在陈松良等人的《关于108阶群的完全分类》一文中证明了G108共有45=10+7+28+0种互不同构的类型.若Sylow子群都正规,则G有10种;若Sylow 2-子群正规而Sylow 3-子群不正规,则G有7种;若Sylow 3-子群正规而Sylow 2-子群不正规,则G有28种;若Sylow子群都不正规,则G不存在。
20151029：4种类型所包含的GAP4编号：
Sylow子群都正规(10个):2、5、12~14、29~31、35、45
Sylow 2-子群正规而Sylow 3-子群不正规(7个):3、18~22、41
Sylow 3-子群正规而Sylow 2-子群不正规(28个):1、4、6~11、15~17、23~28、32~34、36~40、42~44
Sylow子群都不正规(0个):
gap> Factors(108);
[ 2, 2, 3, 3, 3 ]
gap> NumberSmallGroups(108);
45
gap> for n in [1..45] do g:=SmallGroup(108,n);;gid:=IdGroup(g);Print(gid,"是否幂零：",IsNilpotentGroup(g));s:=Elements(g);;sl2:=SylowSubgroup(g,2);;Print(IdGroup(sl2),IsSubnormal(g,sl2));sl3:=SylowSubgroup(g,3);;Print(IdGroup(sl3),IsSubnormal(g,sl3),"\n");od;
[ 108, 1 ]是否幂零：false[ 4, 1 ]false[ 27, 1 ]true
[ 108, 2 ]是否幂零：true[ 4, 1 ]true[ 27, 1 ]true
[ 108, 3 ]是否幂零：false[ 4, 2 ]true[ 27, 1 ]false
[ 108, 4 ]是否幂零：false[ 4, 2 ]false[ 27, 1 ]true
[ 108, 5 ]是否幂零：true[ 4, 2 ]true[ 27, 1 ]true
[ 108, 6 ]是否幂零：false[ 4, 1 ]false[ 27, 2 ]true
[ 108, 7 ]是否幂零：false[ 4, 1 ]false[ 27, 2 ]true
[ 108, 8 ]是否幂零：false[ 4, 1 ]false[ 27, 3 ]true
[ 108, 9 ]是否幂零：false[ 4, 1 ]false[ 27, 4 ]true
[ 108, 10 ]是否幂零：false[ 4, 1 ]false[ 27, 2 ]true
[ 108, 11 ]是否幂零：false[ 4, 1 ]false[ 27, 3 ]true
[ 108, 12 ]是否幂零：true[ 4, 1 ]true[ 27, 2 ]true
[ 108, 13 ]是否幂零：true[ 4, 1 ]true[ 27, 3 ]true
[ 108, 14 ]是否幂零：true[ 4, 1 ]true[ 27, 4 ]true
[ 108, 15 ]是否幂零：false[ 4, 1 ]false[ 27, 3 ]true
[ 108, 16 ]是否幂零：false[ 4, 2 ]false[ 27, 2 ]true
[ 108, 17 ]是否幂零：false[ 4, 2 ]false[ 27, 3 ]true
[ 108, 18 ]是否幂零：false[ 4, 2 ]true[ 27, 2 ]false
[ 108, 19 ]是否幂零：false[ 4, 2 ]true[ 27, 4 ]false
[ 108, 20 ]是否幂零：false[ 4, 2 ]true[ 27, 2 ]false
[ 108, 21 ]是否幂零：false[ 4, 2 ]true[ 27, 4 ]false
[ 108, 22 ]是否幂零：false[ 4, 2 ]true[ 27, 3 ]false
[ 108, 23 ]是否幂零：false[ 4, 2 ]false[ 27, 2 ]true
[ 108, 24 ]是否幂零：false[ 4, 2 ]false[ 27, 2 ]true
[ 108, 25 ]是否幂零：false[ 4, 2 ]false[ 27, 3 ]true
[ 108, 26 ]是否幂零：false[ 4, 2 ]false[ 27, 4 ]true
