reëntrantly

Sporadic groups

(Click on a column heading to sort.)

G |G| ⌊log2|G|⌋ ω(|G|) Ω(|G|) ν2(|G|) gpf(|G|) n! ∣ ∣ n! λ(G) l(G) k(G) Out(G) M(G) min perm deg min irrep min proj irrep min mod irrep min mod proj irrep Year predicted Year constructed
M11 Mathieu group of degree 11 7920 24⋅32⋅5⋅11 12 4 8 4 11 6 11 4 7 10 1 1 11 10 10 53 53 1861 1938
M12 Mathieu group of degree 12 95040 26⋅33⋅5⋅11 16 4 11 6 11 6 11 4 8 15 C2 C2 12 11 11 102,3 63 1861 1938
M22 Mathieu group of degree 22 443520 27⋅32⋅5⋅7⋅11 18 5 12 7 11 8 11 4 10 12 C2 C12 22 21 10 102 62 1861 1938
M23 Mathieu group of degree 23 10200960 27⋅32⋅5⋅7⋅11⋅23 23 6 13 7 23 8 23 3 11 17 1 1 23 22 22 112 112 1861 1938
M24 Mathieu group of degree 24 244823040 210⋅33⋅5⋅7⋅11⋅23 27 6 17 10 23 8 23 4 14 26 1 1 24 23 23 112 112 1861 1938
J1, Ja, J First Janko group 175560 23⋅3⋅5⋅7⋅11⋅19 17 6 8 3 19 5 19 4 6 15 1 1 266 56 56 711 711 1965 1965
J2, HJ, F5−, HaJ, HaJW Second Janko group, Hall‐Janko group 604800 27⋅33⋅52⋅7 19 4 13 7 7 8 10 4 10 21 C2 C2 100 14 6 62 60 1965 1968
J3, HJM, HJMcK Third Janko group, Higman‐Janko‐McKay group 50232960 27⋅35⋅5⋅17⋅19 25 5 15 7 19 6 19 5 10 21 C2 C3 6156 85 18 183 92 1965 1969
J4 Fourth Janko group 86775571046077562880 221⋅33⋅5⋅7⋅113⋅23⋅29⋅31⋅37⋅43 66 10 34 21 43 6 19 4 26 62 1 1 173067389 1333 1333 1122 1122 1976 1980
HS, HiS, ·233 Higman‐Sims group 44352000 29⋅32⋅53⋅7⋅11 25 5 16 9 11 8 15 5 12 24 C2 C2 100 22 22 202 202 1968 1968
McL, Mc, ·223 McLaughlin group 898128000 27⋅36⋅53⋅7⋅11 29 5 18 7 11 9 15 5 12 24 C2 C3 275 22 22 213,5 213,5 1969 1969
He, F7, F7+, H, HHM, HHMcK Held group 4030387200 210⋅33⋅52⋅73⋅17 31 5 19 10 17 8 21 6 13 33 C2 1 2058 51 51 507 507 1969 1973
Ru, Ruv, Rv, Rud, R, RCW Rudvalis group 145926144000 214⋅33⋅53⋅7⋅13⋅29 37 6 23 14 29 8 29 5 17 36 1 C2 4060 378 28 282 280 1973 1973
Suz, Sz, F3− Suzuki group 448345497600 213⋅37⋅52⋅7⋅11⋅13 38 6 25 13 13 13 18 4 17 43 C2 C6 1782 143 12 643 120 1969 1969
O’N, ON, O’S, O’NS O’Nan group, O’Nan‐Sims group 460815505920 29⋅34⋅5⋅73⋅11⋅19⋅31 38 7 20 9 31 9 31 5 13 30 C2 C3 122760 10944 342 1543 457 1976 1976
Co3, C3, ·3 Third Conway group 495766656000 210⋅37⋅53⋅7⋅11⋅23 38 6 23 10 23 12 23 4 14 42 1 1 276 23 23 222,3 222,3 1968 1968
Co2, C2, ·2 Second Conway group 42305421312000 218⋅36⋅53⋅7⋅11⋅23 45 6 30 18 23 12 23 4 22 60 1 1 2300 23 23 222 222 1968 1968
Co1, F2−, C1, ·1 First Conway group 4157776806543360000 221⋅39⋅54⋅72⋅11⋅13⋅23 61 7 39 21 23 16 24 5 26 101 1 C2 98280 276 24 242 240 1968 1968
HN, F5+, F5, F, HaN, HaNo, HaCNS Harada‐Norton group 273030912000000 214⋅36⋅56⋅7⋅11⋅19 47 6 29 14 19 12 25 5 19 54 C2 1 1140000 133 133 1322 1322 1975 1975
Ly, LyS Lyons group, Lyons‐Sims group 51765179004000000 28⋅37⋅56⋅7⋅11⋅31⋅37⋅67 55 8 26 8 67 11 67 4 15 53 1 1 8835156 2480 2480 1115 1115 1972 1973
Th, F3|3, F3, E, T Thompson group 90745943887872000 215⋅310⋅53⋅72⋅13⋅19⋅31 56 7 33 15 31 10 31 4 20 48 1 1 143127000 248 248 2480 2480 1976 1976
Fi22, F22, M(22) First Fischer group 64561751654400 217⋅39⋅52⋅7⋅11⋅13 45 6 31 17 13 13 21 5 21 65 C2 C6 3510 78 78 773 272 1969 1970
Fi23, F23, M(23) Second Fischer group 4089470473293004800 218⋅313⋅52⋅7⋅11⋅13⋅17⋅23 61 8 38 18 23 14 36 4 25 98 1 1 31671 782 782 2533 2533 1969 1970
Fi24′, F24′, F3+, M(24)′ Third Fischer group 1255205709190661721292800 221⋅316⋅52⋅73⋅11⋅13⋅17⋅23⋅29 80 9 47 21 29 14 36 4 28 108 C2 C3 306936 8671 783 7813 7813 1969 1970
B, F2+, F2, 𝔹, F, FLS Baby monster group 4154781481226426191177580544000000 241⋅313⋅56⋅72⋅11⋅13⋅17⋅19⋅23⋅31⋅47 111 11 69 41 47 13 47 3 46 184 1 C2 13571955000 4371 4371 43702 ? 1973 1977
M, F1, FG, 𝕄 Monster group 808017424794512875886459904961710757005754368000000000 246⋅320⋅59⋅76⋅112⋅133⋅17⋅19⋅23⋅29⋅31⋅41⋅47⋅59⋅71 179 15 95 46 71 32 71 4 52 194 1 1 97239461142009186000 196883 196883 1968822,3 1968822,3 1973 1982
T, 2F4(2)′, R(2)′, Tits Tits group 17971200 211⋅33⋅52⋅13 24 4 17 11 13 6 14 4 13 22 C2 1 1600 26 26 260 260 1961 1964

