- List of finite simple groups
In

mathematics , theclassification of finite simple groups states thatevery finitesimple group is cyclic, or alternating, or in one of 16 families ofgroups of Lie type (including theTits group , which strictly speaking is not of Lie type),or one of 26sporadic group s.The list below gives all finite simple groups, together with their order, the size of the

Schur multiplier , the size of theouter automorphism group , usually some small representations, and lists of all duplicates.(In removing duplicates it is useful to note that finite simple groups are determined by their orders, except that the group "B_{n}"("q") has the same order as "C_{n}"("q")for "q" odd, "n" > 2; and the groups A_{8}= "A"_{3}(2) and "A"_{2}(4)both have orders 20160.)**Notation:**"n" is a positive integer, "q" > 1 is a power of a prime number "p", and is the order of some underlyingfinite field .The order of the outer automorphism group is written as "d"·"f"·"g", where "d" is the order of the group of "diagonal automorphisms", "f" is the order of the (cyclic) group of "field automorphisms" (generated by aFrobenius automorphism ), and "g" is the order of the group of "graph automorphisms" (coming from automorphisms of theDynkin diagram ).**Infinite families**Cyclic group s "Z_{p}"**Simplicity:**Always simple.**Order:**"p"**Schur multiplier:**Trivial.**Outer automorphism group:**Cyclic of order "p"-1.**Other names:**"Z/pZ"**Remarks:**These are the only simple groups that are not perfect.**A**_{"n"}, "n" > 4,Alternating group s**Simplicity:**Solvable for "n" < 5, otherwise simple.**Order:**"n"!/2 when "n" > 1.**Schur multiplier:**2 for "n" = 5 or "n" > 7, 6 for "n" = 6 or 7.**Outer automorphism group:**In general 2. Exceptions: for "n" = 1, "n" = 2, it is trivial, and for "n" = 6, it has order 4 (elementary abelian).**Other names:**"Alt_{n}".There is an unfortunate conflict with the notation for the (unrelated) groups "A

