15种24阶群

原题为《15种24阶群的C++程序实现(20140426-20140515)》。

gap> Factors(24);
[ 2, 2, 2, 3 ]
由Sylow第一定理可知24阶群必然存在2、2^2、2^3、3阶子群。
Sylow第一定理：当群的阶可以表示为一个“素数的方幂”与一个与这个素数互素的数的乘积时，可以确定这个群一定含有这个“素数的某些方幂”阶的子群。
四次对称群S_4有1个1阶元，9个2阶元，8个3阶元，6个4阶元，0个6阶元，0个8阶元，0个12阶元，0个24阶元
S_4没有8阶元，|Syl_2(S_4)|=8、Syl_2(S_4)=GAP4[8,3]=D_4!=C_8=GAP4[8,1]
GAP4[8,3]：1个1阶元,5个2阶元,2个4阶元,0个8阶元,
gap> for n in [1..15] do g:=SmallGroup(24,n);;gid:=StructureDescription(g);Print(gid,"是否超可解：",IsSupersolvableGroup(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;
C3 : C8是否超可解：true[ 8, 1 ]false[ 3, 1 ]true
C24是否超可解：true[ 8, 1 ]true[ 3, 1 ]true
SL(2,3)是否超可解：false[ 8, 4 ]true[ 3, 1 ]false
C3 : Q8是否超可解：true[ 8, 4 ]false[ 3, 1 ]true
C4 x S3是否超可解：true[ 8, 2 ]false[ 3, 1 ]true
D24是否超可解：true[ 8, 3 ]false[ 3, 1 ]true
C2 x (C3 : C4)是否超可解：true[ 8, 2 ]false[ 3, 1 ]true
(C6 x C2) : C2是否超可解：true[ 8, 3 ]false[ 3, 1 ]true
C12 x C2是否超可解：true[ 8, 2 ]true[ 3, 1 ]true
C3 x D8是否超可解：true[ 8, 3 ]true[ 3, 1 ]true
C3 x Q8是否超可解：true[ 8, 4 ]true[ 3, 1 ]true
S4是否超可解：false[ 8, 3 ]false[ 3, 1 ]false
C2 x A4是否超可解：false[ 8, 5 ]true[ 3, 1 ]false
C2 x C2 x S3是否超可解：true[ 8, 5 ]false[ 3, 1 ]true
C6 x C2 x C2是否超可解：true[ 8, 5 ]true[ 3, 1 ]true
下列的叙述在有限群中均为等价，表现出一个幂零性的有用性质：
G为一幂零群。
若H为G的纯子群，则H为N(H)(G内H之正规化子)的纯正规子群。
每一个G的最大纯子群均为正规的。
G为其西洛子群的直积。
gap> for n in [1..15] do G:=SmallGroup(24,n);sl2:=SylowSubgroup(G,2);;sl3:=SylowSubgroup(G,3);;idn:=IdGroup(G);Print(idn);Print(":");L:=List(Elements(G),Order);;M:=[1,2,3,4,6,8,12,24];;for i in M do Print(Size(Positions(L,i)),","); od;sl2sl3:=DirectProduct(sl2,sl3);;Print(IdGroup(sl2),"×",IdGroup(sl3),"=",IdGroup(sl2sl3),"是否幂零：",IsNilpotentGroup(G),",",StructureDescription(G),"\n");od;
[ 24, 1 ]:1,1,2,2,2,12,4,0,[ 8, 1 ]×[ 3, 1 ]=[ 24, 2 ]是否幂零：false,C3 : C8
[ 24, 2 ]:1,1,2,2,2,4,4,8,[ 8, 1 ]×[ 3, 1 ]=[ 24, 2 ]是否幂零：true,C24
[ 24, 3 ]:1,1,8,6,8,0,0,0,[ 8, 4 ]×[ 3, 1 ]=[ 24, 11 ]是否幂零：false,SL(2,3)
[ 24, 4 ]:1,1,2,14,2,0,4,0,[ 8, 4 ]×[ 3, 1 ]=[ 24, 11 ]是否幂零：false,C3 : Q8
[ 24, 5 ]:1,7,2,8,2,0,4,0,[ 8, 2 ]×[ 3, 1 ]=[ 24, 9 ]是否幂零：false,C4 x S3
[ 24, 6 ]:1,13,2,2,2,0,4,0,[ 8, 3 ]×[ 3, 1 ]=[ 24, 10 ]是否幂零：false,D24
[ 24, 7 ]:1,3,2,12,6,0,0,0,[ 8, 2 ]×[ 3, 1 ]=[ 24, 9 ]是否幂零：false,C2 x (C3 : C4)
[ 24, 8 ]:1,9,2,6,6,0,0,0,[ 8, 3 ]×[ 3, 1 ]=[ 24, 10 ]是否幂零：false,(C6 x C2) : C2
[ 24, 9 ]:1,3,2,4,6,0,8,0,[ 8, 2 ]×[ 3, 1 ]=[ 24, 9 ]是否幂零：true,C12 x C2
[ 24, 10 ]:1,5,2,2,10,0,4,0,[ 8, 3 ]×[ 3, 1 ]=[ 24, 10 ]是否幂零：true,C3 x D8
[ 24, 11 ]:1,1,2,6,2,0,12,0,[ 8, 4 ]×[ 3, 1 ]=[ 24, 11 ]是否幂零：true,C3 x Q8
[ 24, 12 ]:1,9,8,6,0,0,0,0,[ 8, 3 ]×[ 3, 1 ]=[ 24, 10 ]是否幂零：false,S4
[ 24, 13 ]:1,7,8,0,8,0,0,0,[ 8, 5 ]×[ 3, 1 ]=[ 24, 15 ]是否幂零：false,C2 x A4
[ 24, 14 ]:1,15,2,0,6,0,0,0,[ 8, 5 ]×[ 3, 1 ]=[ 24, 15 ]是否幂零：false,C2 x C2 x S3
[ 24, 15 ]:1,7,2,0,14,0,0,0,[ 8, 5 ]×[ 3, 1 ]=[ 24, 15 ]是否幂零：true,C6 x C2 x C2
http://world.std.com/~jmccarro/math/SmallGroups/g24.4.html

