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49finite group records41Lie group records38Lie algebras22specified maps
266 released records. Individual groups and parameterized families are counted separately.
| Object | Kind | Essential data | Coverage |
|---|---|---|---|
| SU(2) Special unitary group | Group Lie theory | dimℝ 3 | executable 7 linked workbenchs |
| SO(3) Rotation group in three dimensions | Group Lie theory | dimℝ 3 | executable 2 linked workbenchs |
| SU(3) Special unitary group | Group Lie theory | dimℝ 8 | executable 8 linked workbenchs |
| PSU(3) Projective special unitary group | Group Lie theory | dimℝ 8 | executable 1 linked workbench |
| U(1) Circle group | Group Lie theory | dimℝ 1 | reference Sourced record |
| Spin(4) Euclidean spin group in four dimensions | Group Lie theory | dimℝ 6 | reference Sourced record |
| SL(2,ℂ) Lorentz spin group | Group Lie theory | dimℝ 6 | reference Sourced record |
| 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 |
| Aᵣ Simple Lie type Aᵣ | Family Lie theory | Parameterized family | classification Sourced record |
| Bᵣ Simple Lie type Bᵣ | Family Lie theory | Parameterized family | classification Sourced record |
| Cᵣ Simple Lie type Cᵣ | Family Lie theory | Parameterized family | classification Sourced record |
| Dᵣ Simple Lie type Dᵣ | Family Lie theory | Parameterized family | classification Sourced record |
| E₆ Simple Lie type E₆ | Family Lie theory | Parameterized family | classification Sourced record |
| E₇ Simple Lie type E₇ | Family Lie theory | Parameterized family | classification Sourced record |
| E₈ Simple Lie type E₈ | Family Lie theory | Parameterized family | classification Sourced record |
| F₄ Simple Lie type F₄ | Family Lie theory | Parameterized family | classification Sourced record |
| G₂ Simple Lie type G₂ | Family Lie theory | Parameterized family | classification Sourced record |
| sl(2, C) sl(2, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 3 | reference Sourced record |
| sl(3, C) sl(3, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 8 | reference Sourced record |
| sl(4, C) sl(4, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 15 | reference Sourced record |
| sl(5, C) sl(5, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 24 | reference Sourced record |
| so(5, C) ≅ sp(4, C) so(5, C) ≅ sp(4, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 10 | reference Sourced record |
| sp(6, C) sp(6, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 21 | reference Sourced record |
| so(8, C) so(8, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 28 | reference Sourced record |
| so(10, C) so(10, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 45 | reference Sourced record |
| e₆(C) e₆(C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 78 | reference Sourced record |
| e₇(C) e₇(C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 133 | reference Sourced record |
| e₈(C) e₈(C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 248 | reference Sourced record |
| f₄(C) f₄(C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 52 | reference Sourced record |
| g₂(C) g₂(C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 14 | reference Sourced record |
| sl(2, C) ⊕ sl(2, C) sl(2, C) ⊕ sl(2, C) — complex Lie algebra | Lie algebra Lie theory | dimℂ 6 | reference Sourced record |
| C (zero bracket) C (zero bracket) — complex Lie algebra | Lie algebra Lie theory | dimℂ 1 | reference Sourced record |
| sl(2, C) ⊕ C sl(2, C) ⊕ C — complex Lie algebra | Lie algebra Lie theory | dimℂ 4 | reference Sourced record |
| su(2) su(2) — real Lie algebra | Lie algebra Lie theory | dimℝ 3 | reference Sourced record |
| su(3) su(3) — real Lie algebra | Lie algebra Lie theory | dimℝ 8 | reference Sourced record |
