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.

Read the conventions and classification

46 released records · classifications and individual objects are labeled separately.

Exceptional Lie types
Object	Kind	Essential data	Coverage
$G_{2}(q)$ Finite simple groups of type G₂	Family Finite	Parameterized family	classification Sourced record
$F_{4}(q)$ Finite simple groups of type F₄	Family Finite	Parameterized family	classification Sourced record
$E_{6}(q)$ Finite simple groups of type E₆	Family Finite	Parameterized family	classification Sourced record
$E_{7}(q)$ Finite simple groups of type E₇	Family Finite	Parameterized family	classification Sourced record
$E_{8}(q)$ Finite simple groups of type E₈	Family Finite	Parameterized family	classification Sourced record
${}^{2}\!E_{6}(q)$ Finite simple groups of type ²E₆	Family Finite	Parameterized family	classification Sourced record
${}^{2}\!G_{2}(q)$ Small Ree simple groups	Family Finite	Parameterized family	classification Sourced record
${}^{2}\!F_{4}(q)$ Large Ree simple groups	Family Finite	Parameterized family	classification Sourced record
$2T,\quad 2O,\quad 2I$ Exceptional binary polyhedral groups	Family Finite	Parameterized family	reference Sourced record
${}^{2}\!F_{4}(2)'$ Tits group	Group Finite	order 17971200	classification Sourced record
$\mathrm{E}_{6}$ Simple Lie type E₆	Family Lie theory	Parameterized family	classification Sourced record
$\mathrm{E}_{7}$ Simple Lie type E₇	Family Lie theory	Parameterized family	classification Sourced record
$\mathrm{E}_{8}$ Simple Lie type E₈	Family Lie theory	Parameterized family	classification Sourced record
$\mathrm{F}_{4}$ Simple Lie type F₄	Family Lie theory	Parameterized family	classification Sourced record
$\mathrm{G}_{2}$ Simple Lie type G₂	Family Lie theory	Parameterized family	classification Sourced record
$\mathfrak{e}_{6}(\mathbb C)$ e₆(C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 78	reference Sourced record
$\mathfrak{e}_{7}(\mathbb C)$ e₇(C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 133	reference Sourced record
$\mathfrak{e}_{8}(\mathbb C)$ e₈(C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 248	reference Sourced record
$\mathfrak{f}_{4}(\mathbb C)$ f₄(C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 52	reference Sourced record
$\mathfrak{g}_{2}(\mathbb C)$ g₂(C) — complex Lie algebra	Lie algebra Lie theory	dimℂ 14	reference Sourced record
$\mathfrak{e}_{6}\;\text{compact}$ e₆ compact — real Lie algebra	Lie algebra Lie theory	dimℝ 78	reference Sourced record
$\mathfrak{e}_{7}\;\text{compact}$ e₇ compact — real Lie algebra	Lie algebra Lie theory	dimℝ 133	reference Sourced record
$\mathfrak{e}_{8}\;\text{compact}$ e₈ compact — real Lie algebra	Lie algebra Lie theory	dimℝ 248	reference Sourced record
$\mathfrak{f}_{4}\;\text{compact}$ f₄ compact — real Lie algebra	Lie algebra Lie theory	dimℝ 52	reference Sourced record
$\mathfrak{g}_{2}\;\text{compact}$ g₂ compact — real Lie algebra	Lie algebra Lie theory	dimℝ 14	reference Sourced record
$\mathfrak{e}_{6}\;\text{compact}$ e₆ compact real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{e}_{7}\;\text{compact}$ e₇ compact real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{e}_{8}\;\text{compact}$ e₈ compact real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{f}_{4}\;\text{compact}$ f₄ compact real form	Real form Lie theory	Compact	reference Sourced record
$\mathfrak{g}_{2}\;\text{compact}$ g₂ compact real form	Real form Lie theory	Compact	reference Sourced record
$\mathrm{G}_{2}\;\text{compact}$ Compact group of octonion automorphisms	Group Lie theory	dimℝ 14	executable 1 linked workbench
$\mathrm{F}_{4}\;\text{compact}$ Compact exceptional group of type F₄	Group Lie theory	dimℝ 52	executable 1 linked workbench
$\mathrm{E}_{6}\;\text{compact, simply connected}$ Simply connected compact exceptional group of type E₆	Group Lie theory	dimℝ 78	executable 1 linked workbench
$\mathrm{E}_{7}\;\text{compact, simply connected}$ Simply connected compact exceptional group of type E₇	Group Lie theory	dimℝ 133	executable 1 linked workbench
$\mathrm{E}_{8}\;\text{compact}$ Compact exceptional group of type E₈	Group Lie theory	dimℝ 248	executable 1 linked workbench
$\mathrm{E}_{6}\;\text{compact, adjoint}$ Adjoint compact exceptional group of type E₆	Group Lie theory	dimℝ 78	reference Sourced record
$\mathrm{E}_{7}\;\text{compact, adjoint}$ Adjoint compact exceptional group of type E₇	Group Lie theory	dimℝ 133	reference Sourced record
$G_{2}:\quad V_{(1,0)}$ Fundamental of G2 (compact)	Representation Lie theory	dimℂ 7	executable 1 linked workbench
$G_{2}:\quad V_{(0,1)}$ Adjoint of G2 (compact)	Representation Lie theory	dimℂ 14	executable 1 linked workbench
$F_{4}:\quad V_{(0,0,0,1)}$ Minimal representation of F4 (compact)	Representation Lie theory	dimℂ 26	executable 1 linked workbench
$F_{4}:\quad V_{(1,0,0,0)}$ Adjoint of F4 (compact)	Representation Lie theory	dimℂ 52	executable 1 linked workbench
$E_{6}:\quad V_{(1,0,0,0,0,0)}$ 27-dimensional representation of E6 (compact simply connected)	Representation Lie theory	dimℂ 27	executable 1 linked workbench
$E_{6}:\quad V_{(0,1,0,0,0,0)}$ Adjoint of E6 (compact simply connected)	Representation Lie theory	dimℂ 78	executable 1 linked workbench
$E_{7}:\quad V_{(0,0,0,0,0,0,1)}$ 56-dimensional representation of E7 (compact simply connected)	Representation Lie theory	dimℂ 56	executable 1 linked workbench
$E_{7}:\quad V_{(1,0,0,0,0,0,0)}$ Adjoint of E7 (compact simply connected)	Representation Lie theory	dimℂ 133	executable 1 linked workbench
$E_{8}:\quad V_{(0,0,0,0,0,0,0,1)}$ Adjoint of E8 (compact simply connected)	Representation Lie theory	dimℂ 248	executable 1 linked workbench