http://escarbille.free.fr/group.php

http://www.weddslist.com/groups/building/sdp.html

半直积的例子1:

C_3⋊C_2

Aut(C_3)=C_2，C_2到C_2的同态有2个：f_1(非平凡同态，双射),f_2(平凡同态，C_2 {<}ker(f_2))

我们考虑f_1，那么C_3⋊C_2=D_3

半直积的例子2:

K_4⋊C_3

Aut(K_4)=D_3，

D_3有子群C_3，所以C_3到D_3的同态有2个：f_1(非平凡同态，单射),f_2(平凡同态， 此时半直积就是直积K_4×C_3)

我们考虑f_1，那么K_4⋊C_3=A_4

G24_1=C_24有1个1阶元，1个2阶元，2个3阶元，2个4阶元，2个6阶元，4个8阶元，4个12阶元，8个24阶元
G24_2=C_6×C_4有1个1阶元，3个2阶元，2个3阶元，4个4阶元，6个6阶元，0个8阶元，8个12阶元，0个24阶元
G24_3=C_6×K_4有1个1阶元，7个2阶元，2个3阶元，0个4阶元，14个6阶元，0个8阶元，0个12阶元，0个24阶元
G24_4有1个1阶元，1个2阶元，2个3阶元，2个4阶元，2个6阶元，12个8阶元，4个12阶元，0个24阶元
G24_5=SL_2(F_3)有1个1阶元，1个2阶元，8个3阶元，6个4阶元，8个6阶元，0个8阶元，0个12阶元，0个24阶元
G24_6=Q_24有1个1阶元，1个2阶元，2个3阶元，14个4阶元，2个6阶元，0个8阶元，4个12阶元，0个24阶元
G24_7=S_3×C_4=D_3×C_4有1个1阶元，7个2阶元，2个3阶元，8个4阶元，2个6阶元，0个8阶元，4个12阶元，0个24阶元
G24_8=D_12有1个1阶元，13个2阶元，2个3阶元，2个4阶元，2个6阶元，0个8阶元，4个12阶元，0个24阶元
G24_9=Q_12×C_2有1个1阶元，3个2阶元，2个3阶元，12个4阶元，6个6阶元，0个8阶元，0个12阶元，0个24阶元
G24_10有1个1阶元，9个2阶元，2个3阶元，6个4阶元，6个6阶元，0个8阶元，0个12阶元，0个24阶元
G24_11=D_4×C_3有1个1阶元，5个2阶元，2个3阶元，2个4阶元，10个6阶元，0个8阶元，4个12阶元，0个24阶元
G24_12=Q_8×C_3有1个1阶元，1个2阶元，2个3阶元，6个4阶元，2个6阶元，0个8阶元，12个12阶元，0个24阶元
G24_13=S_4有1个1阶元，9个2阶元，8个3阶元，6个4阶元，0个6阶元，0个8阶元，0个12阶元，0个24阶元
G24_14=A_4×C_2有1个1阶元，7个2阶元，8个3阶元，0个4阶元，8个6阶元，0个8阶元，0个12阶元，0个24阶元
G24_15!=S_4有1个1阶元，15个2阶元，2个3阶元，0个4阶元，6个6阶元，0个8阶元，0个12阶元，0个24阶元
----24阶非交换群S_3×K_4=D_6×C_2群元的阶----
//1个1阶元，15个2阶元，2个3阶元，6个6阶元

