20151017補充：定理：51種32階羣與它們的羣元階的分佈、特徵標表、換位子羣、自同構羣、正規子羣的個數一一對應[ 1, 2, 4, 8,16,32]
gap> NumberSmallGroups(32);
51
51個32階羣
Group GAP4(32,1) [C32]1,1,2,4,8,16,
換位子羣：[ 1, 1 ]
自同構羣：[ 16, 5 ]
正規子羣個數：6
Group GAP4(32,2) [(C4 x C2) : C4]1,7,24,0,0,0,
換位子羣：[ 2, 1 ]
自同構羣：[ 64, 202 ]
正規子羣個數：16
Group GAP4(32,3) [C8 x C4]1,3,12,16,0,0,
換位子羣：[ 1, 1 ]
自同構羣：[ 128, 753 ]
正規子羣個數：22
Group GAP4(32,4) [C8 : C4]1,3,12,16,0,0,
換位子羣：[ 2, 1 ]
自同構羣：[ 128, 753 ]
正規子羣個數：18
Group GAP4(32,5) [(C8 x C2) : C2]1,7,8,16,0,0,
換位子羣：[ 2, 1 ]
自同構羣：[ 64, 202 ]
正規子羣個數：16
Group GAP4(32,6) [((C4 x C2) : C2) : C2]1,11,20,0,0,0,
共軛類數：11
中心：[ 2, 1 ]
換位子羣：[ 4, 2 ]
自同構羣：64
正規子羣個數：12
換位子羣：[ 4, 2 ]
自同構羣：[ 64, 138 ]
正規子羣個數：12
Group GAP4(32,7) [(C8 : C2) : C2]1,11,4,16,0,0,
Group GAP4(32,8) [C2 . ((C4 x C2) : C2) = (C2 x C2) . (C4 x C2)]1,3,12,16,0,0,
換位子羣：[ 4, 2 ]
自同構羣：128
正規子羣個數：12
Group GAP4(32,9) [(C8 x C2) : C2]1,11,12,8,0,0,
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 1 ]
自同構羣：[ 64, 261 ]
正規子羣個數：14
Group GAP4(32,10) [Q8 : C4]1,3,20,8,0,0,
秩：2
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 1 ]
自同構羣：[ 64, 261 ]
正規子羣個數：14
Group GAP4(32,11) [(C4 x C4) : C2]1,7,16,8,0,0,
換位子羣：[ 4, 1 ]
自同構羣：32
正規子羣個數：12
Group GAP4(32,12) [C4 : C8]1,3,12,16,0,0,
換位子羣：[ 2, 1 ]
自同構羣：64
正規子羣個數：16
Group GAP4(32,13) [C8 : C4]1,3,20,8,0,0,
秩：2
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 1 ]
自同構羣：[ 128, 1735 ]
正規子羣個數：14
Group GAP4(32,14) [C8 : C4]1,3,20,8,0,0,
秩：2
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 1 ]
自同構羣：[ 128, 1735 ]
正規子羣個數：14
Group GAP4(32,15) [C4 . D8 = C4 . (C4 x C2)]1,3,4,24,0,0,
Group GAP4(32,16) [C16 x C2]1,3,4,8,16,0,
共軛類數：32
中心：[ 32, 16 ]
換位子羣：[ 1, 1 ]
自同構羣：32
正規子羣個數：14
Group GAP4(32,17) [C16 : C2]1,3,4,8,16,0,
共軛類數：20
中心：[ 8, 1 ]
換位子羣：[ 2, 1 ]
自同構羣：32
正規子羣個數：12
Group GAP4(32,18) [D32]1,17,2,4,8,0,
Group GAP4(32,19) [QD32]1,9,10,4,8,0,
gap> F:=FreeGroup(2);;QD_32:=F/[F.1^16,F.2^2,F.2*F.1*F.2*(F.1^7)^(-1)];;IdGroup(QD_32);
[ 32, 19 ]
Group GAP4(32,20) [Q32]1,1,18,4,8,0,
Group GAP4(32,21) [C4 x C4 x C2]1,7,24,0,0,0,
換位子羣：[ 1, 1 ]
自同構羣：1536
正規子羣個數：54
Group GAP4(32,22) [C2 x ((C4 x C2) : C2)]1,15,16,0,0,0,
共軛類數：20
中心：[ 8, 5 ]
換位子羣：[ 2, 1 ]
自同構羣：512
正規子羣個數：38
Group GAP4(32,23) [C2 x (C4 : C4)]1,7,24,0,0,0,
換位子羣：[ 2, 1 ]
自同構羣：512
正規子羣個數：38
Group GAP4(32,24) [(C4 x C4) : C2]1,7,24,0,0,0,
換位子羣：[ 2, 1 ]
自同構羣：256
正規子羣個數：30
Group GAP4(32,25) [C4 x D8]1,11,20,0,0,0,
共軛類數：20
中心：[ 8, 2 ]
換位子羣：[ 2, 1 ]
自同構羣：128
正規子羣個數：32
Group GAP4(32,26) [C4 x Q8]1,3,28,0,0,0,
共軛類數：20
中心：[ 8, 2 ]
換位子羣：[ 2, 1 ]
自同構羣：384
正規子羣個數：32
Group GAP4(32,27) [(C2 x C2 x C2 x C2) : C2]1,19,12,0,0,0,
