Sporadic simple groups

The 26 finite simple groups that belong to none of the infinite families. They are the exceptional cases of the Classification of Finite Simple Groups: everything else is cyclic of prime order, alternating, or of Lie type.

Definition

Every finite simple group is isomorphic to exactly one of:

• Cₚ — cyclic of prime order,
• Aₙ — alternating, n ≥ 5,
• a member of one of the 16 infinite families of Lie type over a finite field (PSL(n,𝔽q), PSp(2n,𝔽q), PSU(n,𝔽q), Ω, and the exceptional types),
• one of 26 sporadic groups.

A simple group is sporadic when it lies in none of the infinite families. The number 26 is final: the classification leaves no room for a 27th.

The 26 groups

Exact orders:

M11       7920                       Fi22      64561751654400
M12       95040                      Fi23      4089470473293004800
M22       443520                     Fi24'     1255205709190661721292800
M23       10200960                   HS        44352000
M24       244823040                  McL       898128000
J1        175560                     He        4030387200
J2        604800                     Ru        145926144000
J3        50232960                   Suz       448345497600
J4        86775571046077562880       O'N       460815505920
Co1       4157776806543360000        HN        273030912000000
Co2       42305421312000             Ly        51765179004000000
Co3       495766656000               Th        90745943887872000

B         4154781481226426191177580544000000
M         808017424794512875886459904961710757005754368000000000

The happy family and the pariahs

20 of the 26 are subquotients of the Monster — Robert Griess's happy family — in three generations:

1. Mathieu groups M11, M12, M22, M23, M24 — inside M24.
2. Leech lattice groups Co1, Co2, Co3, HS, McL, Suz, J2.
3. Monster sections M, B, Fi24', Fi23, Fi22, Th, HN, He.

The remaining 6 are the pariahs, which are not involved in the Monster: J1, J3, J4, Ru, O'N, Ly.

An order names a sporadic group

The 26 orders are pairwise distinct, and no other finite simple group shares one. Only two order coincidences occur among all finite simple groups:

• A₈ ≅ PSL(4,𝔽2) and PSL(3,𝔽4), both of order 20160;
• PSp(2n,𝔽q) and Ω(2n+1,𝔽q) for odd q and n ≥ 3.

Neither involves a sporadic order. So a finite simple group whose order is one of the 26 is that sporadic group — for instance, a simple group of order 7920 is M11.

Shared properties

• Non-abelian, hence perfect and never solvable. Each is its own commutator subgroup, with trivial center.
• Small outer automorphism group. |Out(G)| ≤ 2 for all 26 — trivial for 14 of them and C₂ for the other 12. So each sits in at most one proper almost simple group G.2 over it, and the 14 with trivial Out are complete groups: Aut(G) = G.
• Even order with at least three prime divisors, as for every non-abelian simple group (Feit–Thompson, Burnside). J1 uses six primes at order 175560; the Monster uses fifteen.
• Smallest and largest. M11 at order 7920; the Monster at ≈ 8.08 × 10⁵³.
• Representation size. The Monster's smallest faithful complex representation has dimension 196883, and its smallest faithful permutation action (minimal faithful degree) has degree 97239461142009186000.

The Tits group

²F₄(2)', of order 17971200 = 2¹¹·3³·5²·13, is the derived subgroup of index 2 in ²F₄(2). It is simple, and some authors count it as a 27th sporadic group because it is not a member of the family ²F₄(2²ⁿ⁺¹), n ≥ 1. The usual convention — and the one used here — classifies it as a group of Lie type, leaving 26 sporadic groups.

In this database

M11 is the only sporadic group within reach of the catalog: it has order 7920, while the next smallest, M12, already has order 95040. M11 is complete — Z(M11) = 1 and every automorphism is inner — so its automorphism group is M11 itself and its outer automorphism group is trivial.

See also

• Simple group — the classification these 26 complete.
• Almost simple group — S ≤ G ≤ Aut(S); over a sporadic S there is at most one proper such G.
• Alternating group and matrix groups — the infinite families.
• Perfect group, quasisimple group — the covers n·G a sporadic group's Schur multiplier allows.
• Wikipedia: Sporadic group
• Wikipedia: List of finite simple groups
• Wikipedia: Monster group
• Wikipedia: Mathieu group M11