The GCD of these orders is 5! = 120 = 23⋅3⋅5 (were it not for J1, it would be 6!). All except for Ly and J4 divide the order of the monster, so the LCM is 11⋅37⋅43⋅67⋅|M| and the primes occurring are those up to and including 71, excluding 53 and 61.

Small groups

G |G| ⌊log2|G|⌋ ω(|G|) Ω(|G|) ν2(|G|) max p n! ∣ ∣ n! κ(G) Out(G) M(G) min perm deg min irrep min proj irrep min mod irrep min mod proj irrep
A5 ≅ PSL2(𝔽4) ≅ PSL2(𝔽5) 60 22⋅3⋅5 5 3 4 2 5 3 5 5 C2 C2 5 3 2 22 20
PSL2(𝔽7) ≅ PSL3(𝔽2) 168 23⋅3⋅7 7 3 5 3 7 4 7 6 C2 C2 7 3 3 30 27
A6 ≅ PSL2(𝔽9) ≅ PΩ5(𝔽2)′ 360 23⋅32⋅5 8 3 6 3 5 5 6 7 C2×C2 C6 6 5 3 33 23
PSL2(𝔽8) ≅ Ree(𝔽3)′ 504 23⋅32⋅7 8 3 6 3 7 4 7 9 C3 1 9 7 7 22 22
PSL2(𝔽11) 660 22⋅3⋅5⋅11 9 4 5 2 11 3 11 8 C2 C2 11 5 5 311 211
PSL2(𝔽13) 1092 22⋅3⋅7⋅13 10 4 5 2 13 3 13 9 C2 C2 14 7 6 313 213
PSL2(𝔽17) 2448 24⋅32⋅17 11 3 7 4 17 4 17 11 C2 C2 18 9 8 317 217
A7 2520 23⋅32⋅5⋅7 11 4 7 3 7 5 7 9 C2 C6 7 6 4 42 35
PSL2(𝔽19) 3420 22⋅32⋅5⋅19 11 4 6 2 19 3 19 12 C2 C2 20 9 9 319 219
PSL2(𝔽16) 4080 24⋅3⋅5⋅17 11 4 7 4 17 5 17 17 C4 1 17 15 15 22 22
PSL3(𝔽3) 5616 24⋅33⋅13 12 3 8 4 13 4 13 12 C2 1 13 12 12 33 33
PSU3(𝔽9) ≅ G2(𝔽2)′ 6048 25⋅33⋅7 12 3 9 5 7 4 9 14 C2 1 28 6 6 33 33
PSL2(𝔽23) 6072 23⋅3⋅11⋅23 12 4 6 3 23 4 23 14 C2 C2 24 11 11 323 223
PSL2(𝔽25) 7800 23⋅3⋅52⋅13 12 4 7 3 13 5 13 15 C2×C2 C2 26 13 12 35 25
M11 7920 24⋅32⋅5⋅11 12 4 8 4 11 6 11 10 1 1 11 10 10 53 53
PSL2(𝔽27) 9828 22⋅33⋅7⋅13 13 4 7 2 13 3 13 16 C6 C2 28 13 13 33 23
PSL2(𝔽29) 12180 22⋅3⋅5⋅7⋅29 13 5 6 2 29 3 29 17 C2 C2 30 15 14 329 229
PSL2(𝔽31) 14880 25⋅3⋅5⋅31 13 4 8 5 31 5 31 18 C2 C2 32 15 15 331 231
A8 ≅ PSL4(𝔽2) 20160 26⋅32⋅5⋅7 14 4 10 6 7 7 8 13 C2 C2 8 7 7 42 42
PSL3(𝔽4) 20160 26⋅32⋅5⋅7 14 4 10 6 7 7 8 10 D12 C12×C4 21 20 6 82 32
PSL2(𝔽37) 25308 22⋅32⋅19⋅37 14 4 6 2 37 3 37 21 C2 C2 38 19 18 337 237
PSU4(𝔽4) ≅ PΩ5(𝔽3) 25920 26⋅34⋅5 14 3 11 6 5 6 9 20 C2 C2 27 5 4 42 40
Suz(𝔽8) 29120 26⋅5⋅7⋅13 14 4 9 6 13 2 13 11 C3 C2×C2 65 14 14 42 42
PSL2(𝔽32) 32736 25⋅3⋅11⋅31 14 4 8 5 31 4 31 33 C5 1 33 31 31 22 22
PSL2(𝔽41) 34440 23⋅3⋅5⋅7⋅41 15 5 7 3 41 5 41 23 C2 C2 42 21 20 341 241
PSL2(𝔽43) 39732 22⋅3⋅7⋅11⋅43 15 5 6 2 43 3 43 24 C2 C2 44 21 21 343 243
PSL2(𝔽47) 51888 24⋅3⋅23⋅47 15 4 7 4 47 4 47 26 C2 C2 48 23 23 347 247
PSL2(𝔽49) 58800 24⋅3⋅52⋅72 15 4 9 4 7 5 14 27 C2×C2 C2 50 25 24 37 27
PSU3(𝔽16) 62400 26⋅3⋅52⋅13 15 4 10 6 13 5 13 22 C4 1 65 12 12 32 32
PSL2(𝔽53) 74412 22⋅33⋅13⋅53 16 4 7 2 53 3 53 29 C2 C2 54 27 26 353 253
M12 95040 26⋅33⋅5⋅11 16 4 11 6 11 6 11 15 C2 C2 12 11 11 102,3 63