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 2 ]
自同構羣：384
正規子羣個數：26
Group GAP4(32,28) [(C4 x C2 x C2) : C2]1,15,16,0,0,0,
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 2 ]
自同構羣：128
正規子羣個數：24
Group GAP4(32,29) [(C2 x Q8) : C2]1,7,24,0,0,0,
換位子羣：[ 4, 2 ]
自同構羣：128
正規子羣個數：24
Group GAP4(32,30) [(C4 x C2 x C2) : C2]1,11,20,0,0,0,
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 2 ]
自同構羣：128
正規子羣個數：22
Group GAP4(32,31) [(C4 x C4) : C2]1,11,20,0,0,0,
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 2 ]
自同構羣：256
正規子羣個數：22
Group GAP4(32,32) [(C2 x C2) . (C2 x C2 x C2)]1,3,28,0,0,0,
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 2 ]
自同構羣：256
正規子羣個數：22
Group GAP4(32,33) [(C4 x C4) : C2]1,7,24,0,0,0,
換位子羣：[ 4, 2 ]
自同構羣：192
正規子羣個數：20
Group GAP4(32,34) [(C4 x C4) : C2]1,19,12,0,0,0,
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 2 ]
自同構羣：1536
正規子羣個數：26
Group GAP4(32,35) [C4 : Q8]1,3,28,0,0,0,
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 2 ]
自同構羣：512
正規子羣個數：26
Group GAP4(32,36) [C8 x C2 x C2]1,7,8,16,0,0,
換位子羣：[ 1, 1 ]
自同構羣：384
正規子羣個數：38
Group GAP4(32,37) [C2 x (C8 : C2)]1,7,8,16,0,0,
換位子羣：[ 2, 1 ]
自同構羣：128
正規子羣個數：30
Group GAP4(32,38) [(C8 x C2) : C2]1,7,8,16,0,0,
換位子羣：[ 2, 1 ]
自同構羣：96
正規子羣個數：28
Group GAP4(32,39) [C2 x D16]1,19,4,8,0,0,
Group GAP4(32,40) [C2 x QD16]1,11,12,8,0,0,
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 1 ]
自同構羣：[ 128, 2216 ]
正規子羣個數：22
Group GAP4(32,41) [C2 x Q16]1,3,20,8,0,0,
秩：3
共軛類數：14
中心：[ 4, 2 ]
換位子羣：[ 4, 1 ]
自同構羣：[ 256, 16870 ]
正規子羣個數：22
Group GAP4(32,42) [(C8 x C2) : C2]1,11,12,8,0,0,
共軛類數：14
中心：[ 4, 1 ]
換位子羣：[ 4, 1 ]
自同構羣：[ 64, 254 ]
正規子羣個數：20
Group GAP4(32,43) [(C2 x D8) : C2]1,15,8,8,0,0,
Group GAP4(32,44) [(C2 x Q8) : C2]1,7,16,8,0,0,
換位子羣：[ 4, 1 ]
自同構羣：64
正規子羣個數：20
Group GAP4(32,45) [C4 x C2 x C2 x C2]1,15,16,0,0,0,
共軛類數：32
中心：[ 32, 45 ]
換位子羣：[ 1, 1 ]
自同構羣：21504
正規子羣個數：118
Group GAP4(32,46) [C2 x C2 x D8]1,23,8,0,0,0,
Group GAP4(32,47) [C2 x C2 x Q8]1,7,24,0,0,0,
換位子羣：[ 2, 1 ]
自同構羣：9216
正規子羣個數：78
Group GAP4(32,48) [C2 x ((C4 x C2) : C2)]1,15,16,0,0,0,
共軛類數：20
中心：[ 8, 2 ]
換位子羣：[ 2, 1 ]
自同構羣：[ 768, 1087581 ]
正規子羣個數：70
Group GAP4(32,49) [(C2 x D8) : C2]1,19,12,0,0,0,
共軛類數：17
中心：[ 2, 1 ]
換位子羣：[ 2, 1 ]
自同構羣：1152
正規子羣個數：68
Group GAP4(32,50) [(C2 x Q8) : C2]1,11,20,0,0,0,
共軛類數：17
中心：[ 2, 1 ]
換位子羣：[ 2, 1 ]
自同構羣：1920
正規子羣個數：68
Group GAP4(32,51) [C2 x C2 x C2 x C2 x C2]1,31,0,0,0,0,
http://users.minet.uni-jena.de/~green/Coho_v3/32gps/index.html
The small groups of order 32
All 51 groups of order 32 are listed and completely calculated.
