153種2744階群

GAP4中調用NumberSmallGroups(2744);指令會出錯。

陳松良的《2744階群的構造》一文證明G2744共有153種互不同構的類型,并獲得了G的全部構造:(1)當Sylow子群都正規時,G恰有25個彼此不同構的類型；(2)當Sylow 2-子群正規但Sylow 7-子群不正規時,G恰有8個彼此不同構的類型；(3)當Sylow 2-子群不正規但Sylow 7-子群正規時,G恰有120個彼此不同構的類型；(4)當Sylow子群都不正規時,G不存在.

20151029：4種類型153=25+8+120+0

Sylow子群都正規(25個):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-子群正規而Sylow 7-子群不正規(8個):C7 x ((C2 x C2 x C2) : C49)、C7 x C7 x ((C2 x C2 x C2) : C7)、

Sylow 7-子群正規而Sylow 2-子群不正規(120個):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子群都不正規(0個):

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("與C_2的直積=");Print(StructureDescription(G));Print(":是否可解：",IsSolvableGroup(G),",","是否幂零：",IsNilpotentGroup(G),",","自同構群：",Order(AutomorphismGroup(G)),",");L:=List(Elements(G),Order);;for i in M do Print(Size(Positions(L,i)),","); od;Print("\n");od;

[ 1372, 1 ]與C_2的直積=C2 x (C343 : C4):是否可解：true,是否幂零：false,自同構群：806736,1,3,1372,6,

0,18,0,42,0,126,0,294,0,882,0,0,

[ 1372, 2 ]與C_2的直積=C1372 x C2:是否可解：true,是否幂零：true,自同構群：2352,1,3,4,6,0,18,24,42,

0,126,168,294,0,882,1176,0,

[ 1372, 3 ]與C_2的直積=C2 x C2 x D686:是否可解：true,是否幂零：false,自同構群：2420208,1,1375,0,6,

0,18,0,42,0,126,0,294,0,882,0,0,

[ 1372, 4 ]與C_2的直積=C686 x C2 x C2:是否可解：true,是否幂零：true,自同構群：49392,1,7,0,6,0,42,0,

42,0,294,0,294,0,2058,0,0,

[ 1372, 5 ]與C_2的直積=C14 x (C49 : C4):是否可解：true,是否幂零：false,自同構群：98784,1,3,196,48,

0,144,1176,294,0,882,0,0,0,0,0,0,

[ 1372, 6 ]與C_2的直積=C98 x (C7 : C4):是否可解：true,是否幂零：false,自同構群：14112,1,3,28,48,0,

144,168,294,0,882,1176,0,0,0,0,0,

[ 1372, 7 ]與C_2的直積=C2 x (((C7 x C7) : C7) : C4):是否可解：true,是否幂零：false,自同構群：98784,

1,3,196,342,0,1026,1176,0,0,0,0,0,0,0,0,0,

[ 1372, 8 ]與C_2的直積=C2 x ((C49 : C7) : C4):是否可解：true,是否幂零：false,自同構群：16464,1,3,

196,48,0,144,1176,294,0,882,0,0,0,0,0,0,

[ 1372, 9 ]與C_2的直積=C2 x ((C49 x C7) : C4):是否可解：true,是否幂零：false,自同構群：33882912,1,

3,1372,48,0,144,0,294,0,882,0,0,0,0,0,0,