_{n}"("q"), and some authors use various different fonts for A_{"n"}to distinguish them. In particular,in this article we make the distinction by setting the alternating groups A_{"n"}in Roman font and the Lie-type groups "A_{n}"("q") in italic.**Isomorphisms:**A_{1}and A_{2}are trivial. A_{3}is cyclic of order 3. A_{4}is isomorphic to "A"_{1}(3) (solvable). A_{5}is isomorphic to "A"_{1}(4) and to "A"_{1}(5). A_{6}is isomorphic to "A"_{1}(9) and to the derived group "B"_{2}(2)'. A_{8}is isomorphic to "A"_{3}(2).**Remarks:**An index 2 subgroup of thesymmetric group of permutations of "n" points when "n" > 1.**"A**_{n}"("q")Chevalley group s, linear groups**Simplicity:**"A"_{1}(2) and "A"_{1}(3) aresolvable, the others are simple.**Order:**:$\{1over\; (n+1,q-1)\}q^\{n(n+1)/2\}prod\_\{i=1\}^n(q^\{i+1\}-1)$**Schur multiplier:**For the simple groups it is cyclic of order ("n"+1, "q" − 1) except for "A"_{1}(4) (order 2), "A"_{1}(9) (order 6), "A"_{2}(2) (order 2), "A"_{2}(4) (order 48, product of cyclic groups of orders 3, 4, 4), "A"_{3}(2) (order 2).**Outer automorphism group:**(2, "q" − 1) ·"f"·1 for "n" = 1; ("n"+1, "q" − 1) ·"f"·2 for "n" > 1, where "q" = "p^{f}".**Other names:**Projective special linear group s, "PSL_{n+1}(q)","L"_{"n"+1}("q"), "PSL"("n"+1,"q")**Isomorphisms:**"A"_{1}(2) is isomorphic to the symmetric group on 3 points of order 6. "A"_{1}(3) is isomorphic to the alternating group A_{4}(solvable). "A"_{1}(4) and "A"_{1}(5)are isomorphic, and are both isomorphic to the alternating group A_{5}."A"_{1}(7) and "A"_{2}(2) are isomorphic. "A"_{1}(8) is isomorphic to the derived group^{2}"G"_{2}(3)′. "A"_{1}(9) is isomorphic to A_{6}and to the derived group "B"_{2}(2)′. "A"_{3}(2) is isomorphic to A_{8}.**Remarks:**These groups are obtained from thegeneral linear group s "GL"_{"n"+1}("q") bytaking the elements of determinant 1 (giving thespecial linear group s "SL"_{"n"+1}("q")) and then "quotienting out" by the center.**"B**_{n}"("q") "n" > 1Chevalley group s,orthogonal group **Simplicity:**"B"_{2}(2) is not simple and has a simple subgroup of index 2; the others are simple.**Order:**:$\{1over\; (2,q-1)\}q^\{n^2\}prod\_\{i=1\}^n(q^\{2i\}-1)$**Schur multiplier:**(2,"q" − 1) except for "B"_{2}(2) = S_{6}(order 2 for "B"_{2}(2), order 6 for "B"_{2}(2)′) and "B"_{3}(2)(order 2) and "B"_{3}(3)(order 6).**Outer automorphism group:**(2, "q" − 1) ·"f"·1 for "q" odd or "n">2; (2, "q" − 1) ·"f"·2 if "q" is even and "n"=2, where "q" = "p^{f}".**Other names:**"O"_{2"n"+1}("q"),Ω_{2"n"+1}("q") (for "q" odd).**Isomorphisms:**"B_{n}"(2^{"m"}) is isomorphic to "C_{n}"(2^{"m"}). "B"_{2}(2) is isomorphic to the symmetric group on 6 points, and the derived group "B"_{2}(2)′ is isomorphic to "A"_{1}(9) and toA_{6}. "B"_{2}(3) is isomorphic to^{2}"A"_{3}(2^{2}).