本文主要是介绍153种2744阶群，希望对大家解决编程问题提供一定的参考价值，需要的开发者们随着小编来一起学习吧！
GAP4中调用NumberSmallGroups(2744);命令会出错。
陈松良的《2744阶群的构造》一文证明G2744共有153种互不同构的类型,并获得了G的全部构造:(1)当Sylow子群都正规时,G恰有25个彼此不同构的类型；(2)当Sylow 2-子群正规但Sylow 7-子群不正规时,G恰有8个彼此不同构的类型；(3)当Sylow 2-子群不正规但Sylow 7-子群正规时,G恰有120个彼此不同构的类型；(4)当Sylow子群都不正规时,G不存在.
20151029：4种类型153=25+8+120+0
Sylow子群都正规(25个):C1372 x C2、C686 x C2 x C2、
C196 x C14、C2 x C4 x ((C7 x C7) : C7)、C2 x C4 x (C49 : C7)、C98 x C14 x C2、C2 x C2 x C2 x ((C7 x C7) : C7)、C2 x C2 x C2 x (C49 : C7)、C28 x C14 x C7、C14 x C14 x C14、
C1372 x C2、C196 x C14、C2 x C4 x ((C7 x C7) : C7)、C2 x C4 x (C49 : C7)、C28 x C14 x C7、
C686 x C2 x C2、C98 x C14 x C2、C2 x C2 x C2 x ((C7 x C7) : C7)、C2 x C2 x C2 x (C49 : C7)、C14 x C14 x C14
C392 x C7、C196 x C14、C7 x C49 x D8、C7 x C49 x Q8、C98 x C14 x C2、C56 x C7 x C7、C28 x C14 x C7、C7 x C7 x C7 x D8、C7 x C7 x C7 x Q8、C14 x C14 x C14、
Sylow 2-子群正规而Sylow 7-子群不正规(8个):C7 x ((C2 x C2 x C2) : C49)、C7 x C7 x ((C2 x C2 x C2) : C7)、
Sylow 7-子群正规而Sylow 2-子群不正规(120个):C2 x (C343 : C4)、C2 x C2 x D686、C14 x (C49 : C4)、C98 x (C7 : C4)、C2 x (((C7 x C7) : C7) : C4)、C2 x ((C49 : C7) : C4)、C2 x ((C49 x C7) : C4)、C2 x (((C7 x C7) : C7) : C4)、C2 x (((C7 x C7) : C7) : C4)、C2 x D14 x D98、C2 x ((((C7 x C7) : C7) : C2) : C2)、C2 x C14 x D98、C2 x C98 x D14、C2 x C2 x (((C7 x C7) : C7) : C2)、C2 x C2 x ((C49 x C7) : C2)、C2 x C2 x (((C7 x C7) : C7) : C2)、C7 x C14 x (C7 : C4)、C14 x ((C7 x C7) : C4)、C2 x ((C7 x C7 x C7) : C4)、C14 x ((C7 x C7) : C4)、C2 x ((C7 x C7 x C7) : C4)、C14 x D14 x D14、C2 x (((C7 x C7) : C2) x D14)、C2 x ((C7 x ((C7 x C7) : C2)) : C2)、C14 x C14 x D14、C2 x C14 x ((C7 x C7) : C2)、C2 x C2 x ((C7 x C7 x C7) : C2)、
C4 x D686、C28 x D98、C196 x D14、C4 x (((C7 x C7) : C7) : C2)、C4 x ((C49 : C7) : C2)、C4 x ((C49 x C7) : C2)、C4 x (((C7 x C7) : C7) : C2)、C7 x C28 x D14、C28 x ((C7 x C7) : C2)、C4 x ((C7 x C7 x C7) : C2)、
C2 x C2 x D686、C2 x C14 x D98、C2 x C98 x D14、C2 x C2 x (((C7 x C7) : C7) : C2)、C2 x C2 x ((C49 :
C7) : C2)、C2 x C2 x ((C49 x C7) : C2)、C2 x C2 x (((C7 x C7) : C7) : C2)、C14 x C14 x D14、C2 x C14 x ((C7 x C7) : C2)、C2 x C2 x ((C7 x C7 x C7) : C2)、
C7 x (C49 : C8)、C7 x (C49 : Q8)、C28 x D98、C7 x D392、C14 x (C49 : C4)、C7 x ((C98 x C2) : C2)、C2 x C14 x D98、C7 x C7 x (C7 : C8)、C7 x ((C7 x C7) : C8)、C7 x ((C7 x C7) : C8)、C7 x ((C7 : C4) x D14)、C7 x ((C7 x (C7 : C4)) : C2)、C7 x ((C14 x D14) : C2)、C7 x ((C14 x D14) : C2)、C7 x ((C7 x C7) : Q8)、C7 x C7 x (C7 : Q8)、C7 x C28 x D14、C7 x C7 x D56、C7 x C14 x (C7 : C4)、C7 x C7 x ((C14 x C2) : C2)、C7 x ((C7 x C7) : Q8)、C28 x ((C7 x C7) : C2)、C7 x ((C28 x C7) : C2)、C14 x ((C7 x C7) : C4)、C7 x ((C14 x C14) : C2)、C7 x ((C7 x C7) : C8)、C7 x ((D14 x D14) : C2)、C7 x ((C7 x C7) : Q8)、C14 x ((C7 x C7) : C4)、C14 x D14 x D14、C14 x C14 x D14、C2 x C14 x ((C7 x C7) : C2)、
Sylow子群都不正规(0个):
gap> Factors(2744);
[ 2, 2, 2, 7, 7, 7 ]
gap> n:=2744;;for i in [1..n] do if n mod i=0 then Print(i,",");fi;od;
1,2,4,7,8,14,28,49,56,98,196,343,392,686,1372,2744,
gap> NumberSmallGroups(1372);
38
gap> NumberSmallGroups(686);
15
gap> NumberSmallGroups(392);
44
gap> NumberSmallGroups(343);
5
gap> NumberSmallGroups(196);
12
gap> NumberSmallGroups(98);
5
gap> NumberSmallGroups(56);
13
gap> NumberSmallGroups(49);
2
gap> NumberSmallGroups(28);
4
gap> NumberSmallGroups(14);
2
gap> NumberSmallGroups(8);
5
gap> NumberSmallGroups(4);