[ 1372, 10 ]與C_2的直積=C2 x (((C7 x C7) : C7) : C4):是否可解：true,是否幂零：false,自同構群：

790272,1,3,196,342,0,1026,1176,0,0,0,0,0,0,0,0,0,

[ 1372, 11 ]與C_2的直積=C196 x C14:是否可解：true,是否幂零：true,自同構群：98784,1,3,4,48,0,144,

192,294,0,882,1176,0,0,0,0,0,

[ 1372, 12 ]與C_2的直積=C2 x C4 x ((C7 x C7) : C7):是否可解：true,是否幂零：true,自同構群：790272,

1,3,4,342,0,1026,1368,0,0,0,0,0,0,0,0,0,

[ 1372, 13 ]與C_2的直積=C2 x C4 x (C49 : C7):是否可解：true,是否幂零：true,自同構群：16464,1,3,4,

48,0,144,192,294,0,882,1176,0,0,0,0,0,

[ 1372, 14 ]與C_2的直積=C2 x (((C7 x C7) : C7) : C4):是否可解：true,是否幂零：false,自同構群：9408,

1,99,196,342,0,930,1176,0,0,0,0,0,0,0,0,0,

[ 1372, 15 ]與C_2的直積=C2 x D14 x D98:是否可解：true,是否幂零：false,自同構群：345744,1,799,0,48,

0,720,0,294,0,882,0,0,0,0,0,0,

[ 1372, 16

 ]與C_2的直積=C2 x ((((C7 x C7) : C7) : C2) : C2):是否可解：true,是否幂零：false,自同構群：98784,1,

295,0,342,0,2106,0,0,0,0,0,0,0,0,0,0,

[ 1372, 17 ]與C_2的直積=C2 x C14 x D98:是否可解：true,是否幂零：false,自同構群：296352,1,199,0,48,

0,1320,0,294,0,882,0,0,0,0,0,0,

[ 1372, 18 ]與C_2的直積=C2 x C98 x D14:是否可解：true,是否幂零：false,自同構群：42336,1,31,0,48,0,

312,0,294,0,2058,0,0,0,0,0,0,

[ 1372, 19 ]與C_2的直積=C2 x C2 x (((C7 x C7) : C7) : C2):是否可解：true,是否幂零：false,自同構群：

296352,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,

[ 1372, 20 ]與C_2的直積=C2 x C2 x ((C49 : C7) : C2):是否可解：true,是否幂零：false,自同構群：49392,

1,199,0,48,0,1320,0,294,0,882,0,0,0,0,0,0,

[ 1372, 21 ]與C_2的直積=C2 x C2 x ((C49 x C7) : C2):是否可解：true,是否幂零：false,自同構群：

101648736,1,1375,0,48,0,144,0,294,0,882,0,0,0,0,0,0,

[ 1372, 22 ]與C_2的直積=C2 x C2 x (((C7 x C7) : C7) : C2):是否可解：true,是否幂零：false,自同構群：

2370816,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,

[ 1372, 23 ]與C_2的直積=C98 x C14 x C2:是否可解：true,是否幂零：true,自同構群：2074464,1,7,0,48,0,

336,0,294,0,2058,0,0,0,0,0,0,

[ 1372, 24 ]與C_2的直積=C2 x C2 x C2 x ((C7 x C7) : C7):是否可解：true,是否幂零：true,自同構群：

16595712,1,7,0,342,0,2394,0,0,0,0,0,0,0,0,0,0,

[ 1372, 25 ]與C_2的直積=C2 x C2 x C2 x (C49 : C7):是否可解：true,是否幂零：true,自同構群：345744,1,