**Remarks:**This is the group obtained from theorthogonal group in dimension 2"n"+1 bytaking the kernel of the determinant andspinor norm maps."B_{1}"("q") also exists, but is the same as"A_{1}"("q"). "B_{2}"("q") has a non-trivial graph automorphism when "q" is a power of 2.**"C**_{n}"("q") "n" > 2Chevalley group s,symplectic group s**Simplicity:**All simple.**Order:**:$\{1over\; (2,q-1)\}q^\{n^2\}prod\_\{i=1\}^n(q^\{2i\}-1)$**Schur multiplier:**(2,"q" − 1) except for "C_{3}"(2)(order 2).**Outer automorphism group:**(2, "q" − 1) ·"f"·1 where "q" = "p^{f}".**Other names:**Projective symplectic group, "PSp"_{2"n"}("q"), "PSp"_{"n"}("q") (not recommended), "S"_{2"n"}("q").**Isomorphisms:**"C_{n}"(2^{"m"}) is isomorphic to "B_{n}"(2^{"m"})**Remarks:**This group is obtained from thesymplectic group in 2"n" dimensions by "quotienting out" the center. "C"_{1}("q") also exists, but is the same as"A"_{1}("q"). "C"_{2}("q") also exists, but is the same as "B"_{2}("q").**"D**_{n}"("q") "n" > 3Chevalley group s,orthogonal group s**Simplicity:**All simple.**Order:**:$\{1over\; (4,q^n-1)\}q^\{n(n-1)\}(q^n-1)prod\_\{i=1\}^\{n-1\}(q^\{2i\}-1)$**Schur multiplier:**The order is (4, "q^{n}"-1) (cyclic for "n" odd, elementary abelian for "n" even) except for "D"_{4}(2) (order 4, elementary abelian).**Outer automorphism group:**(2, "q" − 1)^{2}·"f"·"S"_{3}for "n"=4, (2, "q" − 1)^{2}·"f"·2 for "n>4" even, (4, "q^{n}" − 1)^{2}·"f"·2 for "n" odd, where "q" = "p^{f}", and "S"_{3}is the symmetric group on 3 points of order 6.**Other names:**"O"_{2"n"}^{+}("q"), "PΩ"_{2"n"}^{+}("q").**Remarks:**This is the group obtained from the splitorthogonal group in dimension 2"n" bytaking the kernel of the determinant (orDickson invariant in characteristic 2) andspinor norm maps and then killing the center. The groups of type "D"_{4}have an unusually large diagram automorphism group of order 6, containing thetriality automorphism. "D"_{2}("q") also exists, but is the same as"A"_{1}("q")×"A"_{1}("q"). "D"_{3}("q") also exists, but is the same as "A"_{3}("q").**"E"**_{6}("q")Chevalley group s**Simplicity:**All simple.**Order:**"q"^{36}("q"^{12}−1)("q"^{9}−1)("q"^{8}−1)("q"^{6}−1)("q"^{5}−1)("q"^{2}−1)/(3,"q"-1)**Schur multiplier:**(3,"q" − 1)**Outer automorphism group:**(3, "q" − 1) ·"f"·2 where "q" = "p^{f}".**Other names:**Exceptional Chevalley group.**Remarks:**Has two representations of dimension 27, and acts on the Lie algebra of dimension 78.**"E"**_{7}("q")Chevalley group s**Simplicity:**All simple.**Order:**"q"^{63}("q"^{18}−1)("q"^{14}−1)("q"^{12}−1)("q"^{10}−1)("q"^{8}−1)("q"^{6}−1)("q"^{2}−1)/(2,"q"-1)**Schur multiplier:**(2,"q" − 1)**Outer automorphism group:**(2, "q" − 1) ·"f"·1 where "q" = "p^{f}".**Other names:**Exceptional Chevalley group.**Remarks:**Has a representations of dimension 56, and acts on the corresponding Lie algebra of dimension 133.**"E"**_{8}("q")Chevalley group s**Simplicity:**All simple.