--------------------------------------------------------------------------------
Small group number 1 of order 32 Cyclic group of order 32
Small group number 2 of order 32
Small group number 3 of order 32 Abelian group C8 x C4
Small group number 4 of order 32
Small group number 5 of order 32
Small group number 6 of order 32
Small group number 7 of order 32
Small group number 8 of order 32
Small group number 9 of order 32
Small group number 10 of order 32
Small group number 11 of order 32
Small group number 12 of order 32
Small group number 13 of order 32
Small group number 14 of order 32
Small group number 15 of order 32
Small group number 16 of order 32 Abelian group C16 x C2
Small group number 17 of order 32 Modular group of order 32
Small group number 18 of order 32 Dihedral group of order 32
Small group number 19 of order 32 Semidihedral group of order 32
Small group number 20 of order 32 Quaternion group of order 32
Small group number 21 of order 32 Abelian group C4 x C4 x C2
Small group number 22 of order 32 Direct product 16gp3 x C_2
Small group number 23 of order 32 Direct product 16gp4 x C_2
Small group number 24 of order 32
Small group number 25 of order 32 Direct product D8 x C_4
Small group number 26 of order 32 Direct product Q8 x C_4
Small group number 27 of order 32
Small group number 28 of order 32
Small group number 29 of order 32
Small group number 30 of order 32
Small group number 31 of order 32
Small group number 32 of order 32
Small group number 33 of order 32
Small group number 34 of order 32
Small group number 35 of order 32
Small group number 36 of order 32 Abelian group C8 x C2 x C2
Small group number 37 of order 32 Direct product Mod16 x C_2
Small group number 38 of order 32
Small group number 39 of order 32 Direct product D16 x C_2
Small group number 40 of order 32 Direct product SD16 x C_2
Small group number 41 of order 32 Direct product Q16 x C_2
Small group number 42 of order 32
Small group number 43 of order 32
Small group number 44 of order 32
Small group number 45 of order 32 Abelian group C4 x C2 x C2 x C2
Small group number 46 of order 32 Direct product D8 x V_4
Small group number 47 of order 32 Direct product Q8 x V_4
Small group number 48 of order 32 Direct product 16gp13 x C_2
Small group number 49 of order 32 Extraspecial 2-group of order 32 and type +
Small group number 50 of order 32 Extraspecial 2-group of order 32 and type -
Small group number 51 of order 32 Elementary abelian group of order 32
