From: Recognizing by its order and one special conjugacy class size
G | Restrictions on G |
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| 6<p and p, p − 2 both are primes | (p − 3)!/2 | p | p − 2 |
| 3<q≡ϵ(mod4) | q − ϵ | q |
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| q>4 is even | q | (q − 1) | q + 1 |
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| q≡0(mod3) |
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| 2<q is even |
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| 215⋅36⋅5 | 7 | 11 | |
| 263⋅311⋅52⋅73⋅11⋅13⋅17⋅19⋅31⋅43 | 73 | 127 | |
| 223⋅363⋅52⋅73⋅112⋅133⋅19⋅37⋅41⋅61⋅73⋅547 | 757 | 1,093 | |
| 24⋅32 | 5 | 11 | |
| 27⋅32⋅5⋅7 | 11 | 23 | |
| 210⋅33⋅5⋅7 | 11 | 23 | |
| 27⋅35⋅5 | 17 | 19 | |
HS | 29⋅32⋅53 | 7 | 11 | |
Sz | 213⋅37⋅52⋅7 | 11 | 13 | |
| 218⋅36⋅53⋅7 | 11 | 23 | |
| 218⋅313⋅52⋅7⋅11⋅13 | 17 | 23 | |
| 241⋅313⋅56⋅72⋅11⋅13⋅17⋅19⋅23 | 31 | 47 | |
| 215⋅310⋅53⋅72⋅13 | 19 | 31 |