Order formulae for groups of Lie type

Type Dimension (=|roots|+n) Power of q (=|+ve roots|) Denominator Cyclotomic factors Distinct cyclotomic factors Total cyclotomic factors
An(q) (n≥1) n(n+2) ½n(n+1) (n+1,q−1) 1n⋅2⌊(n+1)/2⌋⋅3⌊(n+1)/3⌋⋅…⋅(n+1) n+1 ∑⌊(n+1)/i⌋−1 ∼ n log n
Bn(q) (n≥2) n(2n+1) n2 (2,q−1) 1n⋅2n⋅3⌊n/3⌋⋅4⌊n/2⌋⋅…⋅n1+[2∣n]⋅(2⌊n/2⌋+2)⋅(2⌊n/2⌋+4)⋅…⋅(2n−2)⋅(2n) ⌊(3n+1)/2⌋ 2∑⌊n/i⌋−∑⌊n/2i⌋ ∼ 3/2 n log n
Cn(q) (n≥3) n(2n+1) n2 (2,q−1) 1n⋅2n⋅3⌊n/3⌋⋅4⌊n/2⌋⋅…⋅n1+[2∣n]⋅(2⌊n/2⌋+2)⋅(2⌊n/2⌋+4)⋅…⋅(2n−2)⋅(2n) ⌊(3n+1)/2⌋ 2∑⌊n/i⌋−∑⌊n/2i⌋ ∼ 3/2 n log n
Dn(q) (n≥4) n(2n−1) n(n−1) (4,qn−1) 1n⋅2n−1+[2∣n]⋅3⌊n/3⌋⋅4⌊(n−1)/2⌋+[4∣n]⋅…⋅(n−1)1+[2∣n−1]⋅n1+[2∣n]⋅(2⌊n/2⌋+2)⋅(2⌊n/2⌋+4)⋅…⋅(2n−4)⋅(2n−2) ⌊(3n−1)/2⌋ ∑⌊n/i⌋+∑⌊(n−1)/i⌋−∑⌊(n−1)/2i⌋ ∼ 3/2 n log n
E6(q) 78 36 (3,q−1) 16⋅24⋅33⋅42⋅5⋅62⋅8⋅9⋅12 9 21
E7(q) 133 63 (2,q−1) 17⋅27⋅33⋅42⋅5⋅63⋅7⋅8⋅9⋅10⋅12⋅14⋅18 13 30
E8(q) 248 120 1 18⋅28⋅34⋅44⋅52⋅64⋅7⋅82⋅9⋅102⋅122⋅14⋅15⋅18⋅20⋅24⋅30 17 44
F4(q) 52 24 1 14⋅24⋅32⋅42⋅62⋅8⋅12 7 16
G2(q) 14 6 1 12⋅22⋅3⋅6 4 6
2An(q2) (n≥2) n(n+2) ½n(n+1) (n+1,q+1) 1⌊(n+1)/2⌋⋅2n⋅3⌊(n+1)/6⌋⋅4⌊(n+1)/4⌋⋅…⋅(4⌊n/2⌋+2)
(The general term is [2∣k+1]⌊(n+1)/2k⌋+[2∣k]⌊(n+1)/k⌋+[4∣k+2]⌈⌊2(n+1)/k⌋/2⌉−[k=2].)		 n+1 ∑⌊(n+1)/i⌋−1 ∼ n log n
2Dn(q2) (n≥4) n(2n−1) n(n−1) (4,qn+1) 1n−1⋅2n−1+[2∣n+1]⋅3⌊(n−1)/3⌋⋅4⌊(n−1)/2⌋+[4∣n+2]⋅…⋅(n−1)1+[2∣n−1]⋅n[2∣n]⋅(2⌊n/2⌋+2)⋅(2⌊n/2⌋+4)⋅…⋅(2n−4)⋅(2n−2)⋅(2n) ⌊3n/2⌋ ∑⌊n/i⌋+∑⌊(n−1)/i⌋−∑⌊n/2i⌋ ∼ 3/2 n log n
2E6(q2) 78 36 (3,q+1) 14⋅26⋅32⋅42⋅63⋅8⋅10⋅12⋅18 9 21
3D4(q3) 28 12 1 12⋅22⋅32⋅62⋅12 5 9
2B2(q) (q=22k+1) 5 2 1 1⋅4 2 2
2F4(q) (q=22k+1) 26 12 1 12⋅22⋅42⋅6⋅12 5 8
2G2(q) (q=32k+1) 7 3 1 1⋅2⋅6 3 3