低階羣工具http://wims.unice.fr/wims/cn_tool~algebra~smallgroup.html
存在 32 階的51羣
編號|GAP 序列號|性質 | 指數 |中心 |G/[G,G] |共軛類 | 子羣 | 子羣類 | 正規子羣
1 1 循環 32 C32 C32 32 -- -- --
2 16 阿貝爾 16 C2×C16 C2×C16 32 -- -- --
3 3 阿貝爾 8 C4×C8 C4×C8 32 -- -- --
4 36 阿貝爾 8 C22×C8 C22×C8 32 -- -- --
5 21 阿貝爾 4 C2×C42 C2×C42 32 -- -- --
6 45 阿貝爾 4 C23×C4 C23×C4 32 -- -- --
7 51 阿貝爾 2 C25 C25 32 -- -- --
8 20 冪零 16 C2 C22 11 20 12 8
9 19 冪零 16 C2 C22 11 28 13 8
10 18 冪零 16 C2 C22 11 36 14 8
11 17 冪零 16 C8 C2×C8 20 14 13 12
12 8 冪零 8 C2 C2×C4 11 26 19 12
13 7 冪零 8 C2 C2×C4 11 42 23 12
14 44 冪零 8 C2 C23 11 42 30 20
15 43 冪零 8 C2 C23 11 58 34 20
16 15 冪零 8 C4 C2×C4 14 18 15 12
17 11 冪零 8 C4 C2×C4 14 34 22 12
18 42 冪零 8 C4 C23 14 46 31 20
19 13 冪零 8 C22 C2×C4 14 26 18 14
20 14 冪零 8 C22 C2×C4 14 26 18 14
21 10 冪零 8 C22 C2×C4 14 30 21 14
22 9 冪零 8 C22 C2×C4 14 46 25 14
23 41 冪零 8 C22 C23 14 38 30 22
24 40 冪零 8 C22 C23 14 54 34 22
25 39 冪零 8 C22 C23 14 70 38 22
26 38 冪零 8 C8 C22×C4 20 34 31 28
27 4 冪零 8 C2×C4 C42 20 22 20 18
28 12 冪零 8 C2×C4 C2×C8 20 22 19 16
29 5 冪零 8 C2×C4 C2×C8 20 34 25 16
30 37 冪零 8 C2×C4 C22×C4 20 38 34 30
31 6 冪零 4 C2 C2×C4 11 50 26 12
32 50 冪零 4 C2 C24 17 78 73 68
33 49 冪零 4 C2 C24 17 110 83 68
34 32 冪零 4 C22 C23 14 34 28 22
35 33 冪零 4 C22 C23 14 42 30 20
36 35 冪零 4 C22 C23 14 42 34 26
37 29 冪零 4 C22 C23 14 50 37 24
38 31 冪零 4 C22 C23 14 58 38 22
39 30 冪零 4 C22 C23 14 58 39 22
40 28 冪零 4 C22 C23 14 74 47 24
41 34 冪零 4 C22 C23 14 90 54 26
42 27 冪零 4 C22 C23 14 106 65 26
43 26 冪零 4 C2×C4 C22×C4 20 38 35 32
44 24 冪零 4 C2×C4 C22×C4 20 46 38 30
45 25 冪零 4 C2×C4 C22×C4 20 62 47 32
46 48 冪零 4 C2×C4 C24 20 94 82 70
47 2 冪零 4 C23 C42 20 50 38 26
48 23 冪零 4 C23 C22×C4 20 54 46 38
49 22 冪零 4 C23 C22×C4 20 94 66 38
50 47 冪零 4 C23 C24 20 78 78 78
51 46 冪零 4 C23 C24 20 158 118 78
gap> g:=SmallGroup(32,10);;cl:=ConjugacyClasses(g);;L1:=List(cl,Representative);;L2:=List(cl,Centralizer);;L3:=List(L2,IdGroup);;L4:=List(cl,Size);;tbl:= CharacterTable( g );;Display( tbl );
CT1
2 5 3 3 4 5 5 4 3 3 4 5 4 4 4
1a 4a 4b 4c 2a 2b 8a 4d 4e 4f 2c 8b 8c 8d
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 -1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1
X.3 1 1 -1 1 1 1 -1 1 -1 1 1 -1 -1 -1
X.4 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1
X.5 1 A 1 1 -1 1 A -A -1 -1 -1 -A A -A
X.6 1 -A 1 1 -1 1 -A A -1 -1 -1 A -A A
X.7 1 A -1 1 -1 1 -A -A 1 -1 -1 A -A A
X.8 1 -A -1 1 -1 1 A A 1 -1 -1 -A A -A
X.9 2 . . -2 2 2 . . . -2 2 . . .
X.10 2 . . -2 -2 2 . . . 2 -2 . . .