7,0,48,0,336,0,294,0,2058,0,0,0,0,0,0,

[ 1372, 26 ]與C_2的直積=C7 x C14 x (C7 : C4):是否可解：true,是否幂零：false,自同構群：677376,1,3,

28,342,0,1026,1344,0,0,0,0,0,0,0,0,0,

[ 1372, 27 ]與C_2的直積=C14 x ((C7 x C7) : C4):是否可解：true,是否幂零：false,自同構群：4741632,1,

3,196,342,0,1026,1176,0,0,0,0,0,0,0,0,0,

[ 1372, 28 ]與C_2的直積=C2 x ((C7 x C7 x C7) : C4):是否可解：true,是否幂零：false,自同構群：

92703647232,1,3,1372,342,0,1026,0,0,0,0,0,0,0,0,0,0,

[ 1372, 29 ]與C_2的直積=C28 x C14 x C7:是否可解：true,是否幂零：true,自同構群：270273024,1,3,4,342,

0,1026,1368,0,0,0,0,0,0,0,0,0,

[ 1372, 30 ]與C_2的直積=C14 x ((C7 x C7) : C4):是否可解：true,是否幂零：false,自同構群：56448,1,99,

196,342,0,930,1176,0,0,0,0,0,0,0,0,0,

[ 1372, 31 ]與C_2的直積=C2 x ((C7 x C7 x C7) : C4):是否可解：true,是否幂零：false,自同構群：395136,

1,99,1372,342,0,930,0,0,0,0,0,0,0,0,0,0,

[ 1372, 32 ]與C_2的直積=C14 x D14 x D14:是否可解：true,是否幂零：false,自同構群：84672,1,127,0,342,

0,2274,0,0,0,0,0,0,0,0,0,0,

[ 1372, 33 ]與C_2的直積=C2 x (((C7 x C7) : C2) x D14):是否可解：true,是否幂零：false,自同構群：

16595712,1,799,0,342,0,1602,0,0,0,0,0,0,0,0,0,0,

[ 1372, 34

 ]與C_2的直積=C2 x ((C7 x ((C7 x C7) : C2)) : C2):是否可解：true,是否幂零：false,自同構群：1778112,

1,295,0,342,0,2106,0,0,0,0,0,0,0,0,0,0,

[ 1372, 35 ]與C_2的直積=C14 x C14 x D14:是否可解：true,是否幂零：false,自同構群：2032128,1,31,0,

342,0,2370,0,0,0,0,0,0,0,0,0,0,

[ 1372, 36 ]與C_2的直積=C2 x C14 x ((C7 x C7) : C2):是否可解：true,是否幂零：false,自同構群：

14224896,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,

[ 1372, 37 ]與C_2的直積=C2 x C2 x ((C7 x C7 x C7) : C2):是否可解：true,是否幂零：false,自同構群：

278110941696,1,1375,0,342,0,1026,0,0,0,0,0,0,0,0,0,0,

[ 1372, 38 ]與C_2的直積=C14 x C14 x C14:是否可解：true,是否幂零：true,自同構群：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("與C_4的直積=");Print(StructureDescription(G));Print(":是否可解：",IsSolvableGroup(G),",","是否幂零：",IsNilpotentGroup(G),",","自同構群：",Order(AutomorphismGroup(G)),",");L:=List(Elements(G),Order);;for i in M do Print(Size(Positions(L,i)),","); od;Print("\n");od;