**Order:**"q"^{120}("q"^{30}−1)("q"^{24}−1)("q"^{20}−1)("q"^{18}−1)("q"^{14}−1)("q"^{12}−1)("q"^{8}−1)("q"^{2}−1)**Schur multiplier:**Trivial.**Outer automorphism group:**1·"f"·1 where "q" = "p^{f}".**Other names:**Exceptional Chevalley group.**Remarks:**It acts on the corresponding Lie algebra of dimension 248. "E"_{8}(3) contains the Thompson simple group.**"F"**_{4}("q")Chevalley group s**Simplicity:**All simple.**Order:**"q"^{24}("q"^{12}−1)("q"^{8}−1)("q"^{6}−1)("q"^{2}−1)**Schur multiplier:**Trivial except for "F"_{4}(2) (order 2).**Outer automorphism group:**1·"f"·1 for "q" odd,1·"f"·2 for "q" even,where "q" = "p^{f}".**Other names:**Exceptional Chevalley group.**Remarks:**These groups act on 27 dimensional exceptionalJordan algebra s, which gives them 26 dimensional representations. They also act on the corresponding Lie algebras of dimension 52. "F"_{4}("q") has a non-trivial graph automorphism when "q" is a power of 2.**"G"**_{2}("q")Chevalley group s**Simplicity:**"G"_{2}(2) is not simple but has a simple subgroup of index 2; the others are simple.**Order:**"q"^{6}("q"^{6}−1)("q"^{2}−1)**Schur multiplier:**Trivial for the simple groups except for "G"_{2}(3) (order 3) and "G"_{2}(4) (order 2).**Outer automorphism group:**1·"f"·1 for "q" not a power of 3,1·"f"·2 for "q" a power of 3,where "q" = "p^{f}".**Other names:**Exceptional Chevalley group.**Isomorphisms:**The derived group "G"_{2}(2)′ is isomorphic to^{2}"A"_{2}(3^{2}).**Remarks:**These groups are the automorphism groups of 8-dimensionalCayley algebra s over finite fields, which gives them 7 dimensional representations. They also act on the corresponding Lie algebras of dimension 14. "G"_{2}("q") has a non-trivial graph automorphism when "q" is a power of 3.^{2}"A_{n}"("q"^{2}) "n" > 1Steinberg group s,unitary group s**Simplicity:**^{2}"A"_{2}(2^{2}) is solvable, the others are simple.**Order:**:$\{1over\; (n+1,q+1)\}q^\{n(n+1)/2\}prod\_\{i=1\}^n(q^\{i+1\}-(-1)^\{i+1\})$**Schur multiplier:**Cyclic of order ("n" + 1, "q" + 1) for the simple groups, except for^{2}"A"_{3}(2^{2}) (order 2),^{2}"A"_{3}(3^{2}) (order 36, product of cyclic groups of orders 3,3,4),^{2}"A"_{5}(2^{2}) (order 12, product of cyclic groups of orders 2,2,3)**Outer automorphism group:**("n"+1, "q" + 1) ·"f"·1where "q"^{2}= "p^{f}".**Other names:**Twisted chevalley group, projective special unitary group, "PSU"_{"n"+1}("q"), "PSU"("n"+1, "q"),"U"_{"n"+1}("q"),^{2}"A_{n}"("q"),^{2}"A_{n}"("q", "q"^{2})**Isomorphisms:**The solvable group^{2}"A"_{2}(2^{2}) is isomorphic toan extension of the order 8 quaternion group by an elementary abelian group of order 9.^{2}"A"_{2}(3^{2}) is isomorphic to the derived group "G"_{2}(2)′.^{2}"A"_{3}(2^{2}) is isomorphic to "B"_{2}(3).**Remarks:**This is obtained from theunitary group in "n"+1 dimensions by taking the subgroup of elements of determinant 1 and then "quotienting" out by the center.