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Finite groups.

Character theory, discrete symmetries, finite-simple classification, and the maps that connect finite and continuous groups.

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49finite group records41Lie group records38Lie algebras22specified maps

103 released records. Individual groups and parameterized families are counted separately.

Mathematical records
Object	Kind	Essential data	Coverage
$\mathrm{C}_{3}$ Cyclic group of order three	Group Finite	order 3	executable 1 linked workbench
$\mathrm{S}_{3}$ Symmetric group on three letters	Group Finite	order 6	executable 1 linked workbench
$\mathrm{A}_{4}$ Alternating group on four letters	Group Finite	order 12	executable 2 linked workbenchs
$\mathrm{S}_{4}$ Symmetric group on four letters	Group Finite	order 24	executable 1 linked workbench
$\mathrm{A}_{5}$ Alternating group on five letters	Group Finite	order 60	executable 1 linked workbench
$\mathrm{Q}_{8}$ Quaternion group	Group Finite	order 8	executable 1 linked workbench
$\mathrm{D}_{8}$ Dihedral group of order eight	Group Finite	order 8	executable 1 linked workbench
$\mathrm{C}_{p}$ Cyclic groups of prime order	Family Finite	Parameterized family	classification Sourced record
$\mathrm{A}_{n}$ Simple alternating groups	Family Finite	Parameterized family	classification Sourced record
$\mathrm{PSL}(n,q)$ Projective special linear finite simple groups	Family Finite	Parameterized family	classification Sourced record
$\mathrm{PSU}(n,q)$ Projective special unitary finite simple groups	Family Finite	Parameterized family	classification Sourced record
$\mathrm{PSp}(2n,q)$ Projective symplectic finite simple groups	Family Finite	Parameterized family	classification Sourced record
$\mathrm P\Omega(2n+1,q)$ Odd-dimensional orthogonal finite simple groups	Family Finite	Parameterized family	classification Sourced record
$\mathrm P\Omega^{+}(2n,q)$ Plus-type even orthogonal finite simple groups	Family Finite	Parameterized family	classification Sourced record
$\mathrm P\Omega^{-}(2n,q)$ Minus-type even orthogonal finite simple groups	Family Finite	Parameterized family	classification Sourced record
$G_{2}(q)$ 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
${}^{3}\!D_{4}(q)$ Finite simple groups of type ³D₄	Family Finite	Parameterized family	classification Sourced record
${}^{2}\!B_{2}(q)$ Suzuki simple groups	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
Sporadic The 26 sporadic finite simple groups	Family Finite	Parameterized family	classification Sourced record
$\mathrm{C}_{n}$ Finite cyclic groups	Family Finite	Parameterized family	reference Sourced record
$\mathrm{D}_{2n}$ Finite dihedral groups	Family Finite	Parameterized family	reference Sourced record
$\mathrm{Dic}_{n}$ Dicyclic groups	Family Finite	Parameterized family	reference Sourced record
$2T,\quad 2O,\quad 2I$ Exceptional binary polyhedral groups	Family Finite	Parameterized family	reference Sourced record
$\mathrm{S}_{n}$ Finite symmetric groups	Family Finite	Parameterized family	reference Sourced record
$C_{2}$ Cyclic group of order 2	Group Finite	order 2	reference Sourced record
$C_{4}$ Cyclic group of order 4	Group Finite	order 4	reference Sourced record
$C_{5}$ Cyclic group of order 5	Group Finite	order 5	reference Sourced record
$C_{6}$ Cyclic group of order 6	Group Finite	order 6	reference Sourced record
$\mathrm{V}_{4}$ Klein four-group	Group Finite	order 4	reference Sourced record
$\mathrm{D}_{10}$ Dihedral group of order ten	Group Finite	order 10	reference Sourced record
$\mathrm{D}_{12}$ Dihedral group of order twelve	Group Finite	order 12	reference Sourced record
$\mathrm{Dic}_{3}$ Dicyclic group of order twelve	Group Finite	order 12	reference Sourced record
$\mathrm{Q}_{16}$ 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
$\mathrm{PSL}(2,7)$ Projective special linear group PSL(2,7)	Group Finite	order 168	reference Sourced record
$\Delta(27)$ Delta(27), the finite Heisenberg group	Group Finite	order 27	reference Sourced record
$\mathrm{A}_{6}$ Alternating group on six letters	Group Finite	order 360	reference Sourced record
$\mathrm{M}_{11}$ Mathieu group M11	Group Finite	order 7920	classification Sourced record
$\mathrm{M}_{12}$ Mathieu group M12	Group Finite	order 95040	classification Sourced record
$\mathrm{M}_{22}$ Mathieu group M22	Group Finite	order 443520	classification Sourced record
$\mathrm{M}_{23}$ Mathieu group M23	Group Finite	order 10200960	classification Sourced record
$\mathrm{M}_{24}$ Mathieu group M24	Group Finite	order 244823040	classification Sourced record
$\mathrm{J}_{1}$ First Janko group	Group Finite	order 175560	classification Sourced record
$\mathrm{J}_{2}$ Hall–Janko group	Group Finite	order 604800	classification Sourced record
$\mathrm{J}_{3}$ Third Janko group	Group Finite	order 50232960	classification Sourced record
$\mathrm{J}_{4}$ Fourth Janko group	Group Finite	order 86775571046077562880	classification Sourced record
$\mathrm{Co}_{1}$ First Conway group	Group Finite	order 4157776806543360000	classification Sourced record