2
gap> M:=[1,2,4,7,8,14,28,49,56,98,196,343,392,686,1372,2744];;for n in [1..38] do G1372:=SmallGroup(1372,n);;G:=DirectProduct(G1372,CyclicGroup(2));;Print(IdGroup(G1372));Print("与C_2的直积=");Print(StructureDescription(G));Print(":是否可解：",IsSolvableGroup(G),",","是否幂零：",IsNilpotentGroup(G),",","自同构群：",Order(AutomorphismGroup(G)),",");L:=List(Elements(G),Order);;for i in M do Print(Size(Positions(L,i)),","); od;Print("\n");od;
[ 1372, 1 ]与C_2的直积=C2 x (C343 : C4):是否可解：true,是否幂零：false,自同构群：806736,1,3,1372,6,
0,18,0,42,0,126,0,294,0,882,0,0,
[ 1372, 2 ]与C_2的直积=C1372 x C2:是否可解：true,是否幂零：true,自同构群：2352,1,3,4,6,0,18,24,42,
0,126,168,294,0,882,1176,0,
[ 1372, 3 ]与C_2的直积=C2 x C2 x D686:是否可解：true,是否幂零：false,自同构群：2420208,1,1375,0,6,
0,18,0,42,0,126,0,294,0,882,0,0,
[ 1372, 4 ]与C_2的直积=C686 x C2 x C2:是否可解：true,是否幂零：true,自同构群：49392,1,7,0,6,0,42,0,
42,0,294,0,294,0,2058,0,0,
[ 1372, 5 ]与C_2的直积=C14 x (C49 : C4):是否可解：true,是否幂零：false,自同构群：98784,1,3,196,48,
0,144,1176,294,0,882,0,0,0,0,0,0,
[ 1372, 6 ]与C_2的直积=C98 x (C7 : C4):是否可解：true,是否幂零：false,自同构群：14112,1,3,28,48,0,
144,168,294,0,882,1176,0,0,0,0,0,
[ 1372, 7 ]与C_2的直积=C2 x (((C7 x C7) : C7) : C4):是否可解：true,是否幂零：false,自同构群：98784,
1,3,196,342,0,1026,1176,0,0,0,0,0,0,0,0,0,
[ 1372, 8 ]与C_2的直积=C2 x ((C49 : C7) : C4):是否可解：true,是否幂零：false,自同构群：16464,1,3,
196,48,0,144,1176,294,0,882,0,0,0,0,0,0,
[ 1372, 9 ]与C_2的直积=C2 x ((C49 x C7) : C4):是否可解：true,是否幂零：false,自同构群：33882912,1,
3,1372,48,0,144,0,294,0,882,0,0,0,0,0,0,
[ 1372, 10 ]与C_2的直积=C2 x (((C7 x C7) : C7) : C4):是否可解：true,是否幂零：false,自同构群：
790272,1,3,196,342,0,1026,1176,0,0,0,0,0,0,0,0,0,
[ 1372, 11 ]与C_2的直积=C196 x C14:是否可解：true,是否幂零：true,自同构群：98784,1,3,4,48,0,144,
192,294,0,882,1176,0,0,0,0,0,
[ 1372, 12 ]与C_2的直积=C2 x C4 x ((C7 x C7) : C7):是否可解：true,是否幂零：true,自同构群：790272,
1,3,4,342,0,1026,1368,0,0,0,0,0,0,0,0,0,
[ 1372, 13 ]与C_2的直积=C2 x C4 x (C49 : C7):是否可解：true,是否幂零：true,自同构群：16464,1,3,4,
48,0,144,192,294,0,882,1176,0,0,0,0,0,
[ 1372, 14 ]与C_2的直积=C2 x (((C7 x C7) : C7) : C4):是否可解：true,是否幂零：false,自同构群：9408,
1,99,196,342,0,930,1176,0,0,0,0,0,0,0,0,0,
[ 1372, 15 ]与C_2的直积=C2 x D14 x D98:是否可解：true,是否幂零：false,自同构群：345744,1,799,0,48,
0,720,0,294,0,882,0,0,0,0,0,0,
[ 1372, 16
]与C_2的直积=C2 x ((((C7 x C7) : C7) : C2) : C2):是否可解：true,是否幂零：false,自同构群：98784,1,
295,0,342,0,2106,0,0,0,0,0,0,0,0,0,0,
[ 1372, 17 ]与C_2的直积=C2 x C14 x D98:是否可解：true,是否幂零：false,自同构群：296352,1,199,0,48,
0,1320,0,294,0,882,0,0,0,0,0,0,
[ 1372, 18 ]与C_2的直积=C2 x C98 x D14:是否可解：true,是否幂零：false,自同构群：42336,1,31,0,48,0,
312,0,294,0,2058,0,0,0,0,0,0,
[ 1372, 19 ]与C_2的直积=C2 x C2 x (((C7 x C7) : C7) : C2):是否可解：true,是否幂零：false,自同构群：
296352,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,
[ 1372, 20 ]与C_2的直积=C2 x C2 x ((C49 : C7) : C2):是否可解：true,是否幂零：false,自同构群：49392,
1,199,0,48,0,1320,0,294,0,882,0,0,0,0,0,0,
[ 1372, 21 ]与C_2的直积=C2 x C2 x ((C49 x C7) : C2):是否可解：true,是否幂零：false,自同构群：
101648736,1,1375,0,48,0,144,0,294,0,882,0,0,0,0,0,0,
[ 1372, 22 ]与C_2的直积=C2 x C2 x (((C7 x C7) : C7) : C2):是否可解：true,是否幂零：false,自同构群：
2370816,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,
[ 1372, 23 ]与C_2的直积=C98 x C14 x C2:是否可解：true,是否幂零：true,自同构群：2074464,1,7,0,48,0,
336,0,294,0,2058,0,0,0,0,0,0,
[ 1372, 24 ]与C_2的直积=C2 x C2 x C2 x ((C7 x C7) : C7):是否可解：true,是否幂零：true,自同构群：
16595712,1,7,0,342,0,2394,0,0,0,0,0,0,0,0,0,0,