^{2}"D_{n}"("q"^{2}) "n" > 3Steinberg group s,orthogonal group s**Simplicity:**All simple.**Order:**:$\{1over\; (4,q^n+1)\}q^\{n(n-1)\}(q^n+1)prod\_\{i=1\}^\{n-1\}(q^\{2i\}-1)$**Schur multiplier:**Cyclic of order (4, "q^{n}" + 1).**Outer automorphism group:**(4, "q^{n}" + 1) ·"f"·1where "q"^{2}= "p^{f}".**Other names:**^{2}"D_{n}"("q"),"O"_{2"n"}^{−}("q"), "PΩ"_{2"n"}^{−}("q"), twisted chevalley group.**Remarks:**This is the group obtained from the non-split orthogonal group in dimension 2"n" bytaking the kernel of the determinant (orDickson invariant in characteristic 2) andspinor norm maps and then killing the center.^{2}"D"_{2}("q"^{2}) also exists, but is the same as"A"_{1}("q").^{2}"D"_{3}("q"^{2}) also exists, but is the same as^{2}"A"_{3}("q"^{2}).**"**^{2}E_{6}"("q"^{2})Steinberg group s**Simplicity:**All simple.**Order:**"q"^{36}("q"^{12}−1)("q"^{9}+1)("q"^{8}−1)("q"^{6}−1)("q"^{5}+1)("q"^{2}−1)/(3,"q"+1)**Schur multiplier:**(3, "q" + 1) except for "^{2}E_{6}"(2^{2}) (order 12, product of cyclic groups of orders 2,2,3).**Outer automorphism group:**(3, "q" + 1) ·"f"·1where "q"^{2}= "p^{f}".**Other names:**"^{2}E_{6}"("q"), twisted Chevalley group.**Remarks:**One of the exceptional double covers of "^{2}E_{6}"(2^{2}) is a subgroup of the baby monster group,and the exceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.**"**^{3}D_{4}"("q"^{3})Steinberg group s**Simplicity:**All simple.**Order:**"q"^{12}("q"^{8}+"q"^{4}+1)("q"^{6}−1)("q"^{2}−1)**Schur multiplier:**Trivial.**Outer automorphism group:**1·"f"·1where "q"^{3}= "p^{f}".**Other names:**"^{3}D_{4}"("q"), Twisted Chevalley groups.**Remarks:**"^{3}D_{4}"(2^{3}) acts on the unique even 26 dimensional lattice of determinant 3 with no roots.^{2}"B"_{2}(2^{2"n"+1})Suzuki group s**Simplicity:**Simple for "n">1. The group^{2}"B"_{2}(2) is solvable.**Order:**"q"^{2}("q"^{2}+1)("q"−1)where "q" = 2^{2"n"+1}.**Schur multiplier:**Trivial for "n">2, elementary abelian of order 4for^{2}"B"_{2}(8).**Outer automorphism group:**1·"f"·1where "f" = 2"n"+1.**Other names:**Suz(2^{2"n"+1}), Sz(2^{2"n"+1}).**Isomorphisms:**^{2}"B"_{2}(2) is the Frobenius group of order 20.**Remarks:**Suzuki group areZassenhaus group s acting on sets of size (2^{2"n"+1})^{2}+1, and have 4 dimensional representations over the field with 2^{2"n"+1}elements. They are the only non-cyclic simple groups whose order is not divisible by 3. They are not related to the sporadic Suzuki group.^{2}"F"_{4}(2^{2"n"+1})Ree group s,Tits group **Simplicity:**Simple for "n">1. The derived group^{2}"F"_{4}(2)′ is simple of index 2 in^{2}"F"_{4}(2), and is called theTits group ,named for the Belgian mathematicianJacques Tits .**Order:**"q"^{12}("q"^{6}+1)("q"^{4}−1)("q"^{3}+1)("q"−1)where "q" = 2^{2"n"+1}.The Tits group has order 17971200 = 2