[ 108, 27 ]是否幂零：false[ 4, 2 ]false[ 27, 2 ]true
[ 108, 28 ]是否幂零：false[ 4, 2 ]false[ 27, 3 ]true
[ 108, 29 ]是否幂零：true[ 4, 2 ]true[ 27, 2 ]true
[ 108, 30 ]是否幂零：true[ 4, 2 ]true[ 27, 3 ]true
[ 108, 31 ]是否幂零：true[ 4, 2 ]true[ 27, 4 ]true
[ 108, 32 ]是否幂零：false[ 4, 1 ]false[ 27, 5 ]true
[ 108, 33 ]是否幂零：false[ 4, 1 ]false[ 27, 5 ]true
[ 108, 34 ]是否幂零：false[ 4, 1 ]false[ 27, 5 ]true
[ 108, 35 ]是否幂零：true[ 4, 1 ]true[ 27, 5 ]true
[ 108, 36 ]是否幂零：false[ 4, 1 ]false[ 27, 5 ]true
[ 108, 37 ]是否幂零：false[ 4, 1 ]false[ 27, 5 ]true
[ 108, 38 ]是否幂零：false[ 4, 2 ]false[ 27, 5 ]true
[ 108, 39 ]是否幂零：false[ 4, 2 ]false[ 27, 5 ]true
[ 108, 40 ]是否幂零：false[ 4, 2 ]false[ 27, 5 ]true
[ 108, 41 ]是否幂零：false[ 4, 2 ]true[ 27, 5 ]false
[ 108, 42 ]是否幂零：false[ 4, 2 ]false[ 27, 5 ]true
[ 108, 43 ]是否幂零：false[ 4, 2 ]false[ 27, 5 ]true
[ 108, 44 ]是否幂零：false[ 4, 2 ]false[ 27, 5 ]true
[ 108, 45 ]是否幂零：true[ 4, 2 ]true[ 27, 5 ]true
gap>  for n in [1..45] do G:=SmallGroup(108,n);idn:=IdGroup(G);Print(idn);Print(":");L:=List(Elements(G),Order);;M:=[1,2,3,4,6,9,12,18,27,36,54,108];;for i in M do Print(Size(Positions(L,i)),","); od;Print("\n");od;
[ 108, 1 ]:1,1,2,54,2,6,0,6,18,0,18,0,
[ 108, 2 ]:1,1,2,2,2,6,4,6,18,12,18,36,
[ 108, 3 ]:1,3,2,0,6,6,0,18,72,0,0,0,
[ 108, 4 ]:1,55,2,0,2,6,0,6,18,0,18,0,
[ 108, 5 ]:1,3,2,0,6,6,0,18,18,0,54,0,
[ 108, 6 ]:1,1,8,18,8,18,36,18,0,0,0,0,
[ 108, 7 ]:1,1,8,6,8,18,12,18,0,36,0,0,
[ 108, 8 ]:1,1,26,18,26,0,36,0,0,0,0,0,
[ 108, 9 ]:1,1,8,18,8,18,36,18,0,0,0,0,
[ 108, 10 ]:1,1,8,54,8,18,0,18,0,0,0,0,
[ 108, 11 ]:1,1,26,18,26,0,36,0,0,0,0,0,
[ 108, 12 ]:1,1,8,2,8,18,16,18,0,36,0,0,
[ 108, 13 ]:1,1,26,2,26,0,52,0,0,0,0,0,
[ 108, 14 ]:1,1,8,2,8,18,16,18,0,36,0,0,
[ 108, 15 ]:1,9,26,18,18,0,36,0,0,0,0,0,
[ 108, 16 ]:1,39,8,0,24,18,0,18,0,0,0,0,
[ 108, 17 ]:1,27,26,0,54,0,0,0,0,0,0,0,
[ 108, 18 ]:1,3,26,0,6,54,0,18,0,0,0,0,
[ 108, 19 ]:1,3,26,0,6,54,0,18,0,0,0,0,
