Sporadic Finite Simple Group/Catalog
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Catalog of Sporadic Finite Simple Groups
The sporadic finite simple groups are as follows:
| Name | Abbreviation | Order | Date discovered | Discovered by |
|---|---|---|---|---|
| Mathieu group | $\mathrm M_{11}$ | $2^4 \times 3^2 \times 5 \times 11$ | $1861$ | |
| Mathieu group | $\mathrm M_{12}$ | $2^6 \times 3^3 \times 5 \times 11$ | $1861$ | |
| Mathieu group | $\mathrm M_{22}$ | $2^7 \times 3^2 \times 5 \times 7 \times 11$ | $1873$ | |
| Mathieu group | $\mathrm M_{23}$ | $2^7 \times 3^2 \times 5 \times 7 \times 11 \times 23$ | $1873$ | |
| Mathieu group | $\mathrm M_{24}$ | $2^{10} \times 3^3 \times 5 \times 7 \times 11 \times 23$ | $1873$ | |
| Janko group | $\mathrm J_1$ | $2^3 \times 3 \times 5 \times 7 \times 11 \times 19$ | $1966$ | |
| Janko group | $\mathrm J_2$ | $2^7 \times 3^3 \times 5^2 \times 7$ | $1968$ | |
| Janko group | $\mathrm J_3$ | $2^7 \times 3^5 \times 5 \times 17 \times 19$ | $1968$ | |
| Janko group | $\mathrm J_4$ | $2^{21} \times 3^3 \times 5 \times 7 \times 11^3 \times 23 \times 29 \times 31 \times 37 \times 43$ | $1976$ | |
| $\mathrm {HS}$ | $2^9 \times 3^2 \times 5^3 \times 7 \times 11$ | $1968$ | ||
| $\mathrm {McL}$ | $2^7 \times 3^6 \times 5^3 \times 7 \times 11$ | $1969$ | ||
| $\mathrm {Suz}$ | $2^{13} \times 3^7 \times 5^2 \times 7 \times 11 \times 13$ | $1969$ | ||
| $\mathrm {Ru}$ | $2^{14} \times 3^3 \times 5^3 \times 7 \times 13 \times 29$ | $1973$ | ||
| $\mathrm {He}$ | $2^{10} \times 3^2 \times 5^2 \times 7^3 \times 17$ | $1969$ | ||
| $\mathrm {Ly}$ | $2^8 \times 3^7 \times 5^6 \times 7 \times 11 \times 31 \times 37 \times 67$ | $1972$ | ||
| $\text {O'N}$ | $2^9 \times 3^4 \times 5 \times 7^3 \times 11 \times 19 \times 31$ | $1976$ | ||
| $\mathrm {Co3}$ | $2^{10} \times 3^7 \times 5^3 \times 7 \times 11 \times 23$ | $1969$ | ||
| $\mathrm {Co2}$ | $2^{18} \times 3^6 \times 5^3 \times 7 \times 11 \times 23$ | $1969$ | ||
| $\mathrm {Co1}$ | $2^{21} \times 3^9 \times 5^4 \times 7^2 \times 11 \times 13 \times 23$ | $1969$ | ||
| $\mathrm {Fi} \left({22}\right)$ | $2^{17} \times 3^9 \times 5^2 \times 7 \times 11 \times 13$ | $1971$ | ||
| $\mathrm {Fi} \left({23}\right)$ | $2^{18} \times 3^{13} \times 5^2 \times 7 \times 11 \times 13 \times 17 \times 23$ | $1971$ | ||
| $\mathrm {Fi} \left({24}\right)'$ | $2^{21} \times 3^{16} \times 5^2 \times 7^3 \times 11 \times 13 \times 17 \times 23 \times 29$ | $1971$ | ||
| $\mathrm {HN}$ | $2^{14} \times 3^6 \times 5^6 \times 7 \times 11 \times 19$ | $1976$ | ||
| $\mathrm {Th}$ | $2^{15} \times 3^{10} \times 5^3 \times 7^2 \times 13 \times 19 \times 31$ | $1976$ | ||
| $\mathrm {BM}$ | $2^{41} \times 3^{13} \times 5^6 \times 7^2 \times 11 \times 13 \times 17 \times 19 \times 23 \times 31 \times 47$ | $1976$ | ||
| Fischer-Griess Monster | $\mathrm {M}$ | $2^{46} \times 3^{20} \times 5^9 \times 7^6 \times 11^2 \times 13^3 \times 17 \times 19 \times 23 \times 29 \times 31 \times 41 \times 47 \times 59 \times 71$ | $1981$ |
Sources
- 1996: John F. Humphreys: A Course in Group Theory: $\S 25.3$: The Sporadic Groups