^{11}· 3^{3}· 5^{2}· 13.**Schur multiplier:**Trivial for "n">1 and for the Tits group.**Outer automorphism group:**1·"f"·1where "f" = 2"n"+1. Order 2 for the Tits group.**Remarks:**The Tits group is strictly speaking not a group of Lie type, and in particular it is not the group of points of a connected simple algebraic group with values in some field, nor does it have aBN pair . However most authors count it as a sort of honorary group of Lie type.^{2}"G"_{2}(3^{2"n"+1})Ree group s**Simplicity:**Simple for "n">1. The group "^{2}G_{2}"(3) is not simple, but its derived group "^{2}G_{2}"(3)′ is a simple subgroup of index 3.**Order:**"q"^{3}("q"^{3}+1)("q"−1)where "q" = 3^{2"n"+1}**Schur multiplier:**Trivial for "n">1 and for^{2}"G"_{2}(3)′.**Outer automorphism group:**1·"f"·1where "f" = 2"n"+1.**Other names:**Ree(3^{2"n"+1}), R(3^{2"n"+1}).**Isomorphisms:**The derived group^{2}"G"_{2}(3)′ is isomorphic to "A"_{1}(8).**Remarks:**^{2}"G"_{2}(3^{2"n"+1}) has a doubly transitive permutation representation on 3^{3(2"n"+1)}+1 points and acts on a 7 dimensional vector space over the field with 3^{2"n"+1}elements.**poradic groups**Mathieu group "M"_{11}**Order:**2^{4}· 3^{2}· 5 · 11=7920**Schur multiplier:**Trivial.**Outer automorphism group:**Trivial.**Remarks:**A 4-transitivepermutation group on 11 points, and the point stabilizer in "M"_{12}. The subgroup fixing a point is sometimes called "M"_{10}, and has a subgroup of index 2 isomorphic to the alternating group A_{6}.Mathieu group "M"_{12}**Order:**2^{6}· 3^{3}· 5 · 11=95040**Schur multiplier:**Order 2.**Outer automorphism group:**Order 2.**Remarks:**A 5-transitivepermutation group on 12 points.Mathieu group "M"_{22}**Order:**2^{7}· 3^{2}· 5 · 7 · 11 = 443520**Schur multiplier:**Cyclic of order 12. There were several mistakes made in the initial calculations of the Schur multiplier, so some older books and papers list incorrect values. (This caused an error in the title of Janko's original 1976 paper"A new finite simple group of order 86,775,571,046,077,562,880 which possesses "M"_{24}and the full covering group of "M"_{22}as subgroups.*J. Algebra***42**(1976), 564–596."giving evidence for the existence of the group "J"_{4}. At the time it was thought that the full covering group of"M"_{22}was 6·"M"_{22}. In fact "J"_{4}has no subgroup 12·"M"_{22}.)**Outer automorphism group:**Order 2.**Remarks:**A 3-transitivepermutation group on 22 points.Mathieu group "M"_{23}**Order:**2^{7}· 3^{2}· 5 · 7 · 11 · 23=10200960**Schur multiplier:**Trivial.**Outer automorphism group:**Trivial.**Remarks:**A 4-transitivepermutation group on 23 points, contained in "M"_{24}.Mathieu group "M"_{24}**Order:**2^{10}· 3^{3}· 5 · 7 · 11 · 23= 244823040**Schur multiplier:**Trivial.**Outer automorphism group:**Trivial.**Remarks:**A 5-transitivepermutation group on 24 points.