GAP[24,1]=G24_4=C_3⋊C_8=U_24=U_6*4=<a,x|a^3=x^8=e,xax^(-1)=a^(-1)>：有6,8,12阶元

N和H相对于非平凡群同态φ : H → Aut(N)的半直积N⋊H=H:N=C_3⋊C_8=C_8:C_3=GAP[24,1]

gap> H:=CyclicGroup(IsPermGroup,8);;N:=CyclicGroup(IsPermGroup,3);;A:=AutomorphismGroup(N);;elts := Elements(A);sigma := elts[2];sigma^2;map := GroupHomomorphismByImages(H, A, GeneratorsOfGroup(H), [sigma]);SDP := SemidirectProduct(H, map,N);IdGroup(SDP);List(elts,Order);IdGroup(DirectProduct(H,N));
[ [ (1,2,3) ] -> [ (1,2,3) ], [ (1,2,3) ] -> [ (1,3,2) ] ]
[ (1,2,3) ] -> [ (1,3,2) ]
[ (1,2,3) ] -> [ (1,2,3) ]
[ (1,2,3,4,5,6,7,8) ] -> [ [ (1,2,3) ] -> [ (1,3,2) ] ]
Group([ (9,10,11), (1,2,3,4,5,6,7,8)(10,11) ])
[ 24, 1 ]
[ 1, 2 ]
[ 24, 2 ]
平凡群同态的情形：

gap> H:=CyclicGroup(IsPermGroup,8);;N:=CyclicGroup(IsPermGroup,3);;A:=AutomorphismGroup(N);;elts := Elements(A);sigma := elts[1];sigma^2;map := GroupHomomorphismByImages(H, A, GeneratorsOfGroup(H), [sigma]);SDP := SemidirectProduct(H, map,N);IdGroup(SDP);List(elts,Order);IdGroup(DirectProduct(H,N));
[ [ (1,2,3) ] -> [ (1,2,3) ], [ (1,2,3) ] -> [ (1,3,2) ] ]
[ (1,2,3) ] -> [ (1,2,3) ]
[ (1,2,3) ] -> [ (1,2,3) ]
[ (1,2,3,4,5,6,7,8) ] -> [ [ (1,2,3) ] -> [ (1,2,3) ] ]
Group([ (9,10,11), (1,2,3,4,5,6,7,8) ])
[ 24, 2 ]
[ 1, 2 ]
[ 24, 2 ]

G24_10=V_24=<a,b|a^6=b^4=1,ba=a^(-1)b^(-1),b^(-1)a=a^(-1)b>：有6阶元，没有8,12阶元
G24_13=C_3⋊D_4=S_4：没有6,8,12阶元

#include <fstream>
#include <iostream>
#include <vector>
using namespace std;

vector<int> g_G24_4;

int g_G24_4Mul[24][24]={
 0
};

