57種168階羣

在郭繼東等人的《單羣PSL(2,7)的一個新刻畫》一文中給出單羣PSL(2,7)的一個新刻畫,主要結果是下述定理:如果有限羣G的同階的元素的個數組成的集合是{1,21,56,42,48},則G≌PSL(2,7)。
gap> NumberSmallGroups(168);IdGroup(PSL(2,7));IdGroup(SL(3,2));
57
[ 168, 42 ]
[ 168, 42 ]
gap> for n in [1..57] do G:=SmallGroup(168,n);idn:=IdGroup(G);Print(idn);Print(":");L:=List(Elements(G),Order);;M:=[1,2,3,4,6,7,8,12,14,21,24,28,42,56,84,168];;for i in M do Print(Size(Positions(L,i)),","); od;Print("\n");od;
[ 168, 1 ]:1,1,14,2,14,6,28,28,6,0,56,12,0,0,0,0,
[ 168, 2 ]:1,1,14,2,14,6,4,28,6,0,56,12,0,24,0,0,
[ 168, 3 ]:1,1,2,2,2,6,12,4,6,12,0,12,12,72,24,0,
[ 168, 4 ]:1,1,2,2,2,6,28,4,6,12,56,12,12,0,24,0,
[ 168, 5 ]:1,1,2,2,2,6,84,4,6,12,0,12,12,0,24,0,
[ 168, 6 ]:1,1,2,2,2,6,4,4,6,12,8,12,12,24,24,48,
[ 168, 7 ]:1,1,14,30,14,6,0,84,6,0,0,12,0,0,0,0,
[ 168, 8 ]:1,15,14,16,42,6,0,56,6,0,0,12,0,0,0,0,
[ 168, 9 ]:1,29,14,2,70,6,0,28,6,0,0,12,0,0,0,0,
[ 168, 10 ]:1,3,14,28,42,6,0,56,18,0,0,0,0,0,0,0,
[ 168, 11 ]:1,17,14,14,70,6,0,28,18,0,0,0,0,0,0,0,
[ 168, 12 ]:1,15,2,48,30,6,0,0,6,12,0,36,12,0,0,0,
[ 168, 13 ]:1,7,2,56,2,6,0,28,42,12,0,0,12,0,0,0,
[ 168, 14 ]:1,43,2,20,2,6,0,28,6,12,0,36,12,0,0,0,
[ 168, 15 ]:1,21,2,42,30,6,0,0,42,12,0,0,12,0,0,0,
[ 168, 16 ]:1,57,2,6,30,6,0,0,6,12,0,36,12,0,0,0,
[ 168, 17 ]:1,49,2,14,2,6,0,28,42,12,0,0,12,0,0,0,
[ 168, 18 ]:1,1,2,62,2,6,0,28,6,12,0,36,12,0,0,0,
[ 168, 19 ]:1,3,14,4,42,6,0,56,18,0,0,24,0,0,0,0,
[ 168, 20 ]:1,5,14,2,70,6,0,28,30,0,0,12,0,0,0,0,
[ 168, 21 ]:1,1,14,6,14,6,0,84,6,0,0,36,0,0,0,0,
[ 168, 22 ]:1,1,8,6,8,6,0,0,6,48,0,36,48,0,0,0,
[ 168, 23 ]:1,1,56,6,56,6,0,0,6,0,0,36,0,0,0,0,
[ 168, 24 ]:1,1,2,30,2,6,0,60,6,12,0,12,12,0,24,0,
[ 168, 25 ]:1,15,2,16,30,6,0,32,6,12,0,12,12,0,24,0,
[ 168, 26 ]:1,29,2,2,58,6,0,4,6,12,0,12,12,0,24,0,
[ 168, 27 ]:1,3,2,28,6,6,0,56,18,12,0,0,36,0,0,0,
[ 168, 28 ]:1,17,2,14,34,6,0,28,18,12,0,0,36,0,0,0,
[ 168, 29 ]:1,1,2,14,2,6,0,4,6,12,0,84,12,0,24,0,
[ 168, 30 ]:1,7,2,8,2,6,0,4,42,12,0,48,12,0,24,0,