[ 108, 20 ]:1,3,8,0,24,72,0,0,0,0,0,0,
[ 108, 21 ]:1,3,8,0,24,72,0,0,0,0,0,0,
[ 108, 22 ]:1,3,80,0,24,0,0,0,0,0,0,0,
[ 108, 23 ]:1,19,8,0,44,18,0,18,0,0,0,0,
[ 108, 24 ]:1,7,8,0,20,18,0,54,0,0,0,0,
[ 108, 25 ]:1,19,26,0,62,0,0,0,0,0,0,0,
[ 108, 26 ]:1,19,8,0,44,18,0,18,0,0,0,0,
[ 108, 27 ]:1,55,8,0,8,18,0,18,0,0,0,0,
[ 108, 28 ]:1,19,26,0,62,0,0,0,0,0,0,0,
[ 108, 29 ]:1,3,8,0,24,18,0,54,0,0,0,0,
[ 108, 30 ]:1,3,26,0,78,0,0,0,0,0,0,0,
[ 108, 31 ]:1,3,8,0,24,18,0,54,0,0,0,0,
[ 108, 32 ]:1,1,26,6,26,0,48,0,0,0,0,0,
[ 108, 33 ]:1,1,26,18,26,0,36,0,0,0,0,0,
[ 108, 34 ]:1,1,26,54,26,0,0,0,0,0,0,0,
[ 108, 35 ]:1,1,26,2,26,0,52,0,0,0,0,0,
[ 108, 36 ]:1,9,26,18,18,0,36,0,0,0,0,0,
[ 108, 37 ]:1,9,26,54,18,0,0,0,0,0,0,0,
[ 108, 38 ]:1,15,26,0,66,0,0,0,0,0,0,0,
[ 108, 39 ]:1,39,26,0,42,0,0,0,0,0,0,0,
[ 108, 40 ]:1,27,26,0,54,0,0,0,0,0,0,0,
[ 108, 41 ]:1,3,80,0,24,0,0,0,0,0,0,0,
[ 108, 42 ]:1,7,26,0,74,0,0,0,0,0,0,0,
[ 108, 43 ]:1,19,26,0,62,0,0,0,0,0,0,0,
[ 108, 44 ]:1,55,26,0,26,0,0,0,0,0,0,0,
[ 108, 45 ]:1,3,26,0,78,0,0,0,0,0,0,0,
gap> for n in [1..45] do G:=SmallGroup(108,n);idn:=IdGroup(G);Print(idn);Print(":");L:=List(Elements(G),Order);;M:=[1,2,3,4,6,9,12,18,27,36,54,108];;for i in M do Print(Size(Positions(L,i)),","); od;arr:=[];;idn:=IdGroup(G);cl:=ConjugacyClasses(G);;Append(arr,"共轭类数：");;Append(arr,String(Size(cl)));Append(arr,"中心：");;Append(arr,String(IdGroup(Center(G))));;Append(arr,"换位子群：");;Append(arr,String(IdGroup(DerivedSubgroup(G))));;Append(arr,"自同构群：");;Append(arr,String(Order(AutomorphismGroup(G))));;cl:=NormalSubgroups(G);;Append(arr,"正规子群个数：");;len:=Size(cl);;Append(arr,String(len));;Print(arr);Print("\n");od;
[ 108, 1 ]:1,1,2,54,2,6,0,6,18,0,18,
0,共轭类数：30中心：[ 2, 1 ]换位子群：[ 27, 1 ]自同构群：972正规子群个数：9
[ 108, 2 ]:1,1,2,2,2,6,4,6,18,12,18,
36,共轭类数：108中心：[ 108, 2 ]换位子群：[ 1, 1 ]自同构群：36正规子群个数：12
[ 108, 3 ]:1,3,2,0,6,6,0,18,72,0,0,
0,共轭类数：36中心：[ 9, 1 ]换位子群：[ 4, 2 ]自同构群：216正规子群个数：7