// 未知24阶群的群元的阶
int getG24EOrder(int a)
{
 if(a<0||a>=24)
  return -1;
 int t=0;
 for(int i=1;i<=24;i++)
 {
  t=g_G24_4Mul[t][a];
  if(t==0)
   return i;
 }
 return -1;
}

int main(int argc, char* argv[])
{
 int iElem;
 int sum=0;
 //ifstream fin("G24_4.txt");
 ifstream fin("G24_5.txt");
 while( !fin.eof() && sum<24*24) {
  fin >> iElem;
  sum++;
  int i=(sum-1)/24;
  int j=(sum-1)%24;
  g_G24_4Mul[i][j]=iElem-1;
  g_G24_4.push_back(iElem);
 }
 fin.close();

 //ofstream fout("G24_4_ElementToOrder.txt");
 ofstream fout("G24_5_ElementToOrder.txt");
 int nOrders[8]={1,2,3,4,6,8,12,24};
 int nCounts[25]={0};
 for(int i=0;i<24;i++)
 {
  int ord=getG24EOrder(i);
  printf("G24ElementToOrder(%d)=%d\n",i,ord);
  fout<<"G24ElementToOrder("<<i<<")="<<ord<<endl;
  nCounts[ord]=nCounts[ord]+1;
 }
 char sz[200]={0};
 sprintf(sz,"G24_5有%d个1阶元，%d个2阶元，%d个3阶元，%d个4阶元，%d个6阶元，%d个8阶元，%d个12阶元，%d个24阶元",nCounts[nOrders[0]],nCounts[nOrders[1]],nCounts[nOrders[2]],nCounts[nOrders[3]],nCounts[nOrders[4]],nCounts[nOrders[5]],nCounts[nOrders[6]],nCounts[nOrders[7]]);
 printf("%s\n",sz);
 fout<<sz<<endl;
 fout.close();

 system("pause");
 return 0;
}

证明：S_4!=S_3×K_4
----24阶循环群C_24群元的阶----
//1个1阶元，1个2阶元，2个3阶元，2个4阶元，2个6阶元，4个8阶元，4个12阶元，8个24阶元
G24ElementToOrder(0)=1
G24ElementToOrder(1)=24
G24ElementToOrder(2)=12
G24ElementToOrder(3)=8
G24ElementToOrder(4)=6
G24ElementToOrder(5)=24
G24ElementToOrder(6)=4
G24ElementToOrder(7)=24
G24ElementToOrder(8)=3
G24ElementToOrder(9)=8
G24ElementToOrder(10)=12
G24ElementToOrder(11)=24
G24ElementToOrder(12)=2
G24ElementToOrder(13)=24
G24ElementToOrder(14)=12
G24ElementToOrder(15)=8
G24ElementToOrder(16)=3
G24ElementToOrder(17)=24
G24ElementToOrder(18)=4
G24ElementToOrder(19)=24
G24ElementToOrder(20)=6
G24ElementToOrder(21)=8
G24ElementToOrder(22)=12
G24ElementToOrder(23)=24

----24阶交换群C_6×K_4群元的阶----
//1个1阶元，7个2阶元，2个3阶元，14个6阶元
G24ElementToOrder(0)=1
G24ElementToOrder(1)=2
G24ElementToOrder(2)=2
G24ElementToOrder(3)=2
G24ElementToOrder(4)=6
G24ElementToOrder(5)=6
G24ElementToOrder(6)=6
G24ElementToOrder(7)=6
G24ElementToOrder(8)=3
G24ElementToOrder(9)=6
G24ElementToOrder(10)=6
G24ElementToOrder(11)=6
G24ElementToOrder(12)=2
G24ElementToOrder(13)=2
G24ElementToOrder(14)=2
G24ElementToOrder(15)=2
G24ElementToOrder(16)=3
G24ElementToOrder(17)=6
G24ElementToOrder(18)=6
G24ElementToOrder(19)=6
G24ElementToOrder(20)=6
G24ElementToOrder(21)=6
G24ElementToOrder(22)=6
G24ElementToOrder(23)=6

