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| − | A [[Simple finite group|simple finite group]] that does not belong to any of the known infinite series of simple finite groups. The twenty-six sporadic simple groups are listed in the following table.
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| − | {| class="wikitable" style="margin: 1em auto 1em auto;"
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| − | |+ The twenty-six sporadic simple groups
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| − | ! notation
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| − | ! name
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| − | ! order
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| − | |-
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| − | | $M_{11}$
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| − | | rowspan="5" | Mathieu groups
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| − | | $2^4.3^2.5.11$
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| − | |-
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| − | | $M_{12}$
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| − | | $2^6.3^3.5.11$
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| − | |-
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| − | | $M_{22}$
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| − | | $2^7.3^2.5.7.11$
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| − | |-
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| − | | $M_{23}$
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| − | | $2^7.3^2.5.7.11.23$
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| − | |-
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| − | | $M_{24}$
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| − | | $2^{10}.3^3.5.7.11.23$
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| − | |-
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| − | | $J_1$
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| − | | Janko group
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| − | | $2^3.3.5.7.11.19$
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| − | |-
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| − | | $J_2$, $HJ$
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| − | | Hall–Janko group
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| − | | $2^7.3^3.5^2.7$
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| − | |-
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| − | | $J_3$, $HJM$
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| − | | Hall–Janko–McKay group
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| − | | $2^7.3^5.5.17.19$
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| − | |-
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| − | | $J_4$
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| − | | Janko group
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| − | | $2^{21}.3^3.5.7.11^3.23.29.31.37.43$
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| − | |-
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| − | | $Co_1$
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| − | | rowspan="3" | Conway groups
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| − | | $2^{21}.3^9.5^4.7^2.11.13.23$
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| − | |-
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| − | | $Co_2$
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| − | | $2^{18}.3^6.5^3.7.11.23$
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| − | |-
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| − | | $Co_3$
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| − | | $2^{10}.3^7.5^3.7.11.23$
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| − | |-
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| − | | $F_{22}$, $M(22)$
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| − | | rowspan="3" | Fischer groups
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| − | | $2^{17}.3^9.5^2.7.11.13$
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| − | |-
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| − | | $F_{23}$, $M(23)$
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| − | | $2^{18}.3^{13}.5^2.7.11.13.17.23$
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| − | |-
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| − | | $F_{24}^\prime$, $M(24)^\prime$
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| − | | $2^{21}.3^{16}.5^2.7^3.11.13.17.23.29$
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| − | |-
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| − | | $HS$
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| − | | Higman–Sims group
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| − | | $2^9.3^2.5^3.7.11$
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| − | |-
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| − | | $He$, $HHM$
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| − | | Held–Higman–McKay group
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| − | | $2^{10}.3^3.5^2.7^3.17$
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| − | |-
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| − | | $Suz$
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| − | | Suzuki group
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| − | | $2^{13}.3^7.5^2.7.11.13$
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| − | |-
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| − | | $M^c$
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| − | | McLaughlin group
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| − | | $2^7.3^6.5^3.7.11$
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| − | |-
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| − | | $Ly$
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| − | | Lyons group
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| − | | $2^8.3^7.5^6.7.11.31.37.67$
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| − | |-
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| − | | $Ru$
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| − | | Rudvalis group
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| − | | $2^{14}.3^3.5^3.7.13.29$
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| − | |-
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| − | | $O'N$, $O'NS$
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| − | | O'Nan–Sims group
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| − | | $2^9.3^4.5.7^3.11.19.31$
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| − | |-
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| − | | $F_1$, $M$
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| − | | Monster, Fischer–Griess group
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| − | | $2^{46}.3^{20}.5^9.7^6.11^2.13^3.17.19.23.29.31.41.47.59.71$
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| − | |-
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| − | | $F_2$, $B$
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| − | | Baby monster
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| − | | $2^{41}.3^{13}.5^6.7^2.11.13.17.19.23.31.47$
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| − | |-
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| − | | $F_3$, $E$, $Th$
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| − | | Thompson group
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| − | | $2^{15}.3^{10}.5^3.7^2.13.19.31$
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| − | |-
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| − | | $F_5$, $D$, $HN$
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| − | | Harada–Norton group
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| − | | $2^{14}.3^6.5^6.7.11.19$
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| − | |}
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| − |
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| − | ====References====
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.A. Syskin, "Abstract properties of the simple sporadic groups" ''Russian Math. Surveys'' , '''35''' : 5 (1980) pp. 209–246 ''Uspekhi Mat. Nauk'' , '''35''' : 5 (1980) pp. 181–212</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Aschbacher, "The finite simple groups and their classification" , Yale Univ. Press (1980)</TD></TR></table>
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| − | ====Comments====
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| − | The recent classification of the finite simple groups (1981) has led to the conclusion that — up to a uniqueness proof for the Monster as the only simple group of its order with certain additional properties — every non-Abelian finite simple group is isomorphic to: an [[Alternating group|alternating group]] on at least 5 letters, a group of (twisted or untwisted) Lie type, or one of the above 26 sporadic groups. See [[#References|[a2]]] for a discussion of the proof.
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| − |
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, "Atlas of finite groups" , Clarendon Press (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D. Gorenstein, "Finite simple groups. An introduction to their classification" , Plenum (1982)</TD></TR></table>
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