[ 108, 4 ]:1,55,2,0,2,6,0,6,18,0,18,
0,共轭类数：30中心：[ 2, 1 ]换位子群：[ 27, 1 ]自同构群：972正规子群个数：11
[ 108, 5 ]:1,3,2,0,6,6,0,18,18,0,54,
0,共轭类数：108中心：[ 108, 5 ]换位子群：[ 1, 1 ]自同构群：108正规子群个数：20
[ 108, 6 ]:1,1,8,18,8,18,36,18,0,0,0,
0,共轭类数：36中心：[ 6, 2 ]换位子群：[ 9, 1 ]自同构群：216正规子群个数：14
[ 108, 7 ]:1,1,8,6,8,18,12,18,0,36,0,
0,共轭类数：54中心：[ 18, 2 ]换位子群：[ 3, 1 ]自同构群：72正规子群个数：15
[ 108, 8 ]:1,1,26,18,26,0,36,0,0,0,0,
0,共轭类数：20中心：[ 2, 1 ]换位子群：[ 9, 2 ]自同构群：216正规子群个数：12
[ 108, 9 ]:1,1,8,18,8,18,36,18,0,0,0,
0,共轭类数：20中心：[ 2, 1 ]换位子群：[ 9, 1 ]自同构群：108正规子群个数：12
[ 108, 10 ]:1,1,8,54,8,18,0,18,0,0,0,
0,共轭类数：30中心：[ 2, 1 ]换位子群：[ 27, 2 ]自同构群：5832正规子群个数：21
[ 108, 11 ]:1,1,26,18,26,0,36,0,0,0,0,
0,共轭类数：20中心：[ 6, 2 ]换位子群：[ 27, 3 ]自同构群：864正规子群个数：15
[ 108, 12 ]:1,1,8,2,8,18,16,18,0,36,0,
0,共轭类数：108中心：[ 108, 12 ]换位子群：[ 1, 1 ]自同构群：216正规子群个数：30
[ 108, 13 ]:1,1,26,2,26,0,52,0,0,0,0,
0,共轭类数：44中心：[ 12, 2 ]换位子群：[ 3, 1 ]自同构群：864正规子群个数：21
[ 108, 14 ]:1,1,8,2,8,18,16,18,0,36,0,
0,共轭类数：44中心：[ 12, 2 ]换位子群：[ 3, 1 ]自同构群：108正规子群个数：21
[ 108, 15 ]:1,9,26,18,18,0,36,0,0,0,0,
0,共轭类数：14中心：[ 3, 1 ]换位子群：[ 27, 3 ]自同构群：144正规子群个数：5
[ 108, 16 ]:1,39,8,0,24,18,0,18,0,0,0,
0,共轭类数：18中心：[ 1, 1 ]换位子群：[ 27, 2 ]自同构群：324正规子群个数：13
[ 108, 17 ]:1,27,26,0,54,0,0,0,0,0,0,
0,共轭类数：11中心：[ 1, 1 ]换位子群：[ 27, 3 ]自同构群：216正规子群个数：11
[ 108, 18 ]:1,3,26,0,6,54,0,18,0,0,0,
0,共轭类数：36中心：[ 9, 1 ]换位子群：[ 4, 2 ]自同构群：432正规子群个数：13
[ 108, 19 ]:1,3,26,0,6,54,0,18,0,0,0,
0,共轭类数：20中心：[ 3, 1 ]换位子群：[ 12, 5 ]自同构群：216正规子群个数：10
[ 108, 20 ]:1,3,8,0,24,72,0,0,0,0,0,
0,共轭类数：36中心：[ 9, 2 ]换位子群：[ 4, 2 ]自同构群：1296正规子群个数：16
[ 108, 21 ]:1,3,8,0,24,72,0,0,0,0,0,
0,共轭类数：20中心：[ 3, 1 ]换位子群：[ 12, 5 ]自同构群：648正规子群个数：10
[ 108, 22 ]:1,3,80,0,24,0,0,0,0,0,0,
0,共轭类数：20中心：[ 3, 1 ]换位子群：[ 12, 5 ]自同构群：1296正规子群个数：10
[ 108, 23 ]:1,19,8,0,44,18,0,18,0,0,0,
0,共轭类数：36中心：[ 6, 2 ]换位子群：[ 9, 1 ]自同构群：216正规子群个数：18
[ 108, 24 ]:1,7,8,0,20,18,0,54,0,0,0,
0,共轭类数：54中心：[ 18, 2 ]换位子群：[ 3, 1 ]自同构群：72正规子群个数：21
