Curated collection
Exceptional Lie types
The compact exceptional groups, their complex algebras and real forms, and the finite Lie-type families are distinct objects with related root data.
46 released records · classifications and individual objects are labeled separately.
| Object | Kind | Essential data | Coverage |
|---|---|---|---|
| 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 |
| 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 |
| 2T · 2O · 2I Exceptional binary polyhedral groups | Family Finite | Parameterized family | reference Sourced record |
| ²F₄(2)′ Tits group | Group Finite | order 17971200 | 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |