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

Journal of Inequalities and Applications

Table 3 The order components of simple groups G with $t (G) \geq 4$

$t (G)$	G	Restrictions on G	$m_{1}$	$m_{2}$	$m_{3}$	$m_{4}$	$m_{5}$	$m_{6}$
4	${}^{2}B_{2} (q)$	$2 < q = 2^{2 m + 1}$	$q^{2}$	q − 1	$q + \sqrt{2 q} + 1$	$q - \sqrt{2 q} + 1$
	$E_{8} (q)$	q≡2,3(mod5)	$q^{120} (q^{20} - 1) (q^{18} - 1) (q^{14} - 1) (q^{12} - 1) (q^{8} - 1) (q^{4} + 1) (q^{4} + q^{2} + 1)$	$q^{8} + q^{7} - q^{5} - q^{4} - q^{2} + q + 1$	$q^{8} - q^{7} + q^{5} - q^{4} + q^{2} - q + 1$	$q^{8} - q^{4} + 1$
	$A_{2} (4)$		2⁶	5	7	9
	$M_{22}$		2⁷⋅3²	5	7	11
	$J_{1}$		2³⋅3⋅5	7	11	19
	ON		2⁹⋅3⁴⋅5⋅7³	11	19	31
	Ly		2⁸⋅3⁷⋅5⁶⋅7⋅11	31	37	67
	$F_{24}^{'}$		2²¹⋅3¹⁶⋅5²⋅7³⋅11⋅13	17	23	29
	$F_{1} = M$		2⁴⁶⋅3²⁰⋅5⁹⋅7⁶⋅11²⋅13³⋅17⋅19⋅23⋅29⋅31⋅47	41	59	71
5	$E_{8} (q)$	q≡0,1,4(mod5)	$q^{120} (q^{18} - 1) (q^{14} - 1) {(q^{12} - 1)}^{2} {(q^{10} - 1)}^{2} {(q^{8} - 1)}^{2} (q^{4} + q^{2} + 1)$	$q^{8} + q^{7} - q^{5} - q^{4} - q^{2} + q + 1$	$q^{8} - q^{7} + q^{5} - q^{4} + q^{2} - q + 1$	$q^{8} - q^{6} + q^{4} - q^{2} + 1$	$q^{8} - q^{4} + 1$
6	$J_{4}$		2²¹⋅3³⋅5⋅7⋅11³	23	29	31	37	43