Database of Some Permutation Groups

This database contains a number of permutation representations for finite groups. A particular group is included either if

• It is an 'interesting' group. In practice this means a simple group or a close relative of a simple group; or
• It is representative of some class of groups which is useful for testing conjectures and algorithms.

               AbelianGroup
               DihedralGroup
               AlternatingGroup
               SymmetricGroup
               ProjectiveSpecialLinearGroup
               ProjectiveGeneralLinearGroup
               ProjectiveSigmaLinearGroup
               ProjectiveGammaLinearGroup
               ProjectiveSpecialUnitaryGroup
               ProjectiveGeneralUnitaryGroup
               ProjectiveSigmaUnitaryGroup
               ProjectiveGammaUnitaryGroup
               ProjectiveSymplecticGroup
               ProjectiveSigmaSymplecticGroup
               AffineGeneralLinearGroup
               AffineSpecialLinearGroup
               AffineGammaLinearGroup
               AffineSigmaLinearGroup

               DirectProduct
               WreathProduct
               TensorProduct

Further, permutation representations of various classical groups may be constructed by setting up a matrix group using one of the standard functions

               GeneralLinearGroup
               SpecialLinearGroup
               GeneralUnitaryGroup
               SpecialUnitaryGroup
               SymplecticGroup
               SuzukiGroup

A large number of permutation representations may be found in the companion databases prmgps, which contains primitive groups of degree up to 50, and simgps, which contains permutation generators for all simple groups of order less than a million. See Database of Primitive Groups and Database of Simple Groups: Permutations, Presentations, Conjugacy Classes, Maximal Subgroups and Sylow Subgroups.

Contents:

Groups of Lie type

   psp42          The symplectic group PSP(4, 2) on 15 letters.
   psp43a         The symplectic group PSP(4, 3) on 27 letters.
   psp43b         The symplectic group PSP(4, 3) on 40 letters.
   psp44          The symplectic group PSP(4, 4) on 85 letters.
   psp62a         The symplectic group PSP(6, 2) on 28 letters.
   psp62b         The symplectic group PSP(6, 2) on 63 letters.
   psp63          The symplectic group PSP(6, 3) on 364 letters.
   sz8            Suzuki simple group Sz(8) on 65 letters.
   u33            The unitary group U(3, 3) on 28 letters.
   u34            The unitary group U(3, 4) on 208 letters.
   u35            The unitary group U(3, 5) on 525 letters.
   u42            The unitary group U(4, 2) on 45 letters.
   u43            The unitary group U(4, 3) on 540 letters.
   u52            The unitary group U(5, 2) on 176 letters.
   g24            The Lie group G(2, 4) on 416 letters.
   g25            The Lie group G(2, 5) on 3906 letters.
   twf42          The Lie group twisted F(4, 2 ) on 1755 letters.

Sporadic simple groups

   m11            Mathieu group of degree 11
   m12            Mathieu group of degree 12
   m22            Mathieu group of degree 22
   m23            Mathieu group of degree 23
   m24            Mathieu group of degree 24
   j1             First simple group of Janko on 266 letters
   j2             Second simple group of Janko on 100 letters
   hs100          Simple group of Higman-Sims on 100 letters
   hs176          Simple group of Higman-Sims on 176 letters
   co2            Second simple group of Conway on 2300 letters
   co3            Third simple group of Conway on 276 letters
   mcl            Simple group of McLaughlin on 275 letters
   suz            Simple group of Suzuki on 1782 letters
   he             Simple group of Held on 2058 letters
   tits1600       Simple group TITS on 1600 letters
   tits1755       Simple group TITS on 1755 letters 
                  (Derived subgroup of twf42)

Relatives of simple groups

   m12cover       The two-fold cover of the Mathieu group M12.

   j2cover        The two-fold cover of the second Janko group J2.

   suza2          Simple group of Suzuki extended by an automorphism
                  of order 2.

   psp44a2        The symplectic group PSP(4, 4) extended by an
                  automorphism of order 2.

Imprimitive non-soluble groups

   imp14a         Imprimitive group of degree 14 and
                  order 2^9 * 3 * 7.
   imp14b         Imprimitive group of degree 14 and
                  order 2^4 * 3 * 7.
   imp16a         Imprimitive group of degree 16 and
                  order 2^14 * 3^2 * 5 * 7.
   imp18          Imprimitive group of degree 18 and
                  order 2^7 * 3^4 * 7^2.
   imp21a         Imprimitive group of dgree 21 and
                  order 2^6 * 3^8 * 7.
   imp24a         Imprimitive group of degree 24 and
                  order 2^17 * 3^3 * 5 * 11.
   imp24b         Imprimitive group of degree 24 and
                  order 2^13 * 3 * 5 * 11.
   imp24c         Imprimitive group of degree 24 and
                  order 2^10 * 3^3 * 5.
   imp28          Imprimitive group of degree 28 and
                  order 2^20 * 3^8 * 7.
   imp35          Imprimitive group of degree 35 and
                  order 2^17 * 3^8 * 5^7 * 7.
   imp72          Imprimitive group of degree 72 and
                  order 2^9 * 3^21 * 5.
   imp90          Imprimitive group of degree 90 and
                  order 2^7 * 3^6 * 5.

Imprimitive soluble groups

   imp6           Soluble group of degree 6 and
                  order 2 * 3.
   imp12          Soluble group of degree 12 and
                  order 2^3 * 3^4.
   imp16b         Soluble group of degree 16 and
                  order 2^15 * 3^5.
   imp21b         Soluble group of degree 21 and
                  order 3^4 * 7^3.
   imp100a        Soluble group of degree 100 and
                  order 2^2 * 5^9.
   imp100b        Soluble group of degree 100 and
                  order 2^2 * 5^10.
   f29d190        Homomorphic image of the Fibonacci group F(2, 9)
                  of degree 190 and order 2^3 * 5^18 * 19.
   imp768         Soluble group of degree 768 and
                  order 2^11 * 3^12.

Primitive soluble groups

   sol81          Soluble group of degree 81 and
                  order 2^9 * 3^6.

Intransitive groups

   int288         Intransitive group of degree 288 with orbits of
                  length 32 and 256. The order is 2^18 * 3^2 * 5.
   mink9          Automorphism group of the classical Minkowski plane of
                  order 9.  The degree is 840, and order 
                  2^10 * 3^4 * 5^2.  It has orbits of lengths 
                  20, 100 and 720.

Symmetry groups of geometric objects

   meffert        Automorphism group of Meffert's cube puzzle.
   rubik          Automorphism group of the Rubik cube puzzle.
   rubik444       Automorphism group of the 4x4x4 Rubik cube puzzle.
   tetra          Automorphism group of the tetrahedral version of 
                  Rubik's cube.

Example

> load sz8;
Loading "pergps/sz8"
The first member of the family of simple groups
discovered by Suzuki - Sz(8) - represented as a
permutation group of degree 65. 
Order: 29,120 = 2^6 * 5 * 7 * 13;  Base: 1,2,3. 
Group: G

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