| su(4) ≅ so(6) su(4) ≅ so(6) — real Lie algebra | Lie algebra Lie theory | dimℝ 15 | reference Sourced record |
| su(5) su(5) — real Lie algebra | Lie algebra Lie theory | dimℝ 24 | reference Sourced record |
| sp(2) ≅ so(5) sp(2) ≅ so(5) — real Lie algebra | Lie algebra Lie theory | dimℝ 10 | reference Sourced record |
| sp(3) sp(3) — real Lie algebra | Lie algebra Lie theory | dimℝ 21 | reference Sourced record |
| so(8) so(8) — real Lie algebra | Lie algebra Lie theory | dimℝ 28 | reference Sourced record |
| so(10) so(10) — real Lie algebra | Lie algebra Lie theory | dimℝ 45 | reference Sourced record |
| e₆ compact e₆ compact — real Lie algebra | Lie algebra Lie theory | dimℝ 78 | reference Sourced record |
| e₇ compact e₇ compact — real Lie algebra | Lie algebra Lie theory | dimℝ 133 | reference Sourced record |
| e₈ compact e₈ compact — real Lie algebra | Lie algebra Lie theory | dimℝ 248 | reference Sourced record |
| f₄ compact f₄ compact — real Lie algebra | Lie algebra Lie theory | dimℝ 52 | reference Sourced record |
| g₂ compact g₂ compact — real Lie algebra | Lie algebra Lie theory | dimℝ 14 | reference Sourced record |
| su(2) ⊕ su(2) su(2) ⊕ su(2) — real Lie algebra | Lie algebra Lie theory | dimℝ 6 | reference Sourced record |
| u(1) u(1) — real Lie algebra | Lie algebra Lie theory | dimℝ 1 | reference Sourced record |
| u(2) u(2) — real Lie algebra | Lie algebra Lie theory | dimℝ 4 | reference Sourced record |
| sl(2, R) sl(2, R) — real Lie algebra | Lie algebra Lie theory | dimℝ 3 | reference Sourced record |
| sl(3, R) sl(3, R) — real Lie algebra | Lie algebra Lie theory | dimℝ 8 | reference Sourced record |
| su(2,2) ≅ so(2,4) su(2,2) ≅ so(2,4) — real Lie algebra | Lie algebra Lie theory | dimℝ 15 | reference Sourced record |
| sp(4, R) ≅ so(2,3) sp(4, R) ≅ so(2,3) — real Lie algebra | Lie algebra Lie theory | dimℝ 10 | reference Sourced record |
| sl(2, C) viewed over R sl(2, C) viewed over R — real Lie algebra | Lie algebra Lie theory | dimℝ 6 | reference Sourced record |
| su(2) su(2) real form | Real form Lie theory | Compact | reference Sourced record |
| su(3) su(3) real form | Real form Lie theory | Compact | reference Sourced record |
| su(4) ≅ so(6) su(4) ≅ so(6) real form | Real form Lie theory | Compact | reference Sourced record |
| su(5) su(5) real form | Real form Lie theory | Compact | reference Sourced record |
| sp(2) ≅ so(5) sp(2) ≅ so(5) real form | Real form Lie theory | Compact | reference Sourced record |
| sp(3) sp(3) real form | Real form Lie theory | Compact | reference Sourced record |
| so(8) so(8) real form | Real form Lie theory | Compact | reference Sourced record |
| so(10) so(10) real form | Real form Lie theory | Compact | reference Sourced record |
| e₆ compact e₆ compact real form | Real form Lie theory | Compact | reference Sourced record |
| e₇ compact e₇ compact real form | Real form Lie theory | Compact | reference Sourced record |
| e₈ compact e₈ compact real form | Real form Lie theory | Compact | reference Sourced record |
| f₄ compact f₄ compact real form | Real form Lie theory | Compact | reference Sourced record |
| g₂ compact g₂ compact real form | Real form Lie theory | Compact | reference Sourced record |
| su(2) ⊕ su(2) su(2) ⊕ su(2) real form | Real form Lie theory | Compact | reference Sourced record |
| sl(2, R) sl(2, R) real form | Real form Lie theory | Noncompact | reference Sourced record |
| sl(3, R) sl(3, R) real form | Real form Lie theory | Noncompact | reference Sourced record |
| su(2,2) ≅ so(2,4) su(2,2) ≅ so(2,4) real form | Real form Lie theory | Noncompact | reference Sourced record |
| sp(4, R) ≅ so(2,3) sp(4, R) ≅ so(2,3) real form | Real form Lie theory | Noncompact | reference Sourced record |
