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Table 2 The order components of simple groups G with t(G)=3

From: Recognizing L 2 (p) by its order and one special conjugacy class size

G

Restrictions on G

m 1

m 2

m 3

A p

6<p and p, p − 2 both are primes

(p − 3)!/2

p

p − 2

A 1 (q)

3<qϵ(mod4)

q − ϵ

q

q + ϵ 2

A 1 (q)

q>4 is even

q

(q − 1)

q + 1

D p 2 (3)

p= 2 m +15

2 3 p ( p 1 ) ( 3 p 1 1) i = 1 p 2 ( 3 2 i 1)

3 p 1 + 1 2

3 p + 1 4

G 2 (q)

q≡0(mod3)

q 6 ( q 2 1 ) 2

q 2 +q+1

q 2 q+1

G 2 2 (q)

p= 3 2 m + 1 >3

q 3 ( q 2 1)

q+ 3 q +1

q 3 q +1

F 4 (q)

2<q is even

q 24 ( q 6 1 ) 2 ( q 4 1 ) 2

q 4 +1

q 4 q 2 +1

F 4 2 (q)

2<q= 2 2 m + 1

q 12 ( q 4 1)( q 3 +1)( q 2 +1)(q1)

q 2 2 q 3 + q 2 q +1

q 2 + 2 q 3 +q+ 2 q +1

A 5 2 (2)

 

215365

7

11

E 7 (2)

 

2633115273111317193143

73

127

E 7 (3)

 

22336352731121331937416173547

757

1,093

M 11

 

2432

5

11

M 23

 

273257

11

23

M 24

 

2103357

11

23

J 3

 

27355

17

19

HS

 

293253

7

11

Sz

 

21337527

11

13

Co 2

 

21836537

11

23

F 23

 

2183135271113

17

23

F 2 =B

 

24131356721113171923

31

47

F 3 =Th

 

215310537213

19

31