----24阶交换群C_6×C_4群元的阶----
//1个1阶元，3个2阶元，2个3阶元，4个4阶元，6个6阶元，8个12阶元
G24ElementToOrder(0)=1
G24ElementToOrder(1)=4
G24ElementToOrder(2)=2
G24ElementToOrder(3)=4
G24ElementToOrder(4)=6
G24ElementToOrder(5)=12
G24ElementToOrder(6)=6
G24ElementToOrder(7)=12
G24ElementToOrder(8)=3
G24ElementToOrder(9)=12
G24ElementToOrder(10)=6
G24ElementToOrder(11)=12
G24ElementToOrder(12)=2
G24ElementToOrder(13)=4
G24ElementToOrder(14)=2
G24ElementToOrder(15)=4
G24ElementToOrder(16)=3
G24ElementToOrder(17)=12
G24ElementToOrder(18)=6
G24ElementToOrder(19)=12
G24ElementToOrder(20)=6
G24ElementToOrder(21)=12
G24ElementToOrder(22)=6
G24ElementToOrder(23)=12

----24阶非交换群Q_8×C_3群元的阶----
//1个1阶元，1个2阶元，2个3阶元，6个4阶元，2个6阶元，12个12阶元
G24ElementToOrder(0)=1
G24ElementToOrder(1)=3
G24ElementToOrder(2)=3
G24ElementToOrder(3)=2
G24ElementToOrder(4)=6
G24ElementToOrder(5)=6
G24ElementToOrder(6)=4
G24ElementToOrder(7)=12
G24ElementToOrder(8)=12
G24ElementToOrder(9)=4
G24ElementToOrder(10)=12
G24ElementToOrder(11)=12
G24ElementToOrder(12)=4
G24ElementToOrder(13)=12
G24ElementToOrder(14)=12
G24ElementToOrder(15)=4
G24ElementToOrder(16)=12
G24ElementToOrder(17)=12
G24ElementToOrder(18)=4
G24ElementToOrder(19)=12
G24ElementToOrder(20)=12
G24ElementToOrder(21)=4
G24ElementToOrder(22)=12
G24ElementToOrder(23)=12

----24阶非交换群D_4×C_3群元的阶----
//1个1阶元，5个2阶元，2个3阶元，2个4阶元，10个6阶元，4个12阶元
G24ElementToOrder(0)=1
G24ElementToOrder(1)=3
G24ElementToOrder(2)=3
G24ElementToOrder(3)=2
G24ElementToOrder(4)=6
G24ElementToOrder(5)=6
G24ElementToOrder(6)=2
G24ElementToOrder(7)=6
G24ElementToOrder(8)=6
G24ElementToOrder(9)=2
G24ElementToOrder(10)=6
G24ElementToOrder(11)=6
G24ElementToOrder(12)=2
G24ElementToOrder(13)=6
G24ElementToOrder(14)=6
G24ElementToOrder(15)=2
G24ElementToOrder(16)=6
G24ElementToOrder(17)=6
G24ElementToOrder(18)=4
G24ElementToOrder(19)=12
G24ElementToOrder(20)=12
G24ElementToOrder(21)=4
G24ElementToOrder(22)=12
G24ElementToOrder(23)=12

 ----24阶非交换群S_4群元的阶----
20140606更正后的结果：
G24ElementToOrder(0)=1
G24ElementToOrder(1)=2
G24ElementToOrder(2)=2
G24ElementToOrder(3)=3
G24ElementToOrder(4)=3
G24ElementToOrder(5)=2
G24ElementToOrder(6)=2
G24ElementToOrder(7)=2
G24ElementToOrder(8)=3
G24ElementToOrder(9)=4
G24ElementToOrder(10)=4
G24ElementToOrder(11)=3
G24ElementToOrder(12)=3
G24ElementToOrder(13)=4
G24ElementToOrder(14)=2
G24ElementToOrder(15)=3
G24ElementToOrder(16)=2
G24ElementToOrder(17)=4
G24ElementToOrder(18)=4
G24ElementToOrder(19)=3
G24ElementToOrder(20)=3
G24ElementToOrder(21)=2
G24ElementToOrder(22)=4
G24ElementToOrder(23)=2
S_4有1个1阶元，9个2阶元，8个3阶元，6个4阶元，0个6阶元，0个8阶元，0个12阶元，0个24阶元