[ 1372, 25 ]与C_2的直积=C2 x C2 x C2 x (C49 : C7):是否可解：true,是否幂零：true,自同构群：345744,1,
7,0,48,0,336,0,294,0,2058,0,0,0,0,0,0,
[ 1372, 26 ]与C_2的直积=C7 x C14 x (C7 : C4):是否可解：true,是否幂零：false,自同构群：677376,1,3,
28,342,0,1026,1344,0,0,0,0,0,0,0,0,0,
[ 1372, 27 ]与C_2的直积=C14 x ((C7 x C7) : C4):是否可解：true,是否幂零：false,自同构群：4741632,1,
3,196,342,0,1026,1176,0,0,0,0,0,0,0,0,0,
[ 1372, 28 ]与C_2的直积=C2 x ((C7 x C7 x C7) : C4):是否可解：true,是否幂零：false,自同构群：
92703647232,1,3,1372,342,0,1026,0,0,0,0,0,0,0,0,0,0,
[ 1372, 29 ]与C_2的直积=C28 x C14 x C7:是否可解：true,是否幂零：true,自同构群：270273024,1,3,4,342,
0,1026,1368,0,0,0,0,0,0,0,0,0,
[ 1372, 30 ]与C_2的直积=C14 x ((C7 x C7) : C4):是否可解：true,是否幂零：false,自同构群：56448,1,99,
196,342,0,930,1176,0,0,0,0,0,0,0,0,0,
[ 1372, 31 ]与C_2的直积=C2 x ((C7 x C7 x C7) : C4):是否可解：true,是否幂零：false,自同构群：395136,
1,99,1372,342,0,930,0,0,0,0,0,0,0,0,0,0,
[ 1372, 32 ]与C_2的直积=C14 x D14 x D14:是否可解：true,是否幂零：false,自同构群：84672,1,127,0,342,
0,2274,0,0,0,0,0,0,0,0,0,0,
[ 1372, 33 ]与C_2的直积=C2 x (((C7 x C7) : C2) x D14):是否可解：true,是否幂零：false,自同构群：
16595712,1,799,0,342,0,1602,0,0,0,0,0,0,0,0,0,0,
[ 1372, 34
]与C_2的直积=C2 x ((C7 x ((C7 x C7) : C2)) : C2):是否可解：true,是否幂零：false,自同构群：1778112,
1,295,0,342,0,2106,0,0,0,0,0,0,0,0,0,0,
[ 1372, 35 ]与C_2的直积=C14 x C14 x D14:是否可解：true,是否幂零：false,自同构群：2032128,1,31,0,
342,0,2370,0,0,0,0,0,0,0,0,0,0,
[ 1372, 36 ]与C_2的直积=C2 x C14 x ((C7 x C7) : C2):是否可解：true,是否幂零：false,自同构群：
14224896,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,
[ 1372, 37 ]与C_2的直积=C2 x C2 x ((C7 x C7 x C7) : C2):是否可解：true,是否幂零：false,自同构群：
278110941696,1,1375,0,342,0,1026,0,0,0,0,0,0,0,0,0,0,
[ 1372, 38 ]与C_2的直积=C14 x C14 x C14:是否可解：true,是否幂零：true,自同构群：5675733504,1,7,0,
342,0,2394,0,0,0,0,0,0,0,0,0,0,
gap> for n in [1..38] do G1372:=SmallGroup(1372,n);;g:=DirectProduct(G1372,CyclicGroup(2));;gid:=StructureDescription(g);Print(gid,"是否幂零：",IsNilpotentGroup(g));s:=Elements(g);;sl2:=SylowSubgroup(g,2);;Print(IdGroup(sl2),IsSubnormal(g,sl2));sl7:=SylowSubgroup(g,7);;Print(IdGroup(sl7),IsSubnormal(g,sl7),"\n");od;
C2 x (C343 : C4)是否幂零：false[ 8, 2 ]false[ 343, 1 ]true
C1372 x C2是否幂零：true[ 8, 2 ]true[ 343, 1 ]true
C2 x C2 x D686是否幂零：false[ 8, 5 ]false[ 343, 1 ]true
C686 x C2 x C2是否幂零：true[ 8, 5 ]true[ 343, 1 ]true
C14 x (C49 : C4)是否幂零：false[ 8, 2 ]false[ 343, 2 ]true
C98 x (C7 : C4)是否幂零：false[ 8, 2 ]false[ 343, 2 ]true
C2 x (((C7 x C7) : C7) : C4)是否幂零：false[ 8, 2 ]false[ 343, 3 ]true
C2 x ((C49 : C7) : C4)是否幂零：false[ 8, 2 ]false[ 343, 4 ]true
C2 x ((C49 x C7) : C4)是否幂零：false[ 8, 2 ]false[ 343, 2 ]true
C2 x (((C7 x C7) : C7) : C4)是否幂零：false[ 8, 2 ]false[ 343, 3 ]true
C196 x C14是否幂零：true[ 8, 2 ]true[ 343, 2 ]true
C2 x C4 x ((C7 x C7) : C7)是否幂零：true[ 8, 2 ]true[ 343, 3 ]true
C2 x C4 x (C49 : C7)是否幂零：true[ 8, 2 ]true[ 343, 4 ]true
C2 x (((C7 x C7) : C7) : C4)是否幂零：false[ 8, 2 ]false[ 343, 3 ]true
C2 x D14 x D98是否幂零：false[ 8, 5 ]false[ 343, 2 ]true
C2 x ((((C7 x C7) : C7) : C2) : C2)是否幂零：false[ 8, 5 ]false[ 343, 3 ]true
C2 x C14 x D98是否幂零：false[ 8, 5 ]false[ 343, 2 ]true
C2 x C98 x D14是否幂零：false[ 8, 5 ]false[ 343, 2 ]true
C2 x C2 x (((C7 x C7) : C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 3 ]true
C2 x C2 x ((C49 : C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 4 ]true
C2 x C2 x ((C49 x C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 2 ]true