[ 686, 1 ]與C_4的直積=C4 x D686:是否可解：true,是否幂零：false,自同構群：403368,1,687,688,6,0,6,12,

42,0,42,84,294,0,294,588,0,

[ 686, 2 ]與C_4的直積=C1372 x C2:是否可解：true,是否幂零：true,自同構群：2352,1,3,4,6,0,18,24,42,0,

126,168,294,0,882,1176,0,

[ 686, 3 ]與C_4的直積=C28 x D98:是否可解：true,是否幂零：false,自同構群：49392,1,99,100,48,0,636,

684,294,0,294,588,0,0,0,0,0,

[ 686, 4 ]與C_4的直積=C196 x D14:是否可解：true,是否幂零：false,自同構群：7056,1,15,16,48,0,132,

180,294,0,882,1176,0,0,0,0,0,

[ 686, 5 ]與C_4的直積=C4 x (((C7 x C7) : C7) : C2):是否可解：true,是否幂零：false,自同構群：49392,

1,99,100,342,0,930,1272,0,0,0,0,0,0,0,0,0,

[ 686, 6 ]與C_4的直積=C4 x ((C49 : C7) : C2):是否可解：true,是否幂零：false,自同構群：8232,1,99,

100,48,0,636,684,294,0,294,588,0,0,0,0,0,

[ 686, 7 ]與C_4的直積=C4 x ((C49 x C7) : C2):是否可解：true,是否幂零：false,自同構群：16941456,1,

687,688,48,0,48,96,294,0,294,588,0,0,0,0,0,

[ 686, 8 ]與C_4的直積=C4 x (((C7 x C7) : C7) : C2):是否可解：true,是否幂零：false,自同構群：395136,

1,99,100,342,0,930,1272,0,0,0,0,0,0,0,0,0,

[ 686, 9 ]與C_4的直積=C196 x C14:是否可解：true,是否幂零：true,自同構群：98784,1,3,4,48,0,144,192,

294,0,882,1176,0,0,0,0,0,

[ 686, 10 ]與C_4的直積=C2 x C4 x ((C7 x C7) : C7):是否可解：true,是否幂零：true,自同構群：790272,1,

3,4,342,0,1026,1368,0,0,0,0,0,0,0,0,0,

[ 686, 11 ]與C_4的直積=C2 x C4 x (C49 : C7):是否可解：true,是否幂零：true,自同構群：16464,1,3,4,48,

0,144,192,294,0,882,1176,0,0,0,0,0,

[ 686, 12 ]與C_4的直積=C7 x C28 x D14:是否可解：true,是否幂零：false,自同構群：338688,1,15,16,342,

0,1014,1356,0,0,0,0,0,0,0,0,0,

[ 686, 13 ]與C_4的直積=C28 x ((C7 x C7) : C2):是否可解：true,是否幂零：false,自同構群：2370816,1,

99,100,342,0,930,1272,0,0,0,0,0,0,0,0,0,

[ 686, 14 ]與C_4的直積=C4 x ((C7 x C7 x C7) : C2):是否可解：true,是否幂零：false,自同構群：

46351823616,1,687,688,342,0,342,684,0,0,0,0,0,0,0,0,0,

[ 686, 15 ]與C_4的直積=C28 x C14 x C7:是否可解：true,是否幂零：true,自同構群：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("與K_4的直積=");Print(StructureDescription(G));Print(": 是否可解：",IsSolvableGroup(G),",","是否幂零：",IsNilpotentGroup(G),",","自同構群：",Order(AutomorphismGroup(G)),",");L:=List(Elements(G),Order);;for i in M do Print(Size(Positions(L,i)),","); od;Print("\n");od;