=Janko group "J"_{1}=**Order:**2^{3}· 3 · 5 · 7 · 11 · 19= 175560**Schur multiplier:**Trivial.**Outer automorphism group:**Trivial.**Other names:**J(1), J(11)**Remarks:**It is a subgroup of "G"_{2}(11), and so has a 7 dimensional representation over the field with 11 elements.**Janko group "J"**_{2}**Order:**2^{7}· 3^{3}· 5^{2}· 7 = 604800**Schur multiplier:**Order 2.**Outer automorphism group:**Order 2.**Other names:**Hall-Janko group, HJ**Remarks:**It is the automorphism group of a rank 3 graph on 100 points, and is also contained in "G"_{2}(4).

=Janko group "J"_{3}=**Order:**2^{7}· 3^{5}· 5 · 17 · 19= 50232960**Schur multiplier:**Order 3.**Outer automorphism group:**Order 2.**Other names:**Higman-Janko-McKay group, HJM**Remarks:**"J"_{3}seems unrelated to any other sporadic groups (or to anything else). Its triple cover has a 9 dimensionalunitary representation over the field with 4 elements.

=Janko group "J"_{4}=**Order:**2^{21}· 3^{3}· 5 · 7 · 11^{3}· 23 · 29 · 31 · 37 · 43 = 86775571046077562880**Schur multiplier:**Trivial.**Outer automorphism group:**Trivial.**Remarks:**Has a 112 dimensional representation over the field with 2 elements.Conway group "Co"_{1}**Order:**2^{21}· 3^{9}· 5^{4}· 7^{2}· 11 · 13 · 23 = 4157776806543360000**Schur multiplier:**Order 2.**Outer automorphism group:**Trivial.**Other names:**·1**Remarks:**The perfect double cover of "Co"_{1}is the automorphism group of theLeech lattice , and is sometimes denoted by ·0.Conway group "Co"_{2}**Order:**2^{18}· 3^{6}· 5^{3}· 7 · 11 · 23 = 42305421312000**Schur multiplier:**Trivial.**Outer automorphism group:**Trivial.**Other names:**·2**Remarks:**Subgroup of "Co"_{1}; fixes a norm 4 vector in theLeech lattice .Conway group "Co"_{3}**Order:**2^{10}· 3^{7}· 5^{3}· 7 · 11 · 23 = 495766656000**Schur multiplier:**Trivial.**Outer automorphism group:**Trivial.**Other names:**·3**Remarks:**Subgroup of "Co"_{1}; fixes a norm 6 vector in theLeech lattice .Fischer group "Fi"_{22}**Order:**2^{17}· 3^{9}· 5^{2}· 7 · 11 · 13 = 64561751654400.**Schur multiplier:**Order 6.**Outer automorphism group:**Order 2.**Other names:**"M"(22)**Remarks:**A 3-transposition group whose double cover is contained in "Fi"_{23}.Fischer group "Fi"_{23}**Order:**2^{18}· 3^{13}· 5^{2}· 7 · 11 · 13 · 17 · 23 = 4089470473293004800.**Schur multiplier:**Trivial.**Outer automorphism group:**Trivial.**Other names:**"M"(23)**Remarks:**A 3-transposition group contained in "Fi"_{24}.Fischer group "Fi"_{24}′**Order:**2^{21}· 3^{16}· 5^{2}· 7^{3}· 11 · 13 · 17 · 23 · 29 = 1255205709190661721292800.**Schur multiplier:**Order 3.**Outer automorphism group:**Order 2.**Other names:**"M"(24)′, "F"_{3+}.**Remarks:**The triple cover is contained in the monster group.Higman-Sims group "HS"**Order:**2^{9}· 3^{2}· 5^{3}· 7 · 11 = 44352000**Schur multiplier:**Order 2.**Outer automorphism group:**Order 2.**Remarks:**It acts as a rank 3 permutation group on the Higman Sims graph with 100 points, and is contained in "Co"_{3}.**McLaughlin group "McL"****Order:**2^{7}· 3^{6}· 5^{3}· 7 · 11= 898128000**Schur multiplier:**Order 3.**Outer automorphism group:**Order 2.**Remarks:**Acts as a rank 3 permutation group on the McLauglin graph with 275 points, and is contained in "Co"_{3}.Held group "He"**Order:**2^{10}· 3^{3}· 5^{2}· 7^{3}· 17 = 4030387200**Schur multiplier:**Trivial.**Outer automorphism group:**Order 2.**Other names:**Held-Higman-McKay group, HHM, "F"_{7}**Remarks:**Centralizes an element of order 7 in the monster group.Rudvalis group "Ru"**Order:**2^{14}· 3^{3}· 5^{3}· 7 · 13 · 29= 145926144000**Schur multiplier:**Order 2.**Outer automorphism group:**Trivial.**Remarks:**The double cover acts on a 28 dimensional lattice over theGaussian integer s.**Suzuki sporadic group "Suz"****Order:**2^{13}· 3^{7}· 5^{2}· 7 · 11 · 13 = 448345497600**Schur multiplier:**Order 6.**Outer automorphism group:**Order 2.**Other names:**"Sz"**Remarks:**The 6 fold cover acts on a 12 dimensional lattice over theEisenstein integer s. It is not related to the Suzuki groups of Lie type.O'Nan group "O'N"**Order:**2^{9}· 3^{4}· 5 · 7^{3}· 11 · 19 · 31= 460815505920**Schur multiplier:**Order 3.**Outer automorphism group:**Order 2.**Other names:**O'Nan-Sims group, O'NS**Remarks:**The triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism.Harada-Norton group "HN"**Order:**2^{14}· 3^{6}· 5^{6}· 7 · 11 · 19 = 273030912000000**Schur multiplier:**Trivial.**Outer automorphism group:**Order 2.**Other names:**"F"_{5}, "D"**Remarks:**Centralizes an element of order 5 in the monster group.Lyons group "Ly"**Order:**2^{8}· 3^{7}· 5^{6}· 7 · 11 · 31 · 37 · 67 = 51765179004000000**Schur multiplier:**Trivial.**Outer automorphism group:**Trivial.**Other names:**Lyons-Sims group, "LyS"**Remarks:**Has a 111 dimensional representation over the field with 5 elements.