Prime factorization

By the number of prime factors, counted with multiplicity:

Ω(|G|) |G| G
1 p p Cp
2 —
3 —
4 60 22⋅3⋅5 A5 ≅ PSL2(𝔽4) ≅ PSL2(𝔽5)
5 168 23⋅3⋅7 PSL2(𝔽7) ≅ PSL3(𝔽2)
660 22⋅3⋅5⋅11 PSL2(𝔽11)
1092 22⋅3⋅7⋅13 PSL2(𝔽13)
6 360 23⋅32⋅5 A6 ≅ PSL2(𝔽9) ≅ PΩ5(𝔽2)′
504 23⋅32⋅7 PSL2(𝔽8) ≅ Ree(𝔽3)′
3420 22⋅32⋅5⋅19 PSL2(𝔽19)
6072 23⋅3⋅11⋅23 PSL2(𝔽23)
12180 23⋅5⋅7⋅29 PSL2(𝔽29)
25308 22⋅32⋅19⋅37 PSL2(𝔽37)
39732 22⋅3⋅7⋅11⋅43 PSL2(𝔽43)
102660 22⋅3⋅5⋅29⋅59 PSL2(𝔽59)
113460 22⋅3⋅5⋅31⋅61 PSL2(𝔽61)
150348 22⋅3⋅11⋅17⋅67 PSL2(𝔽67)
285852 22⋅3⋅7⋅41⋅83 PSL2(𝔽83)
22⋅3⋅r⋅s⋅p … various PSL2(𝔽p) … (∞ conjectured)
7 ∞ (conjectured)

The above list for Ω(|G|)≤6 consists of only various PSL2(𝔽q) (this can be proven assuming CFSG, or even just the Gorenstein–Walter theorem). Exactly which and their infinitude depends on open number-theoretic problems as follows:

q Conditions Instances
Sporadic 22, 23, 32, 5, 7, 11, 13
12n−5 2n−1 prime, 3n−1 prime, 12n−5 prime 19, 43, 67, … (∞ conjectured)
12n−1 n prime, 6n−1 prime, 12n−1 prime 23, 59, 83, … (∞ conjectured)
12n+1 n prime, 6n+1 prime, 12n+1 prime 37, 61, 157, … (∞ conjectured)
12n+5 2n+1 prime, 3n+1 prime, 12n+5 prime 29, 173, … (∞ conjectured)

By the number of distinct prime factors:

ω(|G|) |G| G
1 p p Cp
2 —
3 60 22⋅3⋅5 A5 ≅ PSL2(𝔽4) ≅ PSL2(𝔽5)
168 23⋅3⋅7 PSL2(𝔽7) ≅ PSL3(𝔽2)
360 23⋅32⋅5 A6 ≅ PSL2(𝔽9) ≅ PΩ5(𝔽2)′
504 23⋅32⋅7 PSL2(𝔽8) ≅ Ree(𝔽3)′
2448 24⋅32⋅17 PSL2(𝔽17)
5616 24⋅33⋅13 PSL3(𝔽3)
6048 25⋅33⋅7 PSU3(𝔽9) ≅ G2(𝔽2)′
25920 26⋅34⋅5 PSU4(𝔽4) ≅ PΩ5(𝔽3)
4 660 22⋅3⋅5⋅11 PSL2(𝔽11)
1092 22⋅3⋅7⋅13 PSL2(𝔽13)
2520 23⋅32⋅5⋅7 A7
3420 22⋅32⋅5⋅19 PSL2(𝔽19)
4080 24⋅3⋅5⋅17 PSL2(𝔽24)
6072 23⋅3⋅11⋅23 PSL2(𝔽23)
7800 23⋅3⋅52⋅13 PSL2(𝔽52)
7920 24⋅32⋅5⋅11 M11
9828 22⋅33⋅7⋅13 PSL2(𝔽33)
14880 25⋅3⋅5⋅31 PSL2(𝔽31)
20160 26⋅32⋅5⋅7 A8 ≅ PSL4(𝔽2)
20160 26⋅32⋅5⋅7 PSL3(𝔽4)
25308 22⋅32⋅19⋅37 PSL2(𝔽37)
29120 26⋅5⋅7⋅13 Suz(𝔽8)
32736 25⋅3⋅11⋅31 PSL2(𝔽25)
51888 24⋅3⋅23⋅47 PSL2(𝔽47)
58800 24⋅3⋅52⋅72 PSL2(𝔽72)
62400 26⋅3⋅52⋅13 PSU3(𝔽24)
74412 22⋅33⋅13⋅53 PSL2(𝔽53)
95040 26⋅33⋅5⋅11 M12
126000 24⋅32⋅53⋅7 PSU3(𝔽52)
181440 26⋅34⋅5⋅7 A9
194472 23⋅32⋅37⋅73 PSL2(𝔽73)
265680 24⋅34⋅5⋅41 PSL2(𝔽34)
372000 25⋅3⋅53⋅31 PSL3(𝔽5)
456288 25⋅3⋅72⋅97 PSL2(𝔽97)
604800 27⋅33⋅52⋅7 J2
612468 22⋅33⋅53⋅107 PSL2(𝔽107)
979200 28⋅32⋅52⋅17 PΩ5(𝔽4)
1024128 27⋅32⋅7⋅127 PSL2(𝔽127)
1451520 29⋅34⋅5⋅7 PΩ7(𝔽2)
1451520 29⋅34⋅5⋅7 PSp6(𝔽2)
1814400 27⋅34⋅52⋅7 A10
1876896 25⋅32⋅73⋅19 PSL3(𝔽7)
2097024 27⋅3⋅43⋅127 PSL2(𝔽27)
2165292 22⋅34⋅41⋅163 PSL2(𝔽163)
3265920 27⋅36⋅5⋅7 PSU4(𝔽32)
3594432 26⋅3⋅97⋅193 PSL2(𝔽193)
4245696 26⋅36⋅7⋅13 G2(𝔽3)
5515776 29⋅34⋅7⋅19 PSU3(𝔽26)
5663616 27⋅3⋅73⋅43 PSU3(𝔽72)
6065280 27⋅36⋅5⋅13 PSL4(𝔽3)
7174332 22⋅35⋅112⋅61 PSL2(𝔽35)
8487168 28⋅3⋅43⋅257 PSL2(𝔽257)
13685760 210⋅35⋅5⋅11 PSU5(𝔽22)
16482816 29⋅32⋅72⋅73 PSL3(𝔽8)
17971200 211⋅33⋅52⋅13 T ≅ 2F4(𝔽2)′
28090752 27⋅3⋅191⋅383 PSL2(𝔽383)
32537600 210⋅52⋅31⋅41 Suz(𝔽32)
35942400 212⋅33⋅52⋅13 2F4(𝔽23)
42573600 25⋅36⋅52⋅73 PSU3(𝔽34)
57750408 23⋅35⋅61⋅487 PSL2(𝔽487)
74880000 210⋅32⋅54⋅13 PΩ5(𝔽5)
96049728 26⋅32⋅172⋅577 PSL2(𝔽577)
138297600 28⋅32⋅52⋅74 PΩ5(𝔽7)
174182400 212⋅35⋅52⋅7 PΩ8+(𝔽2)
211341312 212⋅34⋅72⋅13 3D4(𝔽23)
321367392 25⋅33⋅431⋅863 PSL2(𝔽863)
766403712 27⋅32⋅577⋅1153 PSL2(𝔽1153)
1721606400 28⋅38⋅52⋅41 PΩ5(𝔽9)
5230175508 22⋅37⋅547⋅1093 PSL2(𝔽37)
6950204928 29⋅32⋅173⋅307 PSL3(𝔽17)
8717209632 25⋅34⋅1297⋅2593 PSL2(𝔽2593)
12410213148 22⋅36⋅1459⋅2917 PSL2(𝔽2917)
41812719372 22⋅37⋅1093⋅4373 PSL2(𝔽4373)