C2 x C2 x (((C7 x C7) : C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 3 ]true
C98 x C14 x C2是否幂零：true[ 8, 5 ]true[ 343, 2 ]true
C2 x C2 x C2 x ((C7 x C7) : C7)是否幂零：true[ 8, 5 ]true[ 343, 3 ]true
C2 x C2 x C2 x (C49 : C7)是否幂零：true[ 8, 5 ]true[ 343, 4 ]true
C7 x C14 x (C7 : C4)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C14 x ((C7 x C7) : C4)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C2 x ((C7 x C7 x C7) : C4)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C28 x C14 x C7是否幂零：true[ 8, 2 ]true[ 343, 5 ]true
C14 x ((C7 x C7) : C4)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C2 x ((C7 x C7 x C7) : C4)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C14 x D14 x D14是否幂零：false[ 8, 5 ]false[ 343, 5 ]true
C2 x (((C7 x C7) : C2) x D14)是否幂零：false[ 8, 5 ]false[ 343, 5 ]true
C2 x ((C7 x ((C7 x C7) : C2)) : C2)是否幂零：false[ 8, 5 ]false[ 343, 5 ]true
C14 x C14 x D14是否幂零：false[ 8, 5 ]false[ 343, 5 ]true
C2 x C14 x ((C7 x C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 5 ]true
C2 x C2 x ((C7 x C7 x C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 5 ]true
C14 x C14 x C14是否幂零：true[ 8, 5 ]true[ 343, 5 ]true
gap> M:=[1,2,4,7,8,14,28,49,56,98,196,343,392,686,1372,2744];;for n in [1..15] do G686:=SmallGroup(686,n);;G:=DirectProduct(G686,CyclicGroup(4));;Print(IdGroup(G686));Print("与C_4的直积=");Print(StructureDescription(G));Print(":是否可解：",IsSolvableGroup(G),",","是否幂零：",IsNilpotentGroup(G),",","自同构群：",Order(AutomorphismGroup(G)),",");L:=List(Elements(G),Order);;for i in M do Print(Size(Positions(L,i)),","); od;Print("\n");od;
[ 686, 1 ]与C_4的直积=C4 x D686:是否可解：true,是否幂零：false,自同构群：403368,1,687,688,6,0,6,12,
42,0,42,84,294,0,294,588,0,
[ 686, 2 ]与C_4的直积=C1372 x C2:是否可解：true,是否幂零：true,自同构群：2352,1,3,4,6,0,18,24,42,0,
126,168,294,0,882,1176,0,
[ 686, 3 ]与C_4的直积=C28 x D98:是否可解：true,是否幂零：false,自同构群：49392,1,99,100,48,0,636,
684,294,0,294,588,0,0,0,0,0,
[ 686, 4 ]与C_4的直积=C196 x D14:是否可解：true,是否幂零：false,自同构群：7056,1,15,16,48,0,132,
180,294,0,882,1176,0,0,0,0,0,
[ 686, 5 ]与C_4的直积=C4 x (((C7 x C7) : C7) : C2):是否可解：true,是否幂零：false,自同构群：49392,
1,99,100,342,0,930,1272,0,0,0,0,0,0,0,0,0,
[ 686, 6 ]与C_4的直积=C4 x ((C49 : C7) : C2):是否可解：true,是否幂零：false,自同构群：8232,1,99,
100,48,0,636,684,294,0,294,588,0,0,0,0,0,
[ 686, 7 ]与C_4的直积=C4 x ((C49 x C7) : C2):是否可解：true,是否幂零：false,自同构群：16941456,1,
687,688,48,0,48,96,294,0,294,588,0,0,0,0,0,
[ 686, 8 ]与C_4的直积=C4 x (((C7 x C7) : C7) : C2):是否可解：true,是否幂零：false,自同构群：395136,
1,99,100,342,0,930,1272,0,0,0,0,0,0,0,0,0,
[ 686, 9 ]与C_4的直积=C196 x C14:是否可解：true,是否幂零：true,自同构群：98784,1,3,4,48,0,144,192,
294,0,882,1176,0,0,0,0,0,
[ 686, 10 ]与C_4的直积=C2 x C4 x ((C7 x C7) : C7):是否可解：true,是否幂零：true,自同构群：790272,1,
3,4,342,0,1026,1368,0,0,0,0,0,0,0,0,0,
[ 686, 11 ]与C_4的直积=C2 x C4 x (C49 : C7):是否可解：true,是否幂零：true,自同构群：16464,1,3,4,48,
0,144,192,294,0,882,1176,0,0,0,0,0,
[ 686, 12 ]与C_4的直积=C7 x C28 x D14:是否可解：true,是否幂零：false,自同构群：338688,1,15,16,342,
0,1014,1356,0,0,0,0,0,0,0,0,0,
[ 686, 13 ]与C_4的直积=C28 x ((C7 x C7) : C2):是否可解：true,是否幂零：false,自同构群：2370816,1,
99,100,342,0,930,1272,0,0,0,0,0,0,0,0,0,
[ 686, 14 ]与C_4的直积=C4 x ((C7 x C7 x C7) : C2):是否可解：true,是否幂零：false,自同构群：
46351823616,1,687,688,342,0,342,684,0,0,0,0,0,0,0,0,0,
[ 686, 15 ]与C_4的直积=C28 x C14 x C7:是否可解：true,是否幂零：true,自同构群：270273024,1,3,4,342,