[ 168, 31 ]:1,13,2,2,2,6,0,4,78,12,0,12,12,0,24,0,
[ 168, 32 ]:1,3,2,12,6,6,0,0,18,12,0,72,36,0,0,0,
[ 168, 33 ]:1,9,2,6,6,6,0,0,54,12,0,36,36,0,0,0,
[ 168, 34 ]:1,1,2,86,2,6,0,4,6,12,0,12,12,0,24,0,
[ 168, 35 ]:1,43,2,44,2,6,0,4,6,12,0,12,12,0,24,0,
[ 168, 36 ]:1,85,2,2,2,6,0,4,6,12,0,12,12,0,24,0,
[ 168, 37 ]:1,3,2,84,6,6,0,0,18,12,0,0,36,0,0,0,
[ 168, 38 ]:1,45,2,42,6,6,0,0,18,12,0,0,36,0,0,0,
[ 168, 39 ]:1,3,2,4,6,6,0,8,18,12,0,24,36,0,48,0,
[ 168, 40 ]:1,5,2,2,10,6,0,4,30,12,0,12,60,0,24,0,
[ 168, 41 ]:1,1,2,6,2,6,0,12,6,12,0,36,12,0,72,0,
[ 168, 42 ]:1,21,56,42,0,48,0,0,0,0,0,0,0,0,0,0,
[ 168, 43 ]:1,7,56,0,56,48,0,0,0,0,0,0,0,0,0,0,
[ 168, 44 ]:1,7,2,0,14,48,0,0,0,96,0,0,0,0,0,0,
[ 168, 45 ]:1,9,8,6,0,6,0,0,54,48,0,36,0,0,0,0,
[ 168, 46 ]:1,45,8,42,0,6,0,0,18,48,0,0,0,0,0,0,
[ 168, 47 ]:1,31,14,0,98,6,0,0,18,0,0,0,0,0,0,0,
[ 168, 48 ]:1,31,8,0,56,6,0,0,18,48,0,0,0,0,0,0,
[ 168, 49 ]:1,31,56,0,56,6,0,0,18,0,0,0,0,0,0,0,
[ 168, 50 ]:1,63,2,0,30,6,0,0,42,12,0,0,12,0,0,0,
[ 168, 51 ]:1,7,14,0,98,6,0,0,42,0,0,0,0,0,0,0,
[ 168, 52 ]:1,7,8,0,8,6,0,0,42,48,0,0,48,0,0,0,
[ 168, 53 ]:1,7,56,0,56,6,0,0,42,0,0,0,0,0,0,0,
[ 168, 54 ]:1,31,2,0,62,6,0,0,18,12,0,0,36,0,0,0,
[ 168, 55 ]:1,15,2,0,6,6,0,0,90,12,0,0,36,0,0,0,
[ 168, 56 ]:1,87,2,0,6,6,0,0,18,12,0,0,36,0,0,0,
[ 168, 57 ]:1,7,2,0,14,6,0,0,42,12,0,0,84,0,0,0,
gap> for n in [1..57] do G:=SmallGroup(168,n);idn:=IdGroup(G);Print(idn);Print(":");L:=List(Elements(G),Order);;M:=[1,2,3,4,6,7,8,12,14,21,24,28,42,56,84,168];;for i in M do Print(Size(Positions(L,i)),","); od;arr:=[];;idn:=IdGroup(G);cl:=ConjugacyClasses(G);;Append(arr,"共軛類數：");;Append(arr,String(Size(cl)));Append(arr,"中心：");;Append(arr,String(IdGroup(Center(G))));;Append(arr,"換位子羣：");;Append(arr,String(IdGroup(DerivedSubgroup(G))));;Append(arr,"自同構羣：");;Append(arr,String(Order(AutomorphismGroup(G))));;cl:=NormalSubgroups(G);;Append(arr,"正規子羣個數：");;len:=Size(cl);;Append(arr,String(len));;Print(arr);Print("\n");od;
