Curated collection
Finite simple groups
Follow the classification through prime-order cyclic groups, alternating groups, finite groups of Lie type, and the sporadic groups. Low parameters and exceptional identifications are part of each family.
52 released records · classifications and individual objects are labeled separately.
| Object | Kind | Essential data | Coverage |
|---|---|---|---|
| C₃ Cyclic group of order three | Group Finite | order 3 | executable 1 linked workbench |
| A₅ Alternating group on five letters | Group Finite | order 60 | executable 1 linked workbench |
| Cₚ Cyclic groups of prime order | Family Finite | Parameterized family | classification Sourced record |
| Aₙ Simple alternating groups | Family Finite | Parameterized family | classification Sourced record |
| PSL(n,q) Projective special linear finite simple groups | Family Finite | Parameterized family | classification Sourced record |
| PSU(n,q) Projective special unitary finite simple groups | Family Finite | Parameterized family | classification Sourced record |
| PSp(2n,q) Projective symplectic finite simple groups | Family Finite | Parameterized family | classification Sourced record |
| PΩ(2n+1,q) Odd-dimensional orthogonal finite simple groups | Family Finite | Parameterized family | classification Sourced record |
| PΩ⁺(2n,q) Plus-type even orthogonal finite simple groups | Family Finite | Parameterized family | classification Sourced record |
| PΩ⁻(2n,q) Minus-type even orthogonal finite simple groups | Family Finite | Parameterized family | classification Sourced record |
| G₂(q) Finite simple groups of type G₂ | Family Finite | Parameterized family | classification Sourced record |
| F₄(q) Finite simple groups of type F₄ | Family Finite | Parameterized family | classification Sourced record |
| E₆(q) Finite simple groups of type E₆ | Family Finite | Parameterized family | classification Sourced record |
| E₇(q) Finite simple groups of type E₇ | Family Finite | Parameterized family | classification Sourced record |
| E₈(q) Finite simple groups of type E₈ | Family Finite | Parameterized family | classification Sourced record |
| ²E₆(q) Finite simple groups of type ²E₆ | Family Finite | Parameterized family | classification Sourced record |
| ³D₄(q) Finite simple groups of type ³D₄ | Family Finite | Parameterized family | classification Sourced record |
| ²B₂(q) Suzuki simple groups | Family Finite | Parameterized family | classification Sourced record |
| ²G₂(q) Small Ree simple groups | Family Finite | Parameterized family | classification Sourced record |
| ²F₄(q) Large Ree simple groups | Family Finite | Parameterized family | classification Sourced record |
| Sporadic The 26 sporadic finite simple groups | Family Finite | Parameterized family | classification Sourced record |
| C2 Cyclic group of order 2 | Group Finite | order 2 | reference Sourced record |
| C5 Cyclic group of order 5 | Group Finite | order 5 | reference Sourced record |
| PSL(2,7) Projective special linear group PSL(2,7) | Group Finite | order 168 | reference Sourced record |
| A₆ Alternating group on six letters | Group Finite | order 360 | reference Sourced record |
| M₁₁ Mathieu group M11 | Group Finite | order 7920 | classification Sourced record |
| M₁₂ Mathieu group M12 | Group Finite | order 95040 | classification Sourced record |
| M₂₂ Mathieu group M22 | Group Finite | order 443520 | classification Sourced record |
| M₂₃ Mathieu group M23 | Group Finite | order 10200960 | classification Sourced record |
| M₂₄ Mathieu group M24 | Group Finite | order 244823040 | classification Sourced record |
| J₁ First Janko group | Group Finite | order 175560 | classification Sourced record |
| J₂ Hall–Janko group | Group Finite | order 604800 | classification Sourced record |
| J₃ Third Janko group | Group Finite | order 50232960 | classification Sourced record |
| J₄ Fourth Janko group | Group Finite | order 86775571046077562880 | classification Sourced record |
| Co₁ First Conway group | Group Finite | order 4157776806543360000 | classification Sourced record |
| Co₂ Second Conway group | Group Finite | order 42305421312000 | classification Sourced record |
| Co₃ Third Conway group | Group Finite | order 495766656000 | classification Sourced record |
| Fi₂₂ Fischer group Fi22 | Group Finite | order 64561751654400 | classification Sourced record |
| Fi₂₃ Fischer group Fi23 | Group Finite | order 4089470473293004800 | classification Sourced record |
| Fi₂₄′ Fischer simple group Fi24 prime | Group Finite | order 1255205709190661721292800 | classification Sourced record |
| HS Higman–Sims group | Group Finite | order 44352000 | classification Sourced record |
| McL McLaughlin group | Group Finite | order 898128000 | classification Sourced record |
| He Held group | Group Finite | order 4030387200 | classification Sourced record |
| Ru Rudvalis group | Group Finite | order 145926144000 | classification Sourced record |
| Suz Sporadic Suzuki group | Group Finite | order 448345497600 | classification Sourced record |
| O’N O’Nan group | Group Finite | order 460815505920 | classification Sourced record |
| HN Harada–Norton group | Group Finite | order 273030912000000 | classification Sourced record |
| Ly Lyons group | Group Finite | order 51765179004000000 | classification Sourced record |
| Th Thompson group | Group Finite | order 90745943887872000 | classification Sourced record |
| B Baby Monster group | Group Finite | order 4154781481226426191177580544000000 | classification Sourced record |
| M Monster group | Group Finite | order 808017424794512875886459904961710757005754368000000000 | classification Sourced record |
| ²F₄(2)′ Tits group | Group Finite | order 17971200 | classification Sourced record |