0,1026,1368,0,0,0,0,0,0,0,0,0,
gap> for n in [1..15] do G:=SmallGroup(686,n);;g:=DirectProduct(G,SmallGroup(4,1));;gid:=StructureDescription(g);Print(gid,"是否幂零：",IsNilpotentGroup(g));s:=Elements(g);;sl2:=SylowSubgroup(g,2);;Print(IdGroup(sl2),IsSubnormal(g,sl2));sl7:=SylowSubgroup(g,7);;Print(IdGroup(sl7),IsSubnormal(g,sl7),"\n");od;
C4 x D686是否幂零：false[ 8, 2 ]false[ 343, 1 ]true
C1372 x C2是否幂零：true[ 8, 2 ]true[ 343, 1 ]true
C28 x D98是否幂零：false[ 8, 2 ]false[ 343, 2 ]true
C196 x D14是否幂零：false[ 8, 2 ]false[ 343, 2 ]true
C4 x (((C7 x C7) : C7) : C2)是否幂零：false[ 8, 2 ]false[ 343, 3 ]true
C4 x ((C49 : C7) : C2)是否幂零：false[ 8, 2 ]false[ 343, 4 ]true
C4 x ((C49 x C7) : C2)是否幂零：false[ 8, 2 ]false[ 343, 2 ]true
C4 x (((C7 x C7) : C7) : C2)是否幂零：false[ 8, 2 ]false[ 343, 3 ]true
C196 x C14是否幂零：true[ 8, 2 ]true[ 343, 2 ]true
C2 x C4 x ((C7 x C7) : C7)是否幂零：true[ 8, 2 ]true[ 343, 3 ]true
C2 x C4 x (C49 : C7)是否幂零：true[ 8, 2 ]true[ 343, 4 ]true
C7 x C28 x D14是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C28 x ((C7 x C7) : C2)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C4 x ((C7 x C7 x C7) : C2)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C28 x C14 x C7是否幂零：true[ 8, 2 ]true[ 343, 5 ]true
gap> M:=[1,2,4,7,8,14,28,49,56,98,196,343,392,686,1372,2744];;for n in [1..15] do G686:=SmallGroup(686,n);;G:=DirectProduct(G686,ElementaryAbelianGroup(4));;Print(IdGroup(G686));Print("与K_4的直积=");Print(StructureDescription(G));Print(": 是否可解：",IsSolvableGroup(G),",","是否幂零：",IsNilpotentGroup(G),",","自同构群：",Order(AutomorphismGroup(G)),",");L:=List(Elements(G),Order);;for i in M do Print(Size(Positions(L,i)),","); od;Print("\n");od;
[ 686, 1 ]与K_4的直积=C2 x C2 x D686:是否可解：true,是否幂零：false,自同构群：2420208,1,1375,0,6,0,
18,0,42,0,126,0,294,0,882,0,0,
[ 686, 2 ]与K_4的直积=C686 x C2 x C2:是否可解：true,是否幂零：true,自同构群：49392,1,7,0,6,0,42,0,
42,0,294,0,294,0,2058,0,0,
[ 686, 3 ]与K_4的直积=C2 x C14 x D98:是否可解：true,是否幂零：false,自同构群：296352,1,199,0,48,0,
1320,0,294,0,882,0,0,0,0,0,0,
[ 686, 4 ]与K_4的直积=C2 x C98 x D14:是否可解：true,是否幂零：false,自同构群：42336,1,31,0,48,0,
312,0,294,0,2058,0,0,0,0,0,0,
[ 686, 5 ]与K_4的直积=C2 x C2 x (((C7 x C7) : C7) : C2):是否可解：true,是否幂零：false,自同构群：
296352,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,
[ 686, 6 ]与K_4的直积=C2 x C2 x ((C49 : C7) : C2):是否可解：true,是否幂零：false,自同构群：49392,1,
199,0,48,0,1320,0,294,0,882,0,0,0,0,0,0,
[ 686, 7 ]与K_4的直积=C2 x C2 x ((C49 x C7) : C2):是否可解：true,是否幂零：false,自同构群：
101648736,1,1375,0,48,0,144,0,294,0,882,0,0,0,0,0,0,
[ 686, 8 ]与K_4的直积=C2 x C2 x (((C7 x C7) : C7) : C2):是否可解：true,是否幂零：false,自同构群：
2370816,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,
[ 686, 9 ]与K_4的直积=C98 x C14 x C2:是否可解：true,是否幂零：true,自同构群：2074464,1,7,0,48,0,
336,0,294,0,2058,0,0,0,0,0,0,
[ 686, 10 ]与K_4的直积=C2 x C2 x C2 x ((C7 x C7) : C7):是否可解：true,是否幂零：true,自同构群：
16595712,1,7,0,342,0,2394,0,0,0,0,0,0,0,0,0,0,
[ 686, 11 ]与K_4的直积=C2 x C2 x C2 x (C49 : C7):是否可解：true,是否幂零：true,自同构群：345744,1,
7,0,48,0,336,0,294,0,2058,0,0,0,0,0,0,
[ 686, 12 ]与K_4的直积=C14 x C14 x D14:是否可解：true,是否幂零：false,自同构群：2032128,1,31,0,342,
0,2370,0,0,0,0,0,0,0,0,0,0,
[ 686, 13 ]与K_4的直积=C2 x C14 x ((C7 x C7) : C2):是否可解：true,是否幂零：false,自同构群：
14224896,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,
[ 686, 14 ]与K_4的直积=C2 x C2 x ((C7 x C7 x C7) : C2):是否可解：true,是否幂零：false,自同构群：