[ 168, 1 ]:1,1,14,2,14,6,28,28,6,0,56,12,0,0,0,
0,共軛類數：28中心：[ 4, 1 ]換位子羣：[ 7, 1 ]自同構羣：168正規子羣個數：11
[ 168, 2 ]:1,1,14,2,14,6,4,28,6,0,56,12,0,24,0,
0,共軛類數：40中心：[ 8, 1 ]換位子羣：[ 7, 1 ]自同構羣：168正規子羣個數：12
[ 168, 3 ]:1,1,2,2,2,6,12,4,6,12,0,12,12,72,24,
0,共軛類數：84中心：[ 28, 2 ]換位子羣：[ 3, 1 ]自同構羣：144正規子羣個數：14
[ 168, 4 ]:1,1,2,2,2,6,28,4,6,12,56,12,12,0,24,
0,共軛類數：60中心：[ 12, 2 ]換位子羣：[ 7, 1 ]自同構羣：336正規子羣個數：14
[ 168, 5 ]:1,1,2,2,2,6,84,4,6,12,0,12,12,0,24,
0,共軛類數：48中心：[ 4, 1 ]換位子羣：[ 21, 2 ]自同構羣：1008正規子羣個數：13
[ 168, 6 ]:1,1,2,2,2,6,4,4,6,12,8,12,12,24,24,
48,共軛類數：168中心：[ 168, 6 ]換位子羣：[ 1, 1 ]自同構羣：48正規子羣個數：16
[ 168, 7 ]:1,1,14,30,14,6,0,84,6,0,0,12,0,0,0,
0,共軛類數：19中心：[ 2, 1 ]換位子羣：[ 14, 2 ]自同構羣：336正規子羣個數：15
[ 168, 8 ]:1,15,14,16,42,6,0,56,6,0,0,12,0,0,0,
0,共軛類數：28中心：[ 4, 1 ]換位子羣：[ 7, 1 ]自同構羣：168正規子羣個數：19
[ 168, 9 ]:1,29,14,2,70,6,0,28,6,0,0,12,0,0,0,
0,共軛類數：19中心：[ 2, 1 ]換位子羣：[ 14, 2 ]自同構羣：336正規子羣個數：15
[ 168, 10 ]:1,3,14,28,42,6,0,56,18,0,0,0,0,0,0,
0,共軛類數：28中心：[ 4, 2 ]換位子羣：[ 7, 1 ]自同構羣：336正規子羣個數：21
[ 168, 11 ]:1,17,14,14,70,6,0,28,18,0,0,0,0,0,0,
0,共軛類數：19中心：[ 2, 1 ]換位子羣：[ 14, 2 ]自同構羣：168正規子羣個數：15
[ 168, 12 ]:1,15,2,48,30,6,0,0,6,12,0,36,12,0,0,
0,共軛類數：30中心：[ 2, 1 ]換位子羣：[ 21, 2 ]自同構羣：1008正規子羣個數：18
[ 168, 13 ]:1,7,2,56,2,6,0,28,42,12,0,0,12,0,0,
0,共軛類數：30中心：[ 2, 1 ]換位子羣：[ 21, 2 ]自同構羣：1008正規子羣個數：18
[ 168, 14 ]:1,43,2,20,2,6,0,28,6,12,0,36,12,0,0,
0,共軛類數：30中心：[ 2, 1 ]換位子羣：[ 21, 2 ]自同構羣：1008正規子羣個數：16
[ 168, 15 ]:1,21,2,42,30,6,0,0,42,12,0,0,12,0,0,
0,共軛類數：27中心：[ 2, 1 ]換位子羣：[ 42, 6 ]自同構羣：1008正規子羣個數：14
[ 168, 16 ]:1,57,2,6,30,6,0,0,6,12,0,36,12,0,0,
0,共軛類數：27中心：[ 2, 1 ]換位子羣：[ 42, 6 ]自同構羣：1008正規子羣個數：14
[ 168, 17 ]:1,49,2,14,2,6,0,28,42,12,0,0,12,0,0,
0,共軛類數：27中心：[ 2, 1 ]換位子羣：[ 42, 6 ]自同構羣：1008正規子羣個數：14
[ 168, 18 ]:1,1,2,62,2,6,0,28,6,12,0,36,12,0,0,
0,共軛類數：27中心：[ 2, 1 ]換位子羣：[ 42, 6 ]自同構羣：1008正規子羣個數：14
[ 168, 19 ]:1,3,14,4,42,6,0,56,18,0,0,24,0,0,0,