334616519988 22⋅37⋅4373⋅8747 PSL2(𝔽8747)
549755805696 213⋅3⋅2731⋅8191 PSL2(𝔽213)
2251799813554176 217⋅3⋅43691⋅131071 PSL2(𝔽217)
144115188075331584 219⋅3⋅174763⋅524287 PSL2(𝔽219)
493023204371017728 212⋅35⋅497663⋅995327 PSL2(𝔽995327)
1663961673594488832 211⋅36⋅746497⋅1492993 PSL2(𝔽1492993)
2026277576508690972 22⋅313⋅398581⋅797161 PSL2(𝔽313)
3944203467163066368 213⋅35⋅995329⋅1990657 PSL2(𝔽1990657)
74793713817969819648 216⋅34⋅2654209⋅5308417 PSL2(𝔽5308417)
11346478189904277798912 220⋅33⋅14155777⋅28311553 PSL2(𝔽28311553)
319065783425611258657932 22⋅316⋅21523361⋅86093443 PSL2(𝔽86093443)
99035203
14283042197045510144		 231⋅3⋅
715827883⋅
2147483647		 PSL2(𝔽231)
1628095431
43540004207861956608		 220⋅38⋅
3439853569⋅
6879707137		 PSL2(𝔽6879707137)
27314908709940346
51591443538507726848		 228⋅38⋅
880602513409⋅
1761205026817		 PSL2(𝔽1761205026817)
64746450275221151
13083087778069086208		 230⋅37⋅
1174136684543⋅
2348273369087		 PSL2(𝔽2348273369087)
862312323374404542
23924776545181237248		 236⋅34⋅
2783138807809⋅
5566277615617		 PSL2(𝔽5566277615617)
2043999581329994019
00747238605122961408		 238⋅33⋅
3710851743743⋅
7421703487487		 PSL2(𝔽7421703487487)
49158532376528723536
70035829256476792832		 211⋅321⋅
10711401679871⋅
21422803359743		 PSL2(𝔽21422803359743)
24
80658454853735404719
73412253386378575872		 243⋅32⋅
39582418599937⋅
79164837199873		 PSL2(𝔽79164837199873)
2089988
76837747497976537813
77896705208280035328		 212⋅325⋅
1735247072139263⋅
3470494144278527		 PSL2(𝔽3470494144278527)
2743409
79839174126149268987
59550695767401299968		 247⋅33⋅
1899956092796929⋅
3799912185593857		 PSL2(𝔽3799912185593857)
76884632929
71057592593542615687
08712970534286000128		 244⋅38⋅
57711166318706689⋅
115422332637413377		 PSL2(𝔽115422332637413377)
118341651390669
77960115338097523460
11701024860850880512		 219⋅326⋅
666334875701477377⋅
1332669751402954753		 PSL2(𝔽1332669751402954753)
1225996432692711
08668667762172024734
66644069968255123456		 261⋅3⋅
768614336404564651⋅
2305843009213693951		 PSL2(𝔽261)
5540072511115634
84371723703047010512
33922169236241571872		 25⋅336⋅
2401514164751985937⋅
4803028329503971873		 PSL2(𝔽4803028329503971873)
2
12259168753892666419
17725225360236985675
