Table 1 The order components of simple groups G with $\mathit{t}\mathbf{(}\mathit{G}\mathbf{)}\mathbf{=}\mathbf{2}$

$A_n$

6<n = p, p + 1, p + 2 and one of n, n − 2 is not a prime

$\frac{n!}{2p}$

p

$A_{p-1}(q)$

(p,q)≠(3,2),(3,4)

$q^{\frac{p(p-1)}{2}}{\prod }_{i=1}^{p-1}(q^i-1)$

$\frac{q^p-1}{(q-1)(p,q-1)}$

$A_p(q)$

(q − 1)|(p + 1)

$q^{\frac{p(p+1)}{2}}(q^{p+1}-1){\prod }_{i=1}^{p-1}(q^i-1)$

$\frac{q^p-1}{q-1}$

${}^{2}A_{p-1}(q)$

$q^{\frac{p(p-1)}{2}}{\prod }_{i=1}^{p-1}(q^i-{(-1)}^i)$

$\frac{q^p+1}{(q+1)(p,q+1)}$

${}^{2}A_p(q)$

(q + 1)|(p + 1) and (p,q)≠(3,3),(5,2)

$q^{\frac{p(p+1)}{2}}(q^p-1){\prod }_{i=1}^{p-1}(q^i-1)$

$\frac{q^p+1}{q+1}$

$B_n(q)$

$n=2^m\ge 4$ and q is odd

$q^{n^2}(q^n-1){\prod }_{i=1}^{n-1}(q^{2i}-1)$

$\frac{q^n+1}{2}$

$B_p(3)$

$3^{p^2}(3^p+1){\prod }_{i=1}^{p-1}(3^{2i}-1)$

$\frac{3^p-1}{2}$

$C_n(q)$

$n=2^m\ge 2$

$q^{n^2}(q^n-1){\prod }_{i=1}^{n-1}(q^{2i}-1)$

$\frac{q^n+1}{(2,q-1)}$

$C_p(q)$

q = 2,3

$q^{p^2}(q^p+1){\prod }_{i=1}^{p-1}(q^{2i}-1)$

$\frac{q^p-1}{(2,q-1)}$

$D_p(q)$

p ≥ 5 and q = 2,3,5

$q^{p(p-1)}{\prod }_{i=1}^{p-1}(q^{2i}-1)$

$\frac{q^p-1}{q-1}$

$D_{p+1}(q)$

q = 2,3

$\frac{1}{(2,q-1)}{q}^{p(p+1)}(q^p+1)(q^{p+1}-1){\prod }_{i=1}^{p-1}(q^{2i}-1)$

$\frac{q^p-1}{(2,q-1)}$

${}^{2}D_n(q)$

$n=2^m\ge 4$

$q^{n(n-1)}{\prod }_{i=1}^{n-1}(q^{2i}-1)$

$\frac{q^n+1}{(2,q+1)}$

${}^{2}D_n(q)$

$n=2^m+1\ge 5$ if q = 2 and $n=2^m+1\ge 9$ if q = 3

$\frac{1}{(2,q-1)}{q}^{n(n-1)}(q^n+1)(q^n-1){\prod }_{i=1}^{n-2}(q^{2i}-1)$

$\frac{q^{n-1}+1}{(2,q-1)}$

${}^{2}D_p(3)$

$p\ne 2^m+1$ and p ≥ 5

$3^{p(p-1)}{\prod }_{i=1}^{p-1}(3^{2i}-1)$

$\frac{3^p+1}{4}$

$G_2(q)$

q≡ϵ(mod3) and q>2

$q^6(q^3-ϵ)(q^2-1)(q+ϵ)$

$q^2-ϵq+1$

${}^{3}D_4(q)$

$q^{12}(q^6-1)(q^2-1)(q^4+q^2+1)$

$q^4-q^2+1$

$F_4(q)$

q is odd

$q^{24}(q^8-1){(q^6-1)}^2(q^4-1)$

$q^4-q^2+1$

$E_6(q)$

$q^{36}(q^{12}-1)(q^8-1)(q^6-1)(q^5-1)(q^3-1)(q^2-1)$

$\frac{q^6+q^3+1}{(3,q-1)}$

${}^{2}E_6(q)$

q>2

$q^{36}(q^{12}-1)(q^8-1)(q^6-1)(q^5+1)(q^3+1)(q^2-1)$

$\frac{q^6-q^3+1}{(3,q-1)}$

${}^{2}A_3(2)$

2⁶⋅3⁴

5

${}^{2}F_4{(2)}^{\prime }$

2¹¹⋅3³⋅5²

13

$M_{12}$

2⁶⋅3³⋅5

11

$J_2$

2⁷⋅3³⋅5²

7

Ru

2¹⁴⋅3³⋅5³⋅7⋅13

29

He

2¹⁰⋅3³⋅5²⋅7³

17

$M^{c}L$

2⁷⋅3⁶⋅5³⋅7

11

${\mathit{Co}}_1$

2²¹⋅3⁹⋅5⁴⋅7²⋅11⋅13

23

${\mathit{Co}}_3$

2¹⁰⋅3⁷⋅5³⋅7⋅11

23

$F_{22}$

2¹⁷⋅3⁹⋅5²⋅7⋅11

13

HN

2¹⁴⋅3⁶⋅5⁶⋅7⋅11

19