错误的结果：//1个1阶元，15个2阶元，2个3阶元，6个6阶元
G24ElementToOrder(0)=1
G24ElementToOrder(1)=6
G24ElementToOrder(2)=2
G24ElementToOrder(3)=2
G24ElementToOrder(4)=2
G24ElementToOrder(5)=6
G24ElementToOrder(6)=2
G24ElementToOrder(7)=2
G24ElementToOrder(8)=6
G24ElementToOrder(9)=6
G24ElementToOrder(10)=2
G24ElementToOrder(11)=2
G24ElementToOrder(12)=6
G24ElementToOrder(13)=2
G24ElementToOrder(14)=2
G24ElementToOrder(15)=3
G24ElementToOrder(16)=2
G24ElementToOrder(17)=3
G24ElementToOrder(18)=2
G24ElementToOrder(19)=2
G24ElementToOrder(20)=6
G24ElementToOrder(21)=2
G24ElementToOrder(22)=2
G24ElementToOrder(23)=2

----24阶非交换群Q_24群元的阶----
//1个1阶元，1个2阶元，2个3阶元，14个4阶元，2个6阶元，4个12阶元
G24ElementToOrder(0)=1
G24ElementToOrder(1)=12
G24ElementToOrder(2)=4
G24ElementToOrder(3)=6
G24ElementToOrder(4)=4
G24ElementToOrder(5)=3
G24ElementToOrder(6)=12
G24ElementToOrder(7)=2
G24ElementToOrder(8)=12
G24ElementToOrder(9)=3
G24ElementToOrder(10)=4
G24ElementToOrder(11)=6
G24ElementToOrder(12)=12
G24ElementToOrder(13)=4
G24ElementToOrder(14)=4
G24ElementToOrder(15)=4
G24ElementToOrder(16)=4
G24ElementToOrder(17)=4
G24ElementToOrder(18)=4
G24ElementToOrder(19)=4
G24ElementToOrder(20)=4
G24ElementToOrder(21)=4
G24ElementToOrder(22)=4
G24ElementToOrder(23)=4

----24阶非交换群A_4×C_2群元的阶----
//1个1阶元，7个2阶元，8个3阶元，8个6阶元
G24ElementToOrder(0)=1
G24ElementToOrder(1)=2
G24ElementToOrder(2)=2
G24ElementToOrder(3)=2
G24ElementToOrder(4)=2
G24ElementToOrder(5)=2
G24ElementToOrder(6)=2
G24ElementToOrder(7)=2
G24ElementToOrder(8)=3
G24ElementToOrder(9)=6
G24ElementToOrder(10)=3
G24ElementToOrder(11)=6
G24ElementToOrder(12)=3
G24ElementToOrder(13)=6
G24ElementToOrder(14)=3
G24ElementToOrder(15)=6
G24ElementToOrder(16)=3
G24ElementToOrder(17)=6
G24ElementToOrder(18)=3
G24ElementToOrder(19)=6
G24ElementToOrder(20)=3
G24ElementToOrder(21)=6
G24ElementToOrder(22)=3
G24ElementToOrder(23)=6

 
----24阶非交换群Q_12×C_2群元的阶----
//1个1阶元，3个2阶元，2个3阶元，12个4阶元，6个6阶元
G24ElementToOrder(0)=1
G24ElementToOrder(1)=2
G24ElementToOrder(2)=3
G24ElementToOrder(3)=6
G24ElementToOrder(4)=3
G24ElementToOrder(5)=6
G24ElementToOrder(6)=2
G24ElementToOrder(7)=2
G24ElementToOrder(8)=6
G24ElementToOrder(9)=6
G24ElementToOrder(10)=6
G24ElementToOrder(11)=6
G24ElementToOrder(12)=4
G24ElementToOrder(13)=4
G24ElementToOrder(14)=4
G24ElementToOrder(15)=4
G24ElementToOrder(16)=4
G24ElementToOrder(17)=4
G24ElementToOrder(18)=4
G24ElementToOrder(19)=4
G24ElementToOrder(20)=4
G24ElementToOrder(21)=4
G24ElementToOrder(22)=4
G24ElementToOrder(23)=4