25548296192777781248		 218⋅331⋅
80959687397729501183⋅
161919374795459002367		 PSL2(𝔽161919374795459002367)
64089297362
35083645443329866199
06407365103349431713
05282675746859049472		 29⋅344⋅
252101350959004475617537⋅
504202701918008951235073		 PSL2(𝔽504202701918008951235073)
3105261350557
33050971258837372554
39653590437632627858
18745595647374655488		 260⋅313⋅
919064635994651045658623⋅
1838129271989302091317247		 PSL2(𝔽1838129271989302091317247)
50974792324
28539846355110493357
17426824433574357467
66164566829837397462
55876486948676698112		 277⋅315⋅
1084172759721818116709104484351⋅
2168345519443636233418208968703		 PSL2(𝔽2168345519443636233418208968703)
48919273458895052
34037559381438805166
46420538991100344572
22245766167162617643
46380629911743234048		 244⋅340⋅
106939956310749872542710767812609⋅
213879912621499745085421535625217		 PSL2(𝔽213879912621499745085421535625217)
57198030301746984
24387215406387759471
87204686343680994043
71043851771465565223
47192896318206902272		 252⋅335⋅
112661023932312622925654142222337⋅
225322047864625245851308284444673		 PSL2(𝔽225322047864625245851308284444673)
1105
85999694296593405584
50420558009710534192
10069860113518379870
73948894323956788963
54184401908192509952		 297⋅311⋅
14035031304914384611683530170171391⋅
28070062609828769223367060340342783		 PSL2(𝔽28070062609828769223367060340342783)
4701594741
08293407168457875996
14158902439999731898
43117419393765335761
91756065747720198078
86958714575984263168		 298⋅315⋅
2273675071396130307092731887567765503⋅
4547350142792260614185463775135531007		 PSL2(𝔽4547350142792260614185463775135531007)
147924363701559
55015855566907713446
62683825151279836108
25382208909239294536
98127141300805379496
14865281577037856768		 2122⋅33⋅
71778311772385457136805581255138607103⋅
143556623544770914273611162510277214207		 PSL2(𝔽143556623544770914273611162510277214207)
492525077454930
99015348800125179517
25634967408808180833
49353667553071522143
69811852433228126288
82767797112614682624		 2127⋅3⋅
56713727820156410577229101238628035243⋅
170141183460469231731687303715884105727		 PSL2(𝔽2127)
2a⋅3b⋅r⋅p … various PSL2(𝔽p) … (∞ conjectured)
5 ∞ (conjectured)