| sl(2, C) viewed over R sl(2, C) viewed over R real form | Real form Lie theory | Noncompact | reference Sourced record |
| SU(4) Special unitary group in four dimensions | Group Lie theory | dimℝ 15 | executable 4 linked workbenchs |
| PSU(4) Adjoint compact group of type A₃ | Group Lie theory | dimℝ 15 | reference Sourced record |
| SU(5) Special unitary group in five dimensions | Group Lie theory | dimℝ 24 | executable 4 linked workbenchs |
| U(2) Unitary group in two dimensions | Group Lie theory | dimℝ 4 | reference Sourced record |
| SO(4) Rotation group in four dimensions | Group Lie theory | dimℝ 6 | reference Sourced record |
| SO(5) Rotation group in five dimensions | Group Lie theory | dimℝ 10 | reference Sourced record |
| SO(6) Rotation group in six dimensions | Group Lie theory | dimℝ 15 | reference Sourced record |
| Sp(2) Compact symplectic group of quaternionic rank two | Group Lie theory | dimℝ 10 | executable 1 linked workbench |
| Sp(3) Compact symplectic group of quaternionic rank three | Group Lie theory | dimℝ 21 | reference Sourced record |
| Spin(8) Simply connected compact spin group in eight dimensions | Group Lie theory | dimℝ 28 | executable 1 linked workbench |
| SO(8) Rotation group in eight dimensions | Group Lie theory | dimℝ 28 | reference Sourced record |
| PSO(8) Adjoint compact group of type D₄ | Group Lie theory | dimℝ 28 | reference Sourced record |
| Spin(10) Simply connected compact spin group in ten dimensions | Group Lie theory | dimℝ 45 | executable 1 linked workbench |
| SO(10) Rotation group in ten dimensions | Group Lie theory | dimℝ 45 | reference Sourced record |
| G₂ compact Compact group of octonion automorphisms | Group Lie theory | dimℝ 14 | executable 1 linked workbench |
| F₄ compact Compact exceptional group of type F₄ | Group Lie theory | dimℝ 52 | executable 1 linked workbench |
| E₆ compact, simply connected Simply connected compact exceptional group of type E₆ | Group Lie theory | dimℝ 78 | executable 1 linked workbench |
| E₇ compact, simply connected Simply connected compact exceptional group of type E₇ | Group Lie theory | dimℝ 133 | executable 1 linked workbench |
| E₈ compact Compact exceptional group of type E₈ | Group Lie theory | dimℝ 248 | executable 1 linked workbench |
| E₆ compact, adjoint Adjoint compact exceptional group of type E₆ | Group Lie theory | dimℝ 78 | reference Sourced record |
| E₇ compact, adjoint Adjoint compact exceptional group of type E₇ | Group Lie theory | dimℝ 133 | reference Sourced record |
| SL(2, R) Real special linear group in two dimensions | Group Lie theory | dimℝ 3 | reference Sourced record |
| PSL(2, R) Adjoint real group of type A₁ | Group Lie theory | dimℝ 3 | reference Sourced record |
| Universal cover of SL(2, R) Simply connected group integrating sl(2,R) | Group Lie theory | dimℝ 3 | reference Sourced record |
| SL(3, R) Real special linear group in three dimensions | Group Lie theory | dimℝ 8 | reference Sourced record |
| SU(2,2) Indefinite special unitary group of signature (2,2) | Group Lie theory | dimℝ 15 | reference Sourced record |
| Sp(4, R) Real symplectic group on a four-dimensional space | Group Lie theory | dimℝ 10 | reference Sourced record |
| SO₀(1,3) Proper orthochronous Lorentz group in four dimensions | Group Lie theory | dimℝ 6 | reference Sourced record |
| SO₀(2,4) Identity component of the orthogonal group of signature (2,4) | Group Lie theory | dimℝ 15 | reference Sourced record |
| SO₀(2,3) Identity component of the orthogonal group of signature (2,3) | Group Lie theory | dimℝ 10 | reference Sourced record |
| SU(2) → SO(3) Spin cover of three-dimensional rotations | Map Lie theory | SO(3) | executable 1 linked workbench |