278110941696,1,1375,0,342,0,1026,0,0,0,0,0,0,0,0,0,0,
[ 686, 15 ]与K_4的直积=C14 x C14 x C14:是否可解：true,是否幂零：true,自同构群：5675733504,1,7,0,
342,0,2394,0,0,0,0,0,0,0,0,0,0,
gap> for n in [1..15] do G:=SmallGroup(686,n);;g:=DirectProduct(G,SmallGroup(4,2));;gid:=StructureDescription(g);Print(gid,"是否幂零：",IsNilpotentGroup(g));s:=Elements(g);;sl2:=SylowSubgroup(g,2);;Print(IdGroup(sl2),IsSubnormal(g,sl2));sl7:=SylowSubgroup(g,7);;Print(IdGroup(sl7),IsSubnormal(g,sl7),"\n");od;
C2 x C2 x D686是否幂零：false[ 8, 5 ]false[ 343, 1 ]true
C686 x C2 x C2是否幂零：true[ 8, 5 ]true[ 343, 1 ]true
C2 x C14 x D98是否幂零：false[ 8, 5 ]false[ 343, 2 ]true
C2 x C98 x D14是否幂零：false[ 8, 5 ]false[ 343, 2 ]true
C2 x C2 x (((C7 x C7) : C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 3 ]true
C2 x C2 x ((C49 : C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 4 ]true
C2 x C2 x ((C49 x C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 2 ]true
C2 x C2 x (((C7 x C7) : C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 3 ]true
C98 x C14 x C2是否幂零：true[ 8, 5 ]true[ 343, 2 ]true
C2 x C2 x C2 x ((C7 x C7) : C7)是否幂零：true[ 8, 5 ]true[ 343, 3 ]true
C2 x C2 x C2 x (C49 : C7)是否幂零：true[ 8, 5 ]true[ 343, 4 ]true
C14 x C14 x D14是否幂零：false[ 8, 5 ]false[ 343, 5 ]true
C2 x C14 x ((C7 x C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 5 ]true
C2 x C2 x ((C7 x C7 x C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 5 ]true
C14 x C14 x C14是否幂零：true[ 8, 5 ]true[ 343, 5 ]true
gap> M:=[1,2,4,7,8,14,28,49,56,98,196,343,392,686,1372,2744];;for n in [1..44] do G392:=SmallGroup(392,n);;G:=DirectProduct(G392,CyclicGroup(7));;Print(IdGroup(G392));Print("与C_7的直积=");Print(StructureDescription(G));Print(":是否可解：",IsSolvableGroup(G),",","是否幂零：",IsNilpotentGroup(G),",","自同构群：",Order(AutomorphismGroup(G)),",");L:=List(Elements(G),Order);;for i in M do Print(Size(Positions(L,i)),","); od;Print("\n");od;
[ 392, 1 ]与C_7的直积=C7 x (C49 : C8):是否可解：true,是否幂零：false,自同构群：49392,1,1,2,48,196,
48,96,294,1176,294,588,0,0,0,0,0,
[ 392, 2 ]与C_7的直积=C392 x C7:是否可解：true,是否幂零：true,自同构群：49392,1,1,2,48,4,48,96,294,
192,294,588,0,1176,0,0,0,
[ 392, 3 ]与C_7的直积=C7 x (C49 : Q8):是否可解：true,是否幂零：false,自同构群：98784,1,1,198,48,0,
48,1272,294,0,294,588,0,0,0,0,0,
[ 392, 4 ]与C_7的直积=C28 x D98:是否可解：true,是否幂零：false,自同构群：49392,1,99,100,48,0,636,
684,294,0,294,588,0,0,0,0,0,
[ 392, 5 ]与C_7的直积=C7 x D392:是否可解：true,是否幂零：false,自同构群：98784,1,197,2,48,0,1224,
96,294,0,294,588,0,0,0,0,0,
[ 392, 6 ]与C_7的直积=C14 x (C49 : C4):是否可解：true,是否幂零：false,自同构群：98784,1,3,196,48,0,
144,1176,294,0,882,0,0,0,0,0,0,
[ 392, 7 ]与C_7的直积=C7 x ((C98 x C2) : C2):是否可解：true,是否幂零：false,自同构群：49392,1,101,
98,48,0,732,588,294,0,882,0,0,0,0,0,0,
[ 392, 8 ]与C_7的直积=C196 x C14:是否可解：true,是否幂零：true,自同构群：98784,1,3,4,48,0,144,192,
294,0,882,1176,0,0,0,0,0,
[ 392, 9 ]与C_7的直积=C7 x C49 x D8:是否可解：true,是否幂零：true,自同构群：98784,1,5,2,48,0,240,
96,294,0,1470,588,0,0,0,0,0,
[ 392, 10 ]与C_7的直积=C7 x C49 x Q8:是否可解：true,是否幂零：true,自同构群：296352,1,1,6,48,0,48,
288,294,0,294,1764,0,0,0,0,0,
[ 392, 11 ]与C_7的直积=C7 x ((C2 x C2 x C2) : C49):是否可解：true,是否幂零：false,自同构群：345744,
1,7,0,48,0,336,0,2352,0,0,0,0,0,0,0,0,
[ 392, 12 ]与C_7的直积=C2 x C14 x D98:是否可解：true,是否幂零：false,自同构群：296352,1,199,0,48,0,
1320,0,294,0,882,0,0,0,0,0,0,
[ 392, 13 ]与C_7的直积=C98 x C14 x C2:是否可解：true,是否幂零：true,自同构群：2074464,1,7,0,48,0,