 ----24阶非交换群SL_2(F_3)群元的阶----
//1个1阶元，1个2阶元，8个3阶元，6个4阶元，8个6阶元
G24ElementToOrder(0)=1
G24ElementToOrder(1)=4
G24ElementToOrder(2)=6
G24ElementToOrder(3)=3
G24ElementToOrder(4)=4
G24ElementToOrder(5)=6
G24ElementToOrder(6)=3
G24ElementToOrder(7)=3
G24ElementToOrder(8)=3
G24ElementToOrder(9)=3
G24ElementToOrder(10)=4
G24ElementToOrder(11)=6
G24ElementToOrder(12)=3
G24ElementToOrder(13)=6
G24ElementToOrder(14)=4
G24ElementToOrder(15)=2
G24ElementToOrder(16)=6
G24ElementToOrder(17)=6
G24ElementToOrder(18)=6
G24ElementToOrder(19)=4
G24ElementToOrder(20)=3
G24ElementToOrder(21)=6
G24ElementToOrder(22)=3
G24ElementToOrder(23)=4

----24阶非交换群D_12群元的阶----
//1个1阶元，13个2阶元，2个3阶元，2个4阶元，2个6阶元，4个12阶元
> euler(12)
4
所以D_12的正规子群C_12中有4个12阶的群元。
G12ElementToOrder(0=I)=1
G12ElementToOrder(r)=12
G12ElementToOrder(r^2)=6
G12ElementToOrder(r^3)=4
G12ElementToOrder(r^4)=3
G12ElementToOrder(r^5)=12
G12ElementToOrder(r^6)=2
G12ElementToOrder(r^7)=12
G12ElementToOrder(r^8)=3
G12ElementToOrder(r^9)=4
G12ElementToOrder(r^10)=6
G12ElementToOrder(r^11)=12
G12ElementToOrder(f)=2
G12ElementToOrder(rf)=2
G12ElementToOrder(r^2f)=2
G12ElementToOrder(r^3f)=2
G12ElementToOrder(r^4f)=2
G12ElementToOrder(r^5f)=2
G12ElementToOrder(r^6f)=2
G12ElementToOrder(r^7f)=2
G12ElementToOrder(r^8f)=2
G12ElementToOrder(r^9f)=2
G12ElementToOrder(r^10f)=2
G12ElementToOrder(r^11f)=2

 
----24阶非交换群S_3×K_4群元的阶----
//1个1阶元，15个2阶元，2个3阶元，6个6阶元
G24ElementToOrder(0)=1
G24ElementToOrder(1)=2
G24ElementToOrder(2)=2
G24ElementToOrder(3)=2
G24ElementToOrder(4)=2
G24ElementToOrder(5)=2
G24ElementToOrder(6)=2
G24ElementToOrder(7)=2
G24ElementToOrder(8)=2
G24ElementToOrder(9)=2
G24ElementToOrder(10)=2
G24ElementToOrder(11)=2
G24ElementToOrder(12)=3
G24ElementToOrder(13)=6
G24ElementToOrder(14)=6
G24ElementToOrder(15)=6
G24ElementToOrder(16)=3
G24ElementToOrder(17)=6
G24ElementToOrder(18)=6
G24ElementToOrder(19)=6
G24ElementToOrder(20)=2
G24ElementToOrder(21)=2
G24ElementToOrder(22)=2
G24ElementToOrder(23)=2
群元编号——(i,j)
0——(0,0)
1——(0,1)
2——(0,2)
3——(0,3)
4——(1,0)
5——(1,1)
……
23——(5,3)

----24阶非交换群S_3×C_4群元的阶----
//1个1阶元，7个2阶元，2个3阶元，8个4阶元，2个6阶元，4个12阶元
G24ElementToOrder(0)=1
G24ElementToOrder(1)=4
G24ElementToOrder(2)=2
G24ElementToOrder(3)=4
G24ElementToOrder(4)=2
G24ElementToOrder(5)=4
G24ElementToOrder(6)=2
G24ElementToOrder(7)=4
G24ElementToOrder(8)=2
G24ElementToOrder(9)=4
G24ElementToOrder(10)=2
G24ElementToOrder(11)=4
G24ElementToOrder(12)=3
G24ElementToOrder(13)=12
G24ElementToOrder(14)=6
G24ElementToOrder(15)=12
G24ElementToOrder(16)=3
G24ElementToOrder(17)=12
G24ElementToOrder(18)=6
G24ElementToOrder(19)=12
G24ElementToOrder(20)=2
G24ElementToOrder(21)=4
G24ElementToOrder(22)=2
G24ElementToOrder(23)=4