X.11 2 . . . 2 -2 B . . . -2 B -B -B
X.12 2 . . . 2 -2 -B . . . -2 -B B B
X.13 2 . . . -2 -2 C . . . 2 -C -C C
X.14 2 . . . -2 -2 -C . . . 2 C C -C
A = E(4)
= Sqrt(-1) = i
B = E(8)+E(8)^3
= Sqrt(-2) = i2
C = -E(8)+E(8)^3
= -Sqrt(2) = -r2
gap> g:=SmallGroup(32,13);;cl:=ConjugacyClasses(g);;L1:=List(cl,Representative);;L2:=List(cl,Centralizer);;L3:=List(L2,IdGroup);;L4:=List(cl,Size);;tbl:= CharacterTable( g );;Display( tbl );
CT2
2 5 3 4 4 5 5 3 3 4 4 4 5 3 4
1a 4a 8a 4b 2a 2b 4c 4d 8b 8c 4e 2c 4f 8d
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 1
X.3 1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 -1
X.4 1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1
X.5 1 A 1 1 -1 1 A -A -1 1 -1 -1 -A -1
X.6 1 -A 1 1 -1 1 -A A -1 1 -1 -1 A -1
X.7 1 A -1 1 -1 1 -A -A 1 -1 -1 -1 A 1
X.8 1 -A -1 1 -1 1 A A 1 -1 -1 -1 -A 1
X.9 2 . . -2 2 2 . . . . -2 2 . .
X.10 2 . . -2 -2 2 . . . . 2 -2 . .
X.11 2 . B . 2 -2 . . B -B . -2 . -B
X.12 2 . -B . 2 -2 . . -B B . -2 . B
X.13 2 . B . -2 -2 . . -B -B . 2 . B
X.14 2 . -B . -2 -2 . . B B . 2 . -B
A = E(4)
= Sqrt(-1) = i
B = E(8)+E(8)^3
= Sqrt(-2) = i2
gap> g:=SmallGroup(32,14);;cl:=ConjugacyClasses(g);;L1:=List(cl,Representative);;L2:=List(cl,Centralizer);;L3:=List(L2,IdGroup);;L4:=List(cl,Size);;tbl:= CharacterTable( g );;Display( tbl );
CT3
2 5 3 4 4 5 5 3 3 4 4 4 5 3 4
1a 4a 8a 4b 2a 2b 4c 4d 8b 8c 4e 2c 4f 8d
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 1
X.3 1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 -1
X.4 1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1
X.5 1 A 1 1 -1 1 A -A -1 1 -1 -1 -A -1
X.6 1 -A 1 1 -1 1 -A A -1 1 -1 -1 A -1
X.7 1 A -1 1 -1 1 -A -A 1 -1 -1 -1 A 1
X.8 1 -A -1 1 -1 1 A A 1 -1 -1 -1 -A 1
X.9 2 . . -2 2 2 . . . . -2 2 . .
X.10 2 . . -2 -2 2 . . . . 2 -2 . .
X.11 2 . B . 2 -2 . . B -B . -2 . -B
X.12 2 . -B . 2 -2 . . -B B . -2 . B
X.13 2 . B . -2 -2 . . -B -B . 2 . B
X.14 2 . -B . -2 -2 . . B B . 2 . -B
A = E(4)
= Sqrt(-1) = i
B = -E(8)+E(8)^3
= -Sqrt(2) = -r2
gap> g:=SmallGroup(32,41);;cl:=ConjugacyClasses(g);;L1:=List(cl,Representative);;L2:=List(cl,Centralizer);;L3:=List(L2,IdGroup);;L4:=List(cl,Size);;tbl:= CharacterTable( g );;Display( tbl );
CT4
2 5 3 3 5 4 5 4 3 3 4 5 4 4 4
1a 4a 4b 2a 4c 2b 8a 4d 4e 4f 2c 8b 8c 8d
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 -1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1
X.3 1 1 -1 1 1 1 -1 1 -1 1 1 -1 -1 -1
X.4 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1
X.5 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 1 -1
X.6 1 -1 1 -1 1 1 -1 1 -1 -1 -1 1 -1 1
X.7 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 1
X.8 1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 1 -1
X.9 2 . . 2 -2 2 . . . -2 2 . . .
X.10 2 . . -2 -2 2 . . . 2 -2 . . .