| SU(3) → PSU(3) Adjoint quotient of SU(3) | Map Lie theory | PSU(3) | executable 1 linked workbench |
| SU(2) ↪ SU(3) Regular block inclusion in SU(3) | Map Lie theory | {diag(A,1) : A ∈ SU(2)} ≅ SU(2) | executable 2 linked workbenchs |
| SU(2) → SU(3), spin 1 Principal spin-one homomorphism into SU(3) | Map Lie theory | An SU(3)-conjugate of the real SO(3) subgroup | executable 2 linked workbenchs |
| U(1) ↪ SU(2) Normalized maximal torus of SU(2) | Map Lie theory | {diag(z,z⁻¹) : |z|=1} | executable 1 linked workbench |
| U(1) ≅ SO(2) ↪ SO(3) Normalized axial rotation subgroup of SO(3) | Map Lie theory | Rotations fixing a chosen oriented axis, isomorphic to SO(2) | executable 1 linked workbench |
| Spin(4) → SO(4) Quaternionic spin cover in four Euclidean dimensions | Map Lie theory | SO(4) | reference Sourced record |
| SU(4) → SO(6) Exterior-square realization of the Spin(6) cover | Map Lie theory | SO(6) | reference Sourced record |
| SU(4) → PSU(4) Full central quotient of SU(4) | Map Lie theory | PSU(4) | reference Sourced record |
| Sp(2) → SO(5) Quaternionic construction of the Spin(5) cover | Map Lie theory | SO(5) | reference Sourced record |
| Spin(8) → SO(8) Vector spin cover in eight Euclidean dimensions | Map Lie theory | SO(8) | reference Sourced record |
| Spin(10) → SO(10) Vector spin cover in ten Euclidean dimensions | Map Lie theory | SO(10) | reference Sourced record |
| SL(2, C) → SO₀(1,3) Hermitian-matrix Lorentz spin cover | Map Lie theory | SO₀(1,3) | reference Sourced record |
| SL(2, R) → PSL(2, R) Adjoint central quotient of SL(2,R) | Map Lie theory | PSL(2,R) ≅ SO₀(2,1) | reference Sourced record |
| Universal SL(2,R) cover → SL(2,R) Universal covering map of SL(2,R) | Map Lie theory | SL(2,R) | reference Sourced record |
| SU(2,2) → SO₀(2,4) Exterior-square spin cover in signature (2,4) | Map Lie theory | SO₀(2,4) | reference Sourced record |
| Sp(4, R) → SO₀(2,3) Primitive-exterior-square spin cover in signature (2,3) | Map Lie theory | SO₀(2,3) | reference Sourced record |
| SU(2)/U(1) ≅ S² Hopf sphere as SU(2)/U(1) | Space Lie theory | Reference | executable 1 linked workbench |
| SO(3)/SO(2) ≅ S² Rotation sphere as SO(3)/SO(2) | Space Lie theory | Reference | executable 1 linked workbench |
| 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 |
| A1 · (1) Doublet of SU(2) | Representation Lie theory | dimℂ 2 | executable 2 linked workbenchs |
| A1 · (2) Adjoint of SU(2) | Representation Lie theory | dimℂ 3 | executable 2 linked workbenchs |
| A2 · (1,0) Fundamental of SU(3) | Representation Lie theory | dimℂ 3 | executable 5 linked workbenchs |
| A2 · (0,1) Conjugate fundamental of SU(3) | Representation Lie theory | dimℂ 3 | executable 5 linked workbenchs |
| A2 · (1,1) Adjoint of SU(3) | Representation Lie theory | dimℂ 8 | executable 5 linked workbenchs |
| A2 · (2,0) Symmetric sextet of SU(3) | Representation Lie theory | dimℂ 6 | executable 5 linked workbenchs |
| A2 · (3,0) Decuplet of SU(3) | Representation Lie theory | dimℂ 10 | executable 5 linked workbenchs |
| A2 · (2,2) 27-dimensional irrep of SU(3) | Representation Lie theory | dimℂ 27 | executable 5 linked workbenchs |
| A3 · (1,0,0) Fundamental of SU(4) | Representation Lie theory | dimℂ 4 | executable 3 linked workbenchs |
| A3 · (0,1,0) Two-index antisymmetric of SU(4) | Representation Lie theory | dimℂ 6 | executable 3 linked workbenchs |
| A3 · (1,0,1) Adjoint of SU(4) | Representation Lie theory | dimℂ 15 | executable 3 linked workbenchs |
| A4 · (1,0,0,0) Fundamental of SU(5) | Representation Lie theory | dimℂ 5 | executable 3 linked workbenchs |
| A4 · (0,1,0,0) Two-index antisymmetric of SU(5) | Representation Lie theory | dimℂ 10 | executable 3 linked workbenchs |
| A4 · (1,0,0,1) Adjoint of SU(5) | Representation Lie theory | dimℂ 24 | executable 3 linked workbenchs |