[ 686, 1 ]與K_4的直積=C2 x C2 x D686:是否可解：true,是否幂零：false,自同構群：2420208,1,1375,0,6,0,

18,0,42,0,126,0,294,0,882,0,0,

[ 686, 2 ]與K_4的直積=C686 x C2 x C2:是否可解：true,是否幂零：true,自同構群：49392,1,7,0,6,0,42,0,

42,0,294,0,294,0,2058,0,0,

[ 686, 3 ]與K_4的直積=C2 x C14 x D98:是否可解：true,是否幂零：false,自同構群：296352,1,199,0,48,0,

1320,0,294,0,882,0,0,0,0,0,0,

[ 686, 4 ]與K_4的直積=C2 x C98 x D14:是否可解：true,是否幂零：false,自同構群：42336,1,31,0,48,0,

312,0,294,0,2058,0,0,0,0,0,0,

[ 686, 5 ]與K_4的直積=C2 x C2 x (((C7 x C7) : C7) : C2):是否可解：true,是否幂零：false,自同構群：

296352,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,

[ 686, 6 ]與K_4的直積=C2 x C2 x ((C49 : C7) : C2):是否可解：true,是否幂零：false,自同構群：49392,1,

199,0,48,0,1320,0,294,0,882,0,0,0,0,0,0,

[ 686, 7 ]與K_4的直積=C2 x C2 x ((C49 x C7) : C2):是否可解：true,是否幂零：false,自同構群：

101648736,1,1375,0,48,0,144,0,294,0,882,0,0,0,0,0,0,

[ 686, 8 ]與K_4的直積=C2 x C2 x (((C7 x C7) : C7) : C2):是否可解：true,是否幂零：false,自同構群：

2370816,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,

[ 686, 9 ]與K_4的直積=C98 x C14 x C2:是否可解：true,是否幂零：true,自同構群：2074464,1,7,0,48,0,

336,0,294,0,2058,0,0,0,0,0,0,

[ 686, 10 ]與K_4的直積=C2 x C2 x C2 x ((C7 x C7) : C7):是否可解：true,是否幂零：true,自同構群：

16595712,1,7,0,342,0,2394,0,0,0,0,0,0,0,0,0,0,

[ 686, 11 ]與K_4的直積=C2 x C2 x C2 x (C49 : C7):是否可解：true,是否幂零：true,自同構群：345744,1,

7,0,48,0,336,0,294,0,2058,0,0,0,0,0,0,

[ 686, 12 ]與K_4的直積=C14 x C14 x D14:是否可解：true,是否幂零：false,自同構群：2032128,1,31,0,342,

0,2370,0,0,0,0,0,0,0,0,0,0,

[ 686, 13 ]與K_4的直積=C2 x C14 x ((C7 x C7) : C2):是否可解：true,是否幂零：false,自同構群：

14224896,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,

[ 686, 14 ]與K_4的直積=C2 x C2 x ((C7 x C7 x C7) : C2):是否可解：true,是否幂零：false,自同構群：

278110941696,1,1375,0,342,0,1026,0,0,0,0,0,0,0,0,0,0,

[ 686, 15 ]與K_4的直積=C14 x C14 x C14:是否可解：true,是否幂零：true,自同構群：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("與C_7的直積=");Print(StructureDescription(G));Print(":是否可解：",IsSolvableGroup(G),",","是否幂零：",IsNilpotentGroup(G),",","自同構群：",Order(AutomorphismGroup(G)),",");L:=List(Elements(G),Order);;for i in M do Print(Size(Positions(L,i)),","); od;Print("\n");od;