0,共軛類數：40中心：[ 8, 2 ]換位子羣：[ 7, 1 ]自同構羣：336正規子羣個數：24
[ 168, 20 ]:1,5,14,2,70,6,0,28,30,0,0,12,0,0,0,
0,共軛類數：25中心：[ 2, 1 ]換位子羣：[ 14, 2 ]自同構羣：336正規子羣個數：18
[ 168, 21 ]:1,1,14,6,14,6,0,84,6,0,0,36,0,0,0,
0,共軛類數：25中心：[ 2, 1 ]換位子羣：[ 14, 2 ]自同構羣：1008正規子羣個數：18
[ 168, 22 ]:1,1,8,6,8,6,0,0,6,48,0,36,48,0,0,
0,共軛類數：49中心：[ 14, 2 ]換位子羣：[ 8, 4 ]自同構羣：144正規子羣個數：8
[ 168, 23 ]:1,1,56,6,56,6,0,0,6,0,0,36,0,0,0,
0,共軛類數：17中心：[ 2, 1 ]換位子羣：[ 56, 10 ]自同構羣：504正規子羣個數：7
[ 168, 24 ]:1,1,2,30,2,6,0,60,6,12,0,12,12,0,24,
0,共軛類數：51中心：[ 6, 2 ]換位子羣：[ 14, 2 ]自同構羣：672正規子羣個數：18
[ 168, 25 ]:1,15,2,16,30,6,0,32,6,12,0,12,12,0,24,
0,共軛類數：60中心：[ 12, 2 ]換位子羣：[ 7, 1 ]自同構羣：336正規子羣個數：22
[ 168, 26 ]:1,29,2,2,58,6,0,4,6,12,0,12,12,0,24,
0,共軛類數：51中心：[ 6, 2 ]換位子羣：[ 14, 2 ]自同構羣：672正規子羣個數：18
[ 168, 27 ]:1,3,2,28,6,6,0,56,18,12,0,0,36,0,0,
0,共軛類數：60中心：[ 12, 5 ]換位子羣：[ 7, 1 ]自同構羣：672正規子羣個數：26
[ 168, 28 ]:1,17,2,14,34,6,0,28,18,12,0,0,36,0,0,
0,共軛類數：51中心：[ 6, 2 ]換位子羣：[ 14, 2 ]自同構羣：336正規子羣個數：18
[ 168, 29 ]:1,1,2,14,2,6,0,4,6,12,0,84,12,0,24,
0,共軛類數：63中心：[ 14, 2 ]換位子羣：[ 6, 2 ]自同構羣：288正規子羣個數：18
[ 168, 30 ]:1,7,2,8,2,6,0,4,42,12,0,48,12,0,24,
0,共軛類數：84中心：[ 28, 2 ]換位子羣：[ 3, 1 ]自同構羣：144正規子羣個數：22
[ 168, 31 ]:1,13,2,2,2,6,0,4,78,12,0,12,12,0,24,
0,共軛類數：63中心：[ 14, 2 ]換位子羣：[ 6, 2 ]自同構羣：288正規子羣個數：18
[ 168, 32 ]:1,3,2,12,6,6,0,0,18,12,0,72,36,0,0,
0,共軛類數：84中心：[ 28, 4 ]換位子羣：[ 3, 1 ]自同構羣：288正規子羣個數：26
[ 168, 33 ]:1,9,2,6,6,6,0,0,54,12,0,36,36,0,0,
0,共軛類數：63中心：[ 14, 2 ]換位子羣：[ 6, 2 ]自同構羣：144正規子羣個數：18
[ 168, 34 ]:1,1,2,86,2,6,0,4,6,12,0,12,12,0,24,
0,共軛類數：45中心：[ 2, 1 ]換位子羣：[ 42, 6 ]自同構羣：2016正規子羣個數：15
[ 168, 35 ]:1,43,2,44,2,6,0,4,6,12,0,12,12,0,24,
0,共軛類數：48中心：[ 4, 1 ]換位子羣：[ 21, 2 ]自同構羣：1008正規子羣個數：17
[ 168, 36 ]:1,85,2,2,2,6,0,4,6,12,0,12,12,0,24,
0,共軛類數：45中心：[ 2, 1 ]換位子羣：[ 42, 6 ]自同構羣：2016正規子羣個數：15
[ 168, 37 ]:1,3,2,84,6,6,0,0,18,12,0,0,36,0,0,
0,共軛類數：48中心：[ 4, 2 ]換位子羣：[ 21, 2 ]自同構羣：2016正規子羣個數：23
[ 168, 38 ]:1,45,2,42,6,6,0,0,18,12,0,0,36,0,0,
