The Wolfram Language 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, the Wolfram Language 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