[ 392, 1 ]與C_7的直積=C7 x (C49 : C8):是否可解：true,是否幂零：false,自同構群：49392,1,1,2,48,196,

48,96,294,1176,294,588,0,0,0,0,0,

[ 392, 2 ]與C_7的直積=C392 x C7:是否可解：true,是否幂零：true,自同構群：49392,1,1,2,48,4,48,96,294,

192,294,588,0,1176,0,0,0,

[ 392, 3 ]與C_7的直積=C7 x (C49 : Q8):是否可解：true,是否幂零：false,自同構群：98784,1,1,198,48,0,

48,1272,294,0,294,588,0,0,0,0,0,

[ 392, 4 ]與C_7的直積=C28 x D98:是否可解：true,是否幂零：false,自同構群：49392,1,99,100,48,0,636,

684,294,0,294,588,0,0,0,0,0,

[ 392, 5 ]與C_7的直積=C7 x D392:是否可解：true,是否幂零：false,自同構群：98784,1,197,2,48,0,1224,

96,294,0,294,588,0,0,0,0,0,

[ 392, 6 ]與C_7的直積=C14 x (C49 : C4):是否可解：true,是否幂零：false,自同構群：98784,1,3,196,48,0,

144,1176,294,0,882,0,0,0,0,0,0,

[ 392, 7 ]與C_7的直積=C7 x ((C98 x C2) : C2):是否可解：true,是否幂零：false,自同構群：49392,1,101,

98,48,0,732,588,294,0,882,0,0,0,0,0,0,

[ 392, 8 ]與C_7的直積=C196 x C14:是否可解：true,是否幂零：true,自同構群：98784,1,3,4,48,0,144,192,

294,0,882,1176,0,0,0,0,0,

[ 392, 9 ]與C_7的直積=C7 x C49 x D8:是否可解：true,是否幂零：true,自同構群：98784,1,5,2,48,0,240,

96,294,0,1470,588,0,0,0,0,0,

[ 392, 10 ]與C_7的直積=C7 x C49 x Q8:是否可解：true,是否幂零：true,自同構群：296352,1,1,6,48,0,48,

288,294,0,294,1764,0,0,0,0,0,

[ 392, 11 ]與C_7的直積=C7 x ((C2 x C2 x C2) : C49):是否可解：true,是否幂零：false,自同構群：345744,

1,7,0,48,0,336,0,2352,0,0,0,0,0,0,0,0,

[ 392, 12 ]與C_7的直積=C2 x C14 x D98:是否可解：true,是否幂零：false,自同構群：296352,1,199,0,48,0,

1320,0,294,0,882,0,0,0,0,0,0,

[ 392, 13 ]與C_7的直積=C98 x C14 x C2:是否可解：true,是否幂零：true,自同構群：2074464,1,7,0,48,0,

336,0,294,0,2058,0,0,0,0,0,0,

[ 392, 14 ]與C_7的直積=C7 x C7 x (C7 : C8):是否可解：true,是否幂零：false,自同構群：338688,1,1,2,

342,28,342,684,0,1344,0,0,0,0,0,0,0,

[ 392, 15 ]與C_7的直積=C7 x ((C7 x C7) : C8):是否可解：true,是否幂零：false,自同構群：2370816,1,1,

2,342,196,342,684,0,1176,0,0,0,0,0,0,0,

[ 392, 16 ]與C_7的直積=C56 x C7 x C7:是否可解：true,是否幂零：true,自同構群：135136512,1,1,2,342,4,

342,684,0,1368,0,0,0,0,0,0,0,

[ 392, 17 ]與C_7的直積=C7 x ((C7 x C7) : C8):是否可解：true,是否幂零：false,自同構群：56448,1,1,98,

342,196,342,588,0,1176,0,0,0,0,0,0,0,

[ 392, 18 ]與C_7的直積=C7 x ((C7 : C4) x D14):是否可解：true,是否幂零：false,自同構群：42336,1,15,

112,342,0,1014,1260,0,0,0,0,0,0,0,0,0,

[ 392, 19 ]與C_7的直積=C7 x ((C7 x (C7 : C4)) : C2):是否可解：true,是否幂零：false,自同構群：84672,

1,99,28,342,0,930,1344,0,0,0,0,0,0,0,0,0,

[ 392, 20 ]與C_7的直積=C7 x ((C14 x D14) : C2):是否可解：true,是否幂零：false,自同構群：84672,1,29,

98,342,0,1686,588,0,0,0,0,0,0,0,0,0,

[ 392, 21 ]與C_7的直積=C7 x ((C14 x D14) : C2):是否可解：true,是否幂零：false,自同構群：42336,1,

113,14,342,0,1602,672,0,0,0,0,0,0,0,0,0,

[ 392, 22 ]與C_7的直積=C7 x ((C7 x C7) : Q8):是否可解：true,是否幂零：false,自同構群：84672,1,1,

126,342,0,342,1932,0,0,0,0,0,0,0,0,0,

