The mathematical reference
Finite groups.
Character theory, discrete symmetries, finite-simple classification, and the maps that connect finite and continuous groups.
49finite group records41Lie group records38Lie algebras22specified maps
103 released records. Individual groups and parameterized families are counted separately.
| Object | Kind | Essential data | Coverage |
|---|---|---|---|
| C₃ Cyclic group of order three | Group Finite | order 3 | executable 1 linked workbench |
| S₃ Symmetric group on three letters | Group Finite | order 6 | executable 1 linked workbench |
| A₄ Alternating group on four letters | Group Finite | order 12 | executable 2 linked workbenchs |
| S₄ Symmetric group on four letters | Group Finite | order 24 | executable 1 linked workbench |
| A₅ Alternating group on five letters | Group Finite | order 60 | executable 1 linked workbench |
| Q₈ Quaternion group | Group Finite | order 8 | executable 1 linked workbench |
| D₈ Dihedral group of order eight | Group Finite | order 8 | 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 |
| Cₙ Finite cyclic groups | Family Finite | Parameterized family | reference Sourced record |
| D₂ₙ Finite dihedral groups | Family Finite | Parameterized family | reference Sourced record |
| Dicₙ Dicyclic groups | Family Finite | Parameterized family | reference Sourced record |
| 2T · 2O · 2I Exceptional binary polyhedral groups | Family Finite | Parameterized family | reference Sourced record |
| Sₙ Finite symmetric groups | Family Finite | Parameterized family | reference Sourced record |
| C2 Cyclic group of order 2 | Group Finite | order 2 | reference Sourced record |
| C4 Cyclic group of order 4 | Group Finite | order 4 | reference Sourced record |
| C5 Cyclic group of order 5 | Group Finite | order 5 | reference Sourced record |
| C6 Cyclic group of order 6 | Group Finite | order 6 | reference Sourced record |
| V₄ Klein four-group | Group Finite | order 4 | reference Sourced record |
| D₁₀ Dihedral group of order ten | Group Finite | order 10 | reference Sourced record |
| D₁₂ Dihedral group of order twelve | Group Finite | order 12 | reference Sourced record |
| Dic₃ Dicyclic group of order twelve | Group Finite | order 12 | reference Sourced record |
| Q₁₆ Generalized quaternion group of order sixteen | Group Finite | order 16 | reference Sourced record |
| 2T Binary tetrahedral group | Group Finite | order 24 | reference Sourced record |
| 2O Binary octahedral group | Group Finite | order 48 | reference Sourced record |
| 2I Binary icosahedral group | Group Finite | order 120 | reference Sourced record |
| PSL(2,7) Projective special linear group PSL(2,7) | Group Finite | order 168 | reference Sourced record |
| Δ(27) Delta(27), the finite Heisenberg group | Group Finite | order 27 | 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 |
| C₃ · 1 C₃ irreducible representation 1 | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| C₃ · χ C₃ irreducible representation χ | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| C₃ · χ² C₃ irreducible representation χ² | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| S₃ · 1 S₃ irreducible representation 1 | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| S₃ · sgn S₃ irreducible representation sgn | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| S₃ · 2 S₃ irreducible representation 2 | Representation Finite | dimℂ 2 | executable 1 linked workbench |
| D₈ · 1 D₈ irreducible representation 1 | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| D₈ · a D₈ irreducible representation a | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| D₈ · b D₈ irreducible representation b | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| D₈ · c D₈ irreducible representation c | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| D₈ · 2 D₈ irreducible representation 2 | Representation Finite | dimℂ 2 | executable 1 linked workbench |
| Q₈ · 1 Q₈ irreducible representation 1 | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| Q₈ · a Q₈ irreducible representation a | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| Q₈ · b Q₈ irreducible representation b | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| Q₈ · c Q₈ irreducible representation c | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| Q₈ · 2 Q₈ irreducible representation 2 | Representation Finite | dimℂ 2 | executable 1 linked workbench |
| A₄ · 1 A₄ irreducible representation 1 | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| A₄ · 1′ A₄ irreducible representation 1′ | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| A₄ · 1″ A₄ irreducible representation 1″ | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| A₄ · 3 A₄ irreducible representation 3 | Representation Finite | dimℂ 3 | executable 1 linked workbench |
| S₄ · 1 S₄ irreducible representation 1 | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| S₄ · sgn S₄ irreducible representation sgn | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| S₄ · 2 S₄ irreducible representation 2 | Representation Finite | dimℂ 2 | executable 1 linked workbench |
| S₄ · 3 S₄ irreducible representation 3 | Representation Finite | dimℂ 3 | executable 1 linked workbench |
| S₄ · 3′ S₄ irreducible representation 3′ | Representation Finite | dimℂ 3 | executable 1 linked workbench |
| A₅ · 1 A₅ irreducible representation 1 | Representation Finite | dimℂ 1 | executable 1 linked workbench |
| A₅ · 3 A₅ irreducible representation 3 | Representation Finite | dimℂ 3 | executable 1 linked workbench |
| A₅ · 3′ A₅ irreducible representation 3′ | Representation Finite | dimℂ 3 | executable 1 linked workbench |
| A₅ · 4 A₅ irreducible representation 4 | Representation Finite | dimℂ 4 | executable 1 linked workbench |
| A₅ · 5 A₅ irreducible representation 5 | Representation Finite | dimℂ 5 | executable 1 linked workbench |
Coverage is specific. Classification identifies a mathematical object or family. Reference supplies stated properties. Executable links to a bounded calculation. Read the coverage policy.