The above list for ω(|G|)≤4 is complete (assuming CFSG) except for the PSL2(𝔽q) case, which depends on various open number-theoretic problems as follows:

q Conditions Instances
Sporadic 22, 24, 32, 34, 5, 52, 7, 72, 17, 31, 97, 127, 577
2p p odd prime, ⅓(2p+1) prime (Wagstaff prime), 2p−1 prime (Mersenne prime) 23, 25, 27, 213, 217, 219, 231, 261, 2127, conjectured no others (see new Mersenne conjecture)
2p p odd prime, ⅓(2p+1) prime power (higher than first), 2p−1 prime Conjectured none (probably provable?)
3p p odd prime, ¼(3p+1) prime (base-3 Wagstaff prime), ½(3p−1) prime (base-3 repunit prime) 33, 37, 313, conjectured no others
3p p odd prime, ¼(3p+1) prime power, ½(3p−1) prime power (at least one power higher than first) 35, conjectured no others (probably provable?)
2⋅3p−1 p≡−1 (mod 4) prime, ½(3p−1) prime (base-3 repunit prime), 2⋅3p−1 prime (base-3 Williams prime) 53, 4373, conjectured no others
2⋅3p−1 p≡−1 (mod 4) prime, ½(3p−1) prime power (higher than first), 2⋅3p−1 prime Conjectured none (probably provable?)
2⋅3p+1 p≡1 (mod 4) prime, ¼(3p+1) prime (base-3 Wagstaff prime), 2⋅3p+1 prime (base-3 Williams prime of the second kind) 487, conjectured no others
2⋅3p+1 p≡1 (mod 4) prime, ¼(3p+1) prime power (higher than first), 2⋅3p+1 prime Conjectured none (probably provable?)
2⋅32n+1 ½(32n+1) prime (base-3 half generalized Fermat prime), 2⋅32n+1 prime (base-3 Williams prime of the second kind) 19, 163, 86093443, conjectured no others
2⋅32n+1 ½(32n+1) prime power (higher than first), 2⋅32n+1 prime Conjectured none (probably provable?)
22p+1 p odd prime, 2p−1 prime (Mersenne prime), ⅓(22p−1+1) prime (Wagstaff prime), 22p+1 prime (Fermat prime) 257, conjectured no others
22p+1 p odd prime, 2p−1 prime, ⅓(22p−1+1) prime power (higher than first), 22p+1 prime Conjectured none (probably provable?)
12n+1 n 3-smooth, 6n+1 prime (Pierpont prime), 12n+1 prime (Pierpont prime, Cunningham chain of the second kind) 13, 37, 73, 193, 1153, 2593, 2917, 1492993, 1990657, 5308417, 28311553, … (∞ conjectured)
12n−1 n 3-smooth, 6n−1 prime (Pierpont prime of the second kind, Sophie Germain prime), 12n−1 prime (Pierpont prime of the second kind, safe prime) 11, 23, 47, 107, 383, 863, 8747, 995327, … (∞ conjectured)