$\mathrm{Co}_{2}$ Second Conway group	Group Finite	order 42305421312000	classification Sourced record
$\mathrm{Co}_{3}$ Third Conway group	Group Finite	order 495766656000	classification Sourced record
$\mathrm{Fi}_{22}$ Fischer group Fi22	Group Finite	order 64561751654400	classification Sourced record
$\mathrm{Fi}_{23}$ Fischer group Fi23	Group Finite	order 4089470473293004800	classification Sourced record
$\mathrm{Fi}_{24}'$ Fischer simple group Fi24 prime	Group Finite	order 1255205709190661721292800	classification Sourced record
$\mathrm{HS}$ Higman–Sims group	Group Finite	order 44352000	classification Sourced record
$\mathrm{McL}$ McLaughlin group	Group Finite	order 898128000	classification Sourced record
$\mathrm{He}$ Held group	Group Finite	order 4030387200	classification Sourced record
$\mathrm{Ru}$ Rudvalis group	Group Finite	order 145926144000	classification Sourced record
$\mathrm{Suz}$ Sporadic Suzuki group	Group Finite	order 448345497600	classification Sourced record
$\mathrm{O'N}$ O’Nan group	Group Finite	order 460815505920	classification Sourced record
$\mathrm{HN}$ Harada–Norton group	Group Finite	order 273030912000000	classification Sourced record
$\mathrm{Ly}$ Lyons group	Group Finite	order 51765179004000000	classification Sourced record
$\mathrm{Th}$ Thompson group	Group Finite	order 90745943887872000	classification Sourced record
$\mathrm{B}$ Baby Monster group	Group Finite	order 4154781481226426191177580544000000	classification Sourced record
$\mathrm{M}$ Monster group	Group Finite	order 808017424794512875886459904961710757005754368000000000	classification Sourced record
${}^{2}\!F_{4}(2)'$ Tits group	Group Finite	order 17971200	classification Sourced record
$C_3:\quad \mathbf{1}$ C₃ irreducible representation 1	Representation Finite	dimℂ 1	executable 1 linked workbench
$C_3:\quad \chi$ C₃ irreducible representation χ	Representation Finite	dimℂ 1	executable 1 linked workbench
$C_3:\quad \chi^2$ C₃ irreducible representation χ²	Representation Finite	dimℂ 1	executable 1 linked workbench
$S_3:\quad \mathbf{1}$ S₃ irreducible representation 1	Representation Finite	dimℂ 1	executable 1 linked workbench
$S_3:\quad \mathrm{sgn}$ S₃ irreducible representation sgn	Representation Finite	dimℂ 1	executable 1 linked workbench
$S_3:\quad \mathbf{2}$ S₃ irreducible representation 2	Representation Finite	dimℂ 2	executable 1 linked workbench
$D_8:\quad \mathbf{1}$ D₈ irreducible representation 1	Representation Finite	dimℂ 1	executable 1 linked workbench
$D_8:\quad a$ D₈ irreducible representation a	Representation Finite	dimℂ 1	executable 1 linked workbench
$D_8:\quad b$ D₈ irreducible representation b	Representation Finite	dimℂ 1	executable 1 linked workbench
$D_8:\quad c$ D₈ irreducible representation c	Representation Finite	dimℂ 1	executable 1 linked workbench
$D_8:\quad \mathbf{2}$ D₈ irreducible representation 2	Representation Finite	dimℂ 2	executable 1 linked workbench
$Q_8:\quad \mathbf{1}$ Q₈ irreducible representation 1	Representation Finite	dimℂ 1	executable 1 linked workbench
$Q_8:\quad a$ Q₈ irreducible representation a	Representation Finite	dimℂ 1	executable 1 linked workbench
$Q_8:\quad b$ Q₈ irreducible representation b	Representation Finite	dimℂ 1	executable 1 linked workbench
$Q_8:\quad c$ Q₈ irreducible representation c	Representation Finite	dimℂ 1	executable 1 linked workbench
$Q_8:\quad \mathbf{2}$ Q₈ irreducible representation 2	Representation Finite	dimℂ 2	executable 1 linked workbench
$A_4:\quad \mathbf{1}$ A₄ irreducible representation 1	Representation Finite	dimℂ 1	executable 1 linked workbench
$A_4:\quad \mathbf{1}'$ A₄ irreducible representation 1′	Representation Finite	dimℂ 1	executable 1 linked workbench
$A_4:\quad \mathbf{1}''$ A₄ irreducible representation 1″	Representation Finite	dimℂ 1	executable 1 linked workbench
$A_4:\quad \mathbf{3}$ A₄ irreducible representation 3	Representation Finite	dimℂ 3	executable 1 linked workbench
$S_4:\quad \mathbf{1}$ S₄ irreducible representation 1	Representation Finite	dimℂ 1	executable 1 linked workbench
$S_4:\quad \mathrm{sgn}$ S₄ irreducible representation sgn	Representation Finite	dimℂ 1	executable 1 linked workbench
$S_4:\quad \mathbf{2}$ S₄ irreducible representation 2	Representation Finite	dimℂ 2	executable 1 linked workbench
$S_4:\quad \mathbf{3}$ S₄ irreducible representation 3	Representation Finite	dimℂ 3	executable 1 linked workbench
$S_4:\quad \mathbf{3}'$ S₄ irreducible representation 3′	Representation Finite	dimℂ 3	executable 1 linked workbench
$A_5:\quad \mathbf{1}$ A₅ irreducible representation 1	Representation Finite	dimℂ 1	executable 1 linked workbench
$A_5:\quad \mathbf{3}$ A₅ irreducible representation 3	Representation Finite	dimℂ 3	executable 1 linked workbench
$A_5:\quad \mathbf{3}'$ A₅ irreducible representation 3′	Representation Finite	dimℂ 3	executable 1 linked workbench
$A_5:\quad \mathbf{4}$ A₅ irreducible representation 4	Representation Finite	dimℂ 4	executable 1 linked workbench
$A_5:\quad \mathbf{5}$ 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.

Finite-simple classification Lie classification