₂, Co₃
• 4. Fischer groups Fi₂₂, Fi₂₃, Fi₂₄′ or F₃₊
• 5. Higman-Sims group HS
• 6. McLaughlin group McL
• 7. Held group He or F₇₊ or F₇
• 8. Rudvalis group Ru
• 9. Suzuki group Suz or F₃₋
• 10. O'Nan group O'N (ON)
• 11. Harada-Norton group HN or F₅₊ or F₅
• 12. Lyons group Ly
• 13. Thompson group Th or F_3|3 or F₃
• 14. Baby Monster group B or F₂₊ or F₂
• 15. Fischer-Griess Monster group M or F₁
• (Undecidedly) the Tits group, ²F₄(2)′.

These are often called the "happy family", a name affectionately coined by Robert Griess, who was one of the leading mathematicians on the ongoing research project about sporadic groups at the time. There are 6 exceptions to the happy family, the "pariahs", namely being J₁, J₃, J₄, O'N, Ru, and Ly.