[ 392, 23 ]與C_7的直積=C7 x C7 x (C7 : Q8):是否可解：true,是否幂零：false,自同構群：677376,1,1,30,

342,0,342,2028,0,0,0,0,0,0,0,0,0,

[ 392, 24 ]與C_7的直積=C7 x C28 x D14:是否可解：true,是否幂零：false,自同構群：338688,1,15,16,342,

0,1014,1356,0,0,0,0,0,0,0,0,0,

[ 392, 25 ]與C_7的直積=C7 x C7 x D56:是否可解：true,是否幂零：false,自同構群：677376,1,29,2,342,0,

1686,684,0,0,0,0,0,0,0,0,0,

[ 392, 26 ]與C_7的直積=C7 x C14 x (C7 : C4):是否可解：true,是否幂零：false,自同構群：677376,1,3,28,

342,0,1026,1344,0,0,0,0,0,0,0,0,0,

[ 392, 27 ]與C_7的直積=C7 x C7 x ((C14 x C2) : C2):是否可解：true,是否幂零：false,自同構群：338688,

1,17,14,342,0,1698,672,0,0,0,0,0,0,0,0,0,

[ 392, 28 ]與C_7的直積=C7 x ((C7 x C7) : Q8):是否可解：true,是否幂零：false,自同構群：4741632,1,1,

198,342,0,342,1860,0,0,0,0,0,0,0,0,0,

[ 392, 29 ]與C_7的直積=C28 x ((C7 x C7) : C2):是否可解：true,是否幂零：false,自同構群：2370816,1,

99,100,342,0,930,1272,0,0,0,0,0,0,0,0,0,

[ 392, 30 ]與C_7的直積=C7 x ((C28 x C7) : C2):是否可解：true,是否幂零：false,自同構群：4741632,1,

197,2,342,0,1518,684,0,0,0,0,0,0,0,0,0,

[ 392, 31 ]與C_7的直積=C14 x ((C7 x C7) : C4):是否可解：true,是否幂零：false,自同構群：4741632,1,3,

196,342,0,1026,1176,0,0,0,0,0,0,0,0,0,

[ 392, 32 ]與C_7的直積=C7 x ((C14 x C14) : C2):是否可解：true,是否幂零：false,自同構群：2370816,1,

101,98,342,0,1614,588,0,0,0,0,0,0,0,0,0,

[ 392, 33 ]與C_7的直積=C28 x C14 x C7:是否可解：true,是否幂零：true,自同構群：270273024,1,3,4,342,

0,1026,1368,0,0,0,0,0,0,0,0,0,

[ 392, 34 ]與C_7的直積=C7 x C7 x C7 x D8:是否可解：true,是否幂零：true,自同構群：270273024,1,5,2,

342,0,1710,684,0,0,0,0,0,0,0,0,0,

[ 392, 35 ]與C_7的直積=C7 x C7 x C7 x Q8:是否可解：true,是否幂零：true,自同構群：810819072,1,1,6,

342,0,342,2052,0,0,0,0,0,0,0,0,0,

[ 392, 36 ]與C_7的直積=C7 x ((C7 x C7) : C8):是否可解：true,是否幂零：false,自同構群：28224,1,49,

98,342,196,294,588,0,1176,0,0,0,0,0,0,0,

[ 392, 37 ]與C_7的直積=C7 x ((D14 x D14) : C2):是否可解：true,是否幂零：false,自同構群：14112,1,77,

98,342,0,1638,588,0,0,0,0,0,0,0,0,0,

[ 392, 38 ]與C_7的直積=C7 x ((C7 x C7) : Q8):是否可解：true,是否幂零：false,自同構群：42336,1,49,

294,342,0,294,1764,0,0,0,0,0,0,0,0,0,

[ 392, 39 ]與C_7的直積=C7 x C7 x ((C2 x C2 x C2) : C7):是否可解：true,是否幂零：false,自同構群：

16595712,1,7,0,2400,0,336,0,0,0,0,0,0,0,0,0,0,

[ 392, 40 ]與C_7的直積=C14 x ((C7 x C7) : C4):是否可解：true,是否幂零：false,自同構群：56448,1,99,

196,342,0,930,1176,0,0,0,0,0,0,0,0,0,

[ 392, 41 ]與C_7的直積=C14 x D14 x D14:是否可解：true,是否幂零：false,自同構群：84672,1,127,0,342,

0,2274,0,0,0,0,0,0,0,0,0,0,

[ 392, 42 ]與C_7的直積=C14 x C14 x D14:是否可解：true,是否幂零：false,自同構群：2032128,1,31,0,342,

0,2370,0,0,0,0,0,0,0,0,0,0,

[ 392, 43 ]與C_7的直積=C2 x C14 x ((C7 x C7) : C2):是否可解：true,是否幂零：false,自同構群：

14224896,1,199,0,342,0,2202,0,0,0,0,0,0,0,0,0,0,

[ 392, 44 ]與C_7的直積=C14 x C14 x C14:是否可解：true,是否幂零：true,自同構群：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