=Thompson group "Th" =**Order:**2^{15}· 3^{10}· 5^{3}· 7^{2}· 13 · 19 · 31 = 90745943887872000**Schur multiplier:**Trivial.**Outer automorphism group:**Trivial.**Other names:**"F"_{3}, "E"**Remarks:**Centralizes an element of order 3 in the monster, and is contained in "E"_{8}(3).Baby Monster group "B"**Order:**: 2^{41}· 3^{13}· 5^{6}· 7^{2}· 11 · 13 · 17 · 19 · 23 · 31 · 47: = 4154781481226426191177580544000000**Schur multiplier:**Order 2.**Outer automorphism group:**Trivial.**Other names:**"F"_{2}**Remarks:**The double cover is contained in the monster group.**Fischer-Griess**Monster group "M"**Order:**: 2^{46}· 3^{20}· 5^{9}· 7^{6}· 11^{2}· 13^{3}· 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 : = 808017424794512875886459904961710757005754368000000000**Schur multiplier:**Trivial.**Outer automorphism group:**Trivial.**Other names:**"F"_{1}, "M"_{1}, Monster group, Friendly giant, Fischer's monster.**Remarks:**Contains all but 6 of the other sporadic groups as subquotients. Related tomonstrous moonshine . The monster is the automorphism group of the 196884 dimensionalGriess algebra and the infinite dimensional monstervertex operator algebra , and acts naturally on themonster Lie algebra .**Non-cyclic simple groups of small order**Order Factorized order Group Schur multiplier Outer automorphism group 60 2 ^{2}· 3 · 5A _{5}= "A"_{1}(4) = "A"_{1}(5)2 2 168 2 ^{3}· 3 · 7"A" _{1}(7) = "A"_{2}(2)2 2 360 2 ^{3}· 3^{2}· 5A _{6}= "A"_{1}(9) = "B"_{2}(2)′6 2×2 504 2 ^{3}· 3^{2}· 7"A" _{1}(8) =^{2}"G"_{2}(3)′1 3 660 2 ^{2}· 3 · 5 · 11"A" _{1}(11)2 2 1092 2 ^{2}· 3 · 7 · 13"A" _{1}(13)2 2 2448 2 ^{4}· 3^{2}· 17"A" _{1}(17)2 2 2520 2 ^{3}· 3^{2}· 5 · 7A _{7}6 2 3420 2 ^{2}· 3^{2}· 5 · 19"A" _{1}(19)2 2 4080 2 ^{4}· 3 · 5 · 17"A" _{1}(16)1 4 5616 2 ^{4}· 3^{3}· 13"A" _{2}(3)1 2 6048 2 ^{5}· 3^{3}· 7^{2}"A"_{2}(9) = "G"_{2}(2)′1 2 6072 2 ^{3}· 3 · 11 · 23"A" _{1}(23)2 2 7800 2 ^{3}· 3 · 5^{2}· 13"A" _{1}(25)2 2×2 7920 2 ^{4}· 3^{2}· 5 · 11"M" _{11}1 1 9828 2 ^{2}· 3^{3}· 7 · 13"A" _{1}(27)2 6 **See also**:List of small groups * [

*http://www.eleves.ens.fr:8080/home/madore/math/simplegroups.html Orders of non abelian simple groups*] up to order 10,000,000,000.**Further reading*** Daniel Gorenstein, Richard Lyons, Ronald Solomon "The Classification of the Finite Simple Groups" [

*http://www.ams.org/online_bks/surv401/ (volume 1)*] , AMS, 1994 [*http://www.ams.org/online_bks/surv402/ (volume 2)*] , AMS,

* Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England1985 .

* [*http://brauer.maths.qmul.ac.uk/Atlas/v3/ Atlas of Finite Group Representations*] : contains representations and other data for many finite simple groups, including the sporadic groups.

*"Simple Groups of Lie Type" by Roger W. Carter, ISBN 0-471-50683-4

* Mark Ronan, "Symmetry and the Monster", Oxford University Press, 2006. (Concise introduction and history written for lay

*Marcus du Sautoy , "Finding Moonshine", Fourth Estate, 2008, ISBN 978-0-00-721461-7 (another introduction for the lay reader)

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