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Named Groups
Mathematica provides access to information on infinite families of groups, as well as sporadic groups of relevance, like the famous 26 sporadic simple groups (27 if the Tits group is included). In particular, Mathematica knows permutation representations for most of them, hence allowing further computation and application to other areas of the system.
Parametrized Infinite Families
SymmetricGroup symmetric group of degree
AlternatingGroup alternating group of degree
CyclicGroup cyclic group of order
DihedralGroup dihedral group of the -gon, of order
AbelianGroup Abelian group isomorphic to a direct product of several cyclic groups
    
Sporadic Simple Groups: The Mathieu Groups
MathieuGroupM11 Mathieu group , with default representation on 11 points
MathieuGroupM12 Mathieu group , with default representation on 12 points
MathieuGroupM22 Mathieu group , with default representation on 22 points
MathieuGroupM23 Mathieu group , with default representation on 23 points
MathieuGroupM24 Mathieu group , with default representation on 24 points
Sporadic Simple Groups: The Second Generation
ConwayGroupCo1 Conway group , with no representation provided
ConwayGroupCo2 Conway group , with default representation on 2300 points
ConwayGroupCo3 Conway group , with default representation on 276 points
HigmanSimsGroupHS Higman-Sims group , with default representation on 100 points
JankoGroupJ2 Janko group , with default representation on 100 points
McLaughlinGroupMcL McLaughlin group , with default representation on 275 points
SuzukiGroupSuz Suzuki group , with default representation on 1782 points
Sporadic Simple Groups: The Third Generation
FischerGroupFi22 Fischer group , with default representation on 3510 points
FischerGroupFi23 Fischer group , with default representation on 31671 points
FischerGroupFi24Prime Fischer group , with no representation provided
HeldGroupHe Held group , with default representation on 2058 points
HaradaNortonGroupHN Harada-Norton group , with no representation provided
ThompsonGroupTh Thompson group , with no representation provided
BabyMonsterGroupB baby monster group , with no representation provided
MonsterGroupM monster group , with no representation provided
Sporadic Simple Groups: The Exceptions or Pariahs, and the Tits Group
JankoGroupJ1 Janko group , with default representation on 266 points
JankoGroupJ3 Janko group , with default representation on 6156 points
JankoGroupJ4 Janko group , with no representation provided
RudvalisGroupRu Rudvalis group , with default representation on 4060 points
ONanGroupON O'Nan group , with no representation provided
LyonsGroupLy Lyons group , with no representation provided
TitsGroupT Tits group , with default representation on 1600 points
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