336,0,294,0,2058,0,0,0,0,0,0,
[ 392, 14 ]与C_7的直积=C7 x C7 x (C7 : C8):是否可解：true,是否幂零：false,自同构群：338688,1,1,2,
342,28,342,684,0,1344,0,0,0,0,0,0,0,
[ 392, 15 ]与C_7的直积=C7 x ((C7 x C7) : C8):是否可解：true,是否幂零：false,自同构群：2370816,1,1,
2,342,196,342,684,0,1176,0,0,0,0,0,0,0,
[ 392, 16 ]与C_7的直积=C56 x C7 x C7:是否可解：true,是否幂零：true,自同构群：135136512,1,1,2,342,4,
342,684,0,1368,0,0,0,0,0,0,0,
[ 392, 17 ]与C_7的直积=C7 x ((C7 x C7) : C8):是否可解：true,是否幂零：false,自同构群：56448,1,1,98,
342,196,342,588,0,1176,0,0,0,0,0,0,0,
[ 392, 18 ]与C_7的直积=C7 x ((C7 : C4) x D14):是否可解：true,是否幂零：false,自同构群：42336,1,15,
112,342,0,1014,1260,0,0,0,0,0,0,0,0,0,
[ 392, 19 ]与C_7的直积=C7 x ((C7 x (C7 : C4)) : C2):是否可解：true,是否幂零：false,自同构群：84672,
1,99,28,342,0,930,1344,0,0,0,0,0,0,0,0,0,
[ 392, 20 ]与C_7的直积=C7 x ((C14 x D14) : C2):是否可解：true,是否幂零：false,自同构群：84672,1,29,
98,342,0,1686,588,0,0,0,0,0,0,0,0,0,
[ 392, 21 ]与C_7的直积=C7 x ((C14 x D14) : C2):是否可解：true,是否幂零：false,自同构群：42336,1,
113,14,342,0,1602,672,0,0,0,0,0,0,0,0,0,
[ 392, 22 ]与C_7的直积=C7 x ((C7 x C7) : Q8):是否可解：true,是否幂零：false,自同构群：84672,1,1,
126,342,0,342,1932,0,0,0,0,0,0,0,0,0,
[ 392, 23 ]与C_7的直积=C7 x C7 x (C7 : Q8):是否可解：true,是否幂零：false,自同构群：677376,1,1,30,
342,0,342,2028,0,0,0,0,0,0,0,0,0,
[ 392, 24 ]与C_7的直积=C7 x C28 x D14:是否可解：true,是否幂零：false,自同构群：338688,1,15,16,342,
0,1014,1356,0,0,0,0,0,0,0,0,0,
[ 392, 25 ]与C_7的直积=C7 x C7 x D56:是否可解：true,是否幂零：false,自同构群：677376,1,29,2,342,0,
1686,684,0,0,0,0,0,0,0,0,0,
[ 392, 26 ]与C_7的直积=C7 x C14 x (C7 : C4):是否可解：true,是否幂零：false,自同构群：677376,1,3,28,
342,0,1026,1344,0,0,0,0,0,0,0,0,0,
[ 392, 27 ]与C_7的直积=C7 x C7 x ((C14 x C2) : C2):是否可解：true,是否幂零：false,自同构群：338688,
1,17,14,342,0,1698,672,0,0,0,0,0,0,0,0,0,
[ 392, 28 ]与C_7的直积=C7 x ((C7 x C7) : Q8):是否可解：true,是否幂零：false,自同构群：4741632,1,1,
198,342,0,342,1860,0,0,0,0,0,0,0,0,0,
[ 392, 29 ]与C_7的直积=C28 x ((C7 x C7) : C2):是否可解：true,是否幂零：false,自同构群：2370816,1,
99,100,342,0,930,1272,0,0,0,0,0,0,0,0,0,
[ 392, 30 ]与C_7的直积=C7 x ((C28 x C7) : C2):是否可解：true,是否幂零：false,自同构群：4741632,1,
197,2,342,0,1518,684,0,0,0,0,0,0,0,0,0,
[ 392, 31 ]与C_7的直积=C14 x ((C7 x C7) : C4):是否可解：true,是否幂零：false,自同构群：4741632,1,3,
196,342,0,1026,1176,0,0,0,0,0,0,0,0,0,
[ 392, 32 ]与C_7的直积=C7 x ((C14 x C14) : C2):是否可解：true,是否幂零：false,自同构群：2370816,1,
101,98,342,0,1614,588,0,0,0,0,0,0,0,0,0,
[ 392, 33 ]与C_7的直积=C28 x C14 x C7:是否可解：true,是否幂零：true,自同构群：270273024,1,3,4,342,
0,1026,1368,0,0,0,0,0,0,0,0,0,
[ 392, 34 ]与C_7的直积=C7 x C7 x C7 x D8:是否可解：true,是否幂零：true,自同构群：270273024,1,5,2,
342,0,1710,684,0,0,0,0,0,0,0,0,0,
[ 392, 35 ]与C_7的直积=C7 x C7 x C7 x Q8:是否可解：true,是否幂零：true,自同构群：810819072,1,1,6,
342,0,342,2052,0,0,0,0,0,0,0,0,0,
[ 392, 36 ]与C_7的直积=C7 x ((C7 x C7) : C8):是否可解：true,是否幂零：false,自同构群：28224,1,49,
98,342,196,294,588,0,1176,0,0,0,0,0,0,0,
[ 392, 37 ]与C_7的直积=C7 x ((D14 x D14) : C2):是否可解：true,是否幂零：false,自同构群：14112,1,77,
98,342,0,1638,588,0,0,0,0,0,0,0,0,0,
[ 392, 38 ]与C_7的直积=C7 x ((C7 x C7) : Q8):是否可解：true,是否幂零：false,自同构群：42336,1,49,
294,342,0,294,1764,0,0,0,0,0,0,0,0,0,
[ 392, 39 ]与C_7的直积=C7 x C7 x ((C2 x C2 x C2) : C7):是否可解：true,是否幂零：false,自同构群：
16595712,1,7,0,2400,0,336,0,0,0,0,0,0,0,0,0,0,
[ 392, 40 ]与C_7的直积=C14 x ((C7 x C7) : C4):是否可解：true,是否幂零：false,自同构群：56448,1,99,
196,342,0,930,1176,0,0,0,0,0,0,0,0,0,
[ 392, 41 ]与C_7的直积=C14 x D14 x D14:是否可解：true,是否幂零：false,自同构群：84672,1,127,0,342,
