Curated collection

Finite simple groups

Follow the classification through prime-order cyclic groups, alternating groups, finite groups of Lie type, and the sporadic groups. Low parameters and exceptional identifications are part of each family.

Read the conventions and classification

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

Finite simple groups
Object	Kind	Essential data	Coverage
$\mathrm{C}_{3}$ Cyclic group of order three	Group Finite	order 3	executable 1 linked workbench
$\mathrm{A}_{5}$ Alternating group on five letters	Group Finite	order 60	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
$C_{2}$ Cyclic group of order 2	Group Finite	order 2	reference Sourced record
$C_{5}$ Cyclic group of order 5	Group Finite	order 5	reference Sourced record
$\mathrm{PSL}(2,7)$ Projective special linear group PSL(2,7)	Group Finite	order 168	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