[ 108, 25 ]:1,19,26,0,62,0,0,0,0,0,0,
0,共轭类数：20中心：[ 2, 1 ]换位子群：[ 9, 2 ]自同构群：216正规子群个数：16
[ 108, 26 ]:1,19,8,0,44,18,0,18,0,0,0,
0,共轭类数：20中心：[ 2, 1 ]换位子群：[ 9, 1 ]自同构群：108正规子群个数：16
[ 108, 27 ]:1,55,8,0,8,18,0,18,0,0,0,
0,共轭类数：30中心：[ 2, 1 ]换位子群：[ 27, 2 ]自同构群：5832正规子群个数：23
[ 108, 28 ]:1,19,26,0,62,0,0,0,0,0,0,
0,共轭类数：20中心：[ 6, 2 ]换位子群：[ 27, 3 ]自同构群：864正规子群个数：17
[ 108, 29 ]:1,3,8,0,24,18,0,54,0,0,0,
0,共轭类数：108中心：[ 108, 29 ]换位子群：[ 1, 1 ]自同构群：648正规子群个数：50
[ 108, 30 ]:1,3,26,0,78,0,0,0,0,0,0,
0,共轭类数：44中心：[ 12, 5 ]换位子群：[ 3, 1 ]自同构群：2592正规子群个数：35
[ 108, 31 ]:1,3,8,0,24,18,0,54,0,0,0,
0,共轭类数：44中心：[ 12, 5 ]换位子群：[ 3, 1 ]自同构群：324正规子群个数：35
[ 108, 32 ]:1,1,26,6,26,0,48,0,0,0,0,
0,共轭类数：54中心：[ 18, 5 ]换位子群：[ 3, 1 ]自同构群：576正规子群个数：30
[ 108, 33 ]:1,1,26,18,26,0,36,0,0,0,0,
0,共轭类数：36中心：[ 6, 2 ]换位子群：[ 9, 2 ]自同构群：1728正规子群个数：26
[ 108, 34 ]:1,1,26,54,26,0,0,0,0,0,0,
0,共轭类数：30中心：[ 2, 1 ]换位子群：[ 27, 5 ]自同构群：606528正规子群个数：57
[ 108, 35 ]:1,1,26,2,26,0,52,0,0,0,0,
0,共轭类数：108中心：[ 108, 35 ]换位子群：[ 1, 1 ]自同构群：22464正规子群个数：84
[ 108, 36 ]:1,9,26,18,18,0,36,0,0,0,0,
0,共轭类数：18中心：[ 3, 1 ]换位子群：[ 9, 2 ]自同构群：288正规子群个数：8
[ 108, 37 ]:1,9,26,54,18,0,0,0,0,0,0,
0,共轭类数：12中心：[ 1, 1 ]换位子群：[ 27, 5 ]自同构群：864正规子群个数：7
[ 108, 38 ]:1,15,26,0,66,0,0,0,0,0,0,
0,共轭类数：27中心：[ 3, 1 ]换位子群：[ 9, 2 ]自同构群：144正规子群个数：20
[ 108, 39 ]:1,39,26,0,42,0,0,0,0,0,0,
0,共轭类数：18中心：[ 1, 1 ]换位子群：[ 27, 5 ]自同构群：2592正规子群个数：22
[ 108, 40 ]:1,27,26,0,54,0,0,0,0,0,0,
0,共轭类数：15中心：[ 1, 1 ]换位子群：[ 27, 5 ]自同构群：1296正规子群个数：15
[ 108, 41 ]:1,3,80,0,24,0,0,0,0,0,0,
0,共轭类数：36中心：[ 9, 2 ]换位子群：[ 4, 2 ]自同构群：10368正规子群个数：34
[ 108, 42 ]:1,7,26,0,74,0,0,0,0,0,0,
0,共轭类数：54中心：[ 18, 5 ]换位子群：[ 3, 1 ]自同构群：576正规子群个数：42
[ 108, 43 ]:1,19,26,0,62,0,0,0,0,0,0,
0,共轭类数：36中心：[ 6, 2 ]换位子群：[ 9, 2 ]自同构群：1728正规子群个数：30
[ 108, 44 ]:1,55,26,0,26,0,0,0,0,0,0,
0,共轭类数：30中心：[ 2, 1 ]换位子群：[ 27, 5 ]自同构群：606528正规子群个数：59
[ 108, 45 ]:1,3,26,0,78,0,0,0,0,0,0,
0,共轭类数：108中心：[ 108, 45 ]换位子群：[ 1, 1 ]自同构群：67392正规子群个数：140