0,共軛類數：45中心：[ 2, 1 ]換位子羣：[ 42, 6 ]自同構羣：1008正規子羣個數：15
[ 168, 39 ]:1,3,2,4,6,6,0,8,18,12,0,24,36,0,48,
0,共軛類數：168中心：[ 168, 39 ]換位子羣：[ 1, 1 ]自同構羣：96正規子羣個數：32
[ 168, 40 ]:1,5,2,2,10,6,0,4,30,12,0,12,60,0,24,
0,共軛類數：105中心：[ 42, 6 ]換位子羣：[ 2, 1 ]自同構羣：96正規子羣個數：24
[ 168, 41 ]:1,1,2,6,2,6,0,12,6,12,0,36,12,0,72,
0,共軛類數：105中心：[ 42, 6 ]換位子羣：[ 2, 1 ]自同構羣：288正規子羣個數：24
[ 168, 42 ]:1,21,56,42,0,48,0,0,0,0,0,0,0,0,0,
0,共軛類數：6中心：[ 1, 1 ]換位子羣：[ 168, 42 ]自同構羣：336正規子羣個數：2
[ 168, 43 ]:1,7,56,0,56,48,0,0,0,0,0,0,0,0,0,
0,共軛類數：8中心：[ 1, 1 ]換位子羣：[ 56, 11 ]自同構羣：168正規子羣個數：4
[ 168, 44 ]:1,7,2,0,14,48,0,0,0,96,0,0,0,0,0,
0,共軛類數：24中心：[ 3, 1 ]換位子羣：[ 8, 5 ]自同構羣：336正規子羣個數：6
[ 168, 45 ]:1,9,8,6,0,6,0,0,54,48,0,36,0,0,0,
0,共軛類數：35中心：[ 7, 1 ]換位子羣：[ 12, 3 ]自同構羣：144正規子羣個數：8
[ 168, 46 ]:1,45,8,42,0,6,0,0,18,48,0,0,0,0,0,
0,共軛類數：17中心：[ 1, 1 ]換位子羣：[ 84, 10 ]自同構羣：1008正規子羣個數：7
[ 168, 47 ]:1,31,14,0,98,6,0,0,18,0,0,0,0,0,0,
0,共軛類數：28中心：[ 4, 2 ]換位子羣：[ 7, 1 ]自同構羣：1008正規子羣個數：37
[ 168, 48 ]:1,31,8,0,56,6,0,0,18,48,0,0,0,0,0,
0,共軛類數：20中心：[ 1, 1 ]換位子羣：[ 28, 4 ]自同構羣：1008正規子羣個數：9
[ 168, 49 ]:1,31,56,0,56,6,0,0,18,0,0,0,0,0,0,
0,共軛類數：12中心：[ 1, 1 ]換位子羣：[ 28, 4 ]自同構羣：504正規子羣個數：8
[ 168, 50 ]:1,63,2,0,30,6,0,0,42,12,0,0,12,0,0,
0,共軛類數：30中心：[ 2, 1 ]換位子羣：[ 21, 2 ]自同構羣：1008正規子羣個數：28
[ 168, 51 ]:1,7,14,0,98,6,0,0,42,0,0,0,0,0,0,
0,共軛類數：40中心：[ 8, 5 ]換位子羣：[ 7, 1 ]自同構羣：7056正規子羣個數：48
[ 168, 52 ]:1,7,8,0,8,6,0,0,42,48,0,0,48,0,0,
0,共軛類數：56中心：[ 14, 2 ]換位子羣：[ 4, 2 ]自同構羣：144正規子羣個數：12
[ 168, 53 ]:1,7,56,0,56,6,0,0,42,0,0,0,0,0,0,
0,共軛類數：24中心：[ 2, 1 ]換位子羣：[ 28, 4 ]自同構羣：504正規子羣個數：10
[ 168, 54 ]:1,31,2,0,62,6,0,0,18,12,0,0,36,0,0,
0,共軛類數：60中心：[ 12, 5 ]換位子羣：[ 7, 1 ]自同構羣：2016正規子羣個數：42
[ 168, 55 ]:1,15,2,0,6,6,0,0,90,12,0,0,36,0,0,
0,共軛類數：84中心：[ 28, 4 ]換位子羣：[ 3, 1 ]自同構羣：864正規子羣個數：42
[ 168, 56 ]:1,87,2,0,6,6,0,0,18,12,0,0,36,0,0,
0,共軛類數：48中心：[ 4, 2 ]換位子羣：[ 21, 2 ]自同構羣：6048正規子羣個數：31
[ 168, 57 ]:1,7,2,0,14,6,0,0,42,12,0,0,84,0,0,
0,共軛類數：168中心：[ 168, 57 ]換位子羣：[ 1, 1 ]自同構羣：2016正規子羣個數：64