0,2274,0,0,0,0,0,0,0,0,0,0,
[ 392, 42 ]与C_7的直积=C14 x C14 x D14:是否可解：true,是否幂零：false,自同构群：2032128,1,31,0,342,
0,2370,0,0,0,0,0,0,0,0,0,0,
[ 392, 43 ]与C_7的直积=C2 x C14 x ((C7 x C7) : C2):是否可解：true,是否幂零：false,自同构群：
14224896,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,
[ 392, 44 ]与C_7的直积=C14 x C14 x C14:是否可解：true,是否幂零：true,自同构群：5675733504,1,7,0,
342,0,2394,0,0,0,0,0,0,0,0,0,0,
gap> for n in [1..44] do G:=SmallGroup(392,n);;g:=DirectProduct(G,SmallGroup(7,1));;gid:=StructureDescription(g);Print(gid,"是否幂零：",IsNilpotentGroup(g));s:=Elements(g);;sl2:=SylowSubgroup(g,2);;Print(IdGroup(sl2),IsSubnormal(g,sl2));sl7:=SylowSubgroup(g,7);;Print(IdGroup(sl7),IsSubnormal(g,sl7),"\n");od;
C7 x (C49 : C8)是否幂零：false[ 8, 1 ]false[ 343, 2 ]true
C392 x C7是否幂零：true[ 8, 1 ]true[ 343, 2 ]true
C7 x (C49 : Q8)是否幂零：false[ 8, 4 ]false[ 343, 2 ]true
C28 x D98是否幂零：false[ 8, 2 ]false[ 343, 2 ]true
C7 x D392是否幂零：false[ 8, 3 ]false[ 343, 2 ]true
C14 x (C49 : C4)是否幂零：false[ 8, 2 ]false[ 343, 2 ]true
C7 x ((C98 x C2) : C2)是否幂零：false[ 8, 3 ]false[ 343, 2 ]true
C196 x C14是否幂零：true[ 8, 2 ]true[ 343, 2 ]true
C7 x C49 x D8是否幂零：true[ 8, 3 ]true[ 343, 2 ]true
C7 x C49 x Q8是否幂零：true[ 8, 4 ]true[ 343, 2 ]true
C7 x ((C2 x C2 x C2) : C49)是否幂零：false[ 8, 5 ]true[ 343, 2 ]false
C2 x C14 x D98是否幂零：false[ 8, 5 ]false[ 343, 2 ]true
C98 x C14 x C2是否幂零：true[ 8, 5 ]true[ 343, 2 ]true
C7 x C7 x (C7 : C8)是否幂零：false[ 8, 1 ]false[ 343, 5 ]true
C7 x ((C7 x C7) : C8)是否幂零：false[ 8, 1 ]false[ 343, 5 ]true
C56 x C7 x C7是否幂零：true[ 8, 1 ]true[ 343, 5 ]true
C7 x ((C7 x C7) : C8)是否幂零：false[ 8, 1 ]false[ 343, 5 ]true
C7 x ((C7 : C4) x D14)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C7 x ((C7 x (C7 : C4)) : C2)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C7 x ((C14 x D14) : C2)是否幂零：false[ 8, 3 ]false[ 343, 5 ]true
C7 x ((C14 x D14) : C2)是否幂零：false[ 8, 3 ]false[ 343, 5 ]true
C7 x ((C7 x C7) : Q8)是否幂零：false[ 8, 4 ]false[ 343, 5 ]true
C7 x C7 x (C7 : Q8)是否幂零：false[ 8, 4 ]false[ 343, 5 ]true
C7 x C28 x D14是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C7 x C7 x D56是否幂零：false[ 8, 3 ]false[ 343, 5 ]true
C7 x C14 x (C7 : C4)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C7 x C7 x ((C14 x C2) : C2)是否幂零：false[ 8, 3 ]false[ 343, 5 ]true
C7 x ((C7 x C7) : Q8)是否幂零：false[ 8, 4 ]false[ 343, 5 ]true
C28 x ((C7 x C7) : C2)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C7 x ((C28 x C7) : C2)是否幂零：false[ 8, 3 ]false[ 343, 5 ]true
C14 x ((C7 x C7) : C4)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C7 x ((C14 x C14) : C2)是否幂零：false[ 8, 3 ]false[ 343, 5 ]true
C28 x C14 x C7是否幂零：true[ 8, 2 ]true[ 343, 5 ]true
C7 x C7 x C7 x D8是否幂零：true[ 8, 3 ]true[ 343, 5 ]true
C7 x C7 x C7 x Q8是否幂零：true[ 8, 4 ]true[ 343, 5 ]true
C7 x ((C7 x C7) : C8)是否幂零：false[ 8, 1 ]false[ 343, 5 ]true
C7 x ((D14 x D14) : C2)是否幂零：false[ 8, 3 ]false[ 343, 5 ]true
C7 x ((C7 x C7) : Q8)是否幂零：false[ 8, 4 ]false[ 343, 5 ]true
C7 x C7 x ((C2 x C2 x C2) : C7)是否幂零：false[ 8, 5 ]true[ 343, 5 ]false
C14 x ((C7 x C7) : C4)是否幂零：false[ 8, 2 ]false[ 343, 5 ]true
C14 x D14 x D14是否幂零：false[ 8, 5 ]false[ 343, 5 ]true
C14 x C14 x D14是否幂零：false[ 8, 5 ]false[ 343, 5 ]true
C2 x C14 x ((C7 x C7) : C2)是否幂零：false[ 8, 5 ]false[ 343, 5 ]true
C14 x C14 x C14是否幂零：true[ 8, 5 ]true[ 343, 5 ]true
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