X.11 2 . . 2 . -2 A . . . -2 A -A -A
X.12 2 . . 2 . -2 -A . . . -2 -A A A
X.13 2 . . -2 . -2 A . . . 2 -A -A A
X.14 2 . . -2 . -2 -A . . . 2 A A -A
A = -E(8)+E(8)^3
= -Sqrt(2) = -r2
1個1階元，3個2階元，20個4階元，8個8階元，0個16階元，0個32階元
Group GAP4(32,10) [Q8 : C4]1,3,20,8,0,0,
Group GAP4(32,13) [C8 : C4]1,3,20,8,0,0,
Group GAP4(32,14) [C8 : C4]1,3,20,8,0,0,
Group GAP4(32,41) [C2 x Q16]1,3,20,8,0,0,
GAP4[32,10]=G32_15
http://world.std.com/~jmccarro/math/SmallGroups/g32.15.html
秩2
gap> G:=Group((1,17,3,19)(2,18,4,20)(5,22,7,24)(6,21,8,23)(9,29,11,31)(10,30,12,32)(13,25,15,27)(14,26,16,28), (1,9,2,10)(3,11,4,12)(5,14,6,13)(7,16,8,15)(17,25,18,26)(19,27,20,28)(21,30,22,29)(23,32,24,31), (1,5,2,6)(3,7,4,8)(9,13,10,14)(11,15,12,16)(17,21,18,22)(19,23,20,24)(25,29,26,30)(27,31,28,32), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32));;IdGroup(G);RankPGroup(G);
[ 32, 10 ]
2
G:=Group((1,17,2,18)(3,20,4,19)(5,21,6,22)(7,24,8,23)(9,27,10,28)(11,25,12,26)(13,31,14,32)(15,29,16,30), (1,9,2,10)(3,12,4,11)(5,13,6,14)(7,16,8,15)(17,25,18,26)(19,28,20,27)(21,29,22,30)(23,32,24,31), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,3,2,4)(5,7,6,8)(9,11,10,12)(13,15,14,16)(17,19,18,20)(21,23,22,24)(25,27,26,28)(29,31,30,32), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32));;IdGroup(G);RankPGroup(G);
GAP4[32,13]=G32_18
http://world.std.com/~jmccarro/math/SmallGroups/g32.18.html
秩2
gap> G:=Group((1,17,3,19)(2,18,4,20)(5,22,7,24)(6,21,8,23)(9,29,11,31)(10,30,12,32)(13,25,15,27)(14,26,16,28), (1,9,5,13,2,10,6,14)(3,11,7,15,4,12,8,16)(17,25,21,29,18,26,22,30)(19,27,23,31,20,28,24,32), (1,5,2,6)(3,7,4,8)(9,13,10,14)(11,15,12,16)(17,21,18,22)(19,23,20,24)(25,29,26,30)(27,31,28,32), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32));;IdGroup(G);RankPGroup(G);
[ 32, 13 ]
2
GAP4[32,14]=G32_19
http://world.std.com/~jmccarro/math/SmallGroups/g32.19.html
秩2
gap> G:=Group((1,17,3,19)(2,18,4,20)(5,22,7,24)(6,21,8,23)(9,29,11,31)(10,30,12,32)(13,25,15,27)(14,26,16,28), (1,9,6,14,2,10,5,13)(3,11,8,16,4,12,7,15)(17,25,22,30,18,26,21,29)(19,27,24,32,20,28,23,31), (1,5,2,6)(3,7,4,8)(9,13,10,14)(11,15,12,16)(17,21,18,22)(19,23,20,24)(25,29,26,30)(27,31,28,32), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32));;IdGroup(G);RankPGroup(G);
[ 32, 14 ]
2
GAP4[32,41]=G32_43
http://world.std.com/~jmccarro/math/SmallGroups/g32.43.html
秩3
gap> G:=Group((1,17,2,18)(3,20,4,19)(5,21,6,22)(7,24,8,23)(9,27,10,28)(11,25,12,26)(13,31,14,32)(15,29,16,30), (1,9,2,10)(3,12,4,11)(5,13,6,14)(7,16,8,15)(17,25,18,26)(19,28,20,27)(21,29,22,30)(23,32,24,31), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,3,2,4)(5,7,6,8)(9,11,10,12)(13,15,14,16)(17,19,18,20)(21,23,22,24)(25,27,26,28)(29,31,30,32), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32));;IdGroup(G);RankPGroup(G);
[ 32, 41 ]
3