| C2 · (1,0) Defining representation of Sp(2) = USp(4) | Representation Lie theory | dimℂ 4 | executable 1 linked workbench |
| C2 · (0,1) Five-dimensional irrep of Sp(2) = USp(4) | Representation Lie theory | dimℂ 5 | executable 1 linked workbench |
| D4 · (1,0,0,0) Vector of Spin(8) | Representation Lie theory | dimℂ 8 | executable 1 linked workbench |
| D4 · (0,0,1,0) Half-spin representation of Spin(8) | Representation Lie theory | dimℂ 8 | executable 1 linked workbench |
| D4 · (0,0,0,1) Other half-spin representation of Spin(8) | Representation Lie theory | dimℂ 8 | executable 1 linked workbench |
| D5 · (1,0,0,0,0) Vector of Spin(10) | Representation Lie theory | dimℂ 10 | executable 1 linked workbench |
| D5 · (0,0,0,0,1) Chiral spinor of Spin(10) | Representation Lie theory | dimℂ 16 | executable 1 linked workbench |
| D5 · (0,0,0,1,0) Conjugate chiral spinor of Spin(10) | Representation Lie theory | dimℂ 16 | executable 1 linked workbench |
| D5 · (0,1,0,0,0) Adjoint of Spin(10) | Representation Lie theory | dimℂ 45 | executable 1 linked workbench |
| G2 · (1,0) Fundamental of G2 (compact) | Representation Lie theory | dimℂ 7 | executable 1 linked workbench |
| G2 · (0,1) Adjoint of G2 (compact) | Representation Lie theory | dimℂ 14 | executable 1 linked workbench |
| F4 · (0,0,0,1) Minimal representation of F4 (compact) | Representation Lie theory | dimℂ 26 | executable 1 linked workbench |
| F4 · (1,0,0,0) Adjoint of F4 (compact) | Representation Lie theory | dimℂ 52 | executable 1 linked workbench |
| E6 · (1,0,0,0,0,0) 27-dimensional representation of E6 (compact simply connected) | Representation Lie theory | dimℂ 27 | executable 1 linked workbench |
| E6 · (0,1,0,0,0,0) Adjoint of E6 (compact simply connected) | Representation Lie theory | dimℂ 78 | executable 1 linked workbench |
| E7 · (0,0,0,0,0,0,1) 56-dimensional representation of E7 (compact simply connected) | Representation Lie theory | dimℂ 56 | executable 1 linked workbench |
| E7 · (1,0,0,0,0,0,0) Adjoint of E7 (compact simply connected) | Representation Lie theory | dimℂ 133 | executable 1 linked workbench |
| E8 · (0,0,0,0,0,0,0,1) Adjoint of E8 (compact simply connected) | Representation Lie theory | dimℂ 248 | executable 1 linked workbench |
| su(3) ⊕ su(2) ⊕ u(1) Compact Standard Model gauge algebra | Lie algebra Lie theory | dimℝ 12 | reference Sourced record |
| SU(3) × SU(2) × U(1) Standard Model compact covering product | Group Lie theory | dimℝ 12 | executable 3 linked workbenchs |
| G_SM^(2) Standard Model global group with diagonal Z2 quotient | Group Lie theory | dimℝ 12 | executable 3 linked workbenchs |
| G_SM^(3) Standard Model global group with diagonal Z3 quotient | Group Lie theory | dimℝ 12 | executable 3 linked workbenchs |
| G_SM^(6) Standard Model global group with diagonal Z6 quotient | Group Lie theory | dimℝ 12 | executable 3 linked workbenchs |
| G_SM^(1) → G_SM^(2) Standard Model quotient map of order 2 | Map Lie theory | G_SM^(2) | executable 1 linked workbench |
| G_SM^(1) → G_SM^(3) Standard Model quotient map of order 3 | Map Lie theory | G_SM^(3) | executable 1 linked workbench |
| G_SM^(1) → G_SM^(6) Standard Model quotient map of order 6 | Map Lie theory | G_SM^(6) | executable 1 linked workbench |
| SU(3) × SU(2) × U(1)_X → SU(5) Color-first SU(5) block homomorphism | Map Lie theory | S(U(3) × U(2)) | executable 1 linked workbench |
| G_SM^(6) ↪ SU(5) The diagonal-Z6 Standard Model group inside SU(5) | Map Lie theory | S(U(3) × U(2)) | reference Sourced record |
Coverage is specific. Classification identifies a mathematical object or family. Reference supplies stated properties. Executable links to a bounded calculation. Read the coverage policy.