Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
2*z4.1 = 0
1
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
2*z6.1 = 0
1
true
Mapping from: GrpAb: z4 to GrpAb: z4
Mapping from: GrpAb: z4 to GrpAb: z6
Mapping from: GrpAb: z4 to GrpAb: z6
Mapping from: GrpAb: z6 to GrpAb: z4
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
4*z4.1 = 0
Abelian Group isomorphic to Z/6
Defined on 1 generator
Relations:
6*z6.1 = 0
Abelian Group isomorphic to Z/6
Defined on 1 generator
Relations:
6*z6.1 = 0
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
4*z4.1 = 0
Abelian Group isomorphic to Z/2
Defined on 1 generator in supergroup z6:
$.1 = 3*z6.1
Relations:
2*$.1 = 0
Abelian Group isomorphic to Z/2
Defined on 1 generator in supergroup z4:
$.1 = 2*z4.1
Relations:
2*$.1 = 0
Abelian Group isomorphic to Z/2
Defined on 1 generator in supergroup z4:
$.1 = 2*z4.1
Relations:
2*$.1 = 0
Abelian Group isomorphic to Z/3
Defined on 1 generator in supergroup z6:
$.1 = 2*z6.1
Relations:
3*$.1 = 0
Mapping from: GrpAb: z4 to GrpAb: z4
Mapping from: GrpAb: z4 to GrpAb: z6
Mapping from: GrpAb: z4 to GrpAb: z6
Mapping from: GrpAb: z6 to GrpAb: z4
Abelian Group isomorphic to Z/4
Defined on 1 generator in supergroup z4:
$.1 = z4.1
Relations:
4*$.1 = 0
Abelian Group isomorphic to Z/2
Defined on 1 generator in supergroup z6:
$.1 = 3*z6.1
Relations:
2*$.1 = 0
Abelian Group isomorphic to Z/2
Defined on 1 generator in supergroup z4:
$.1 = 2*z4.1
Relations:
2*$.1 = 0
Abelian Group of order 1
Abelian Group isomorphic to Z/2
Defined on 1 generator in supergroup z4:
$.1 = 2*z4.1
Relations:
2*$.1 = 0
Abelian Group isomorphic to Z/3
Defined on 1 generator in supergroup z6:
$.1 = 2*z6.1
Relations:
3*$.1 = 0
H642:= hom< z6 -> z4 | z6.1 -> 2*z4.1 >;
H642;
AutomorphismGroup(z4);
AutomorphismGroup(z6);
Mapping from: GrpAb: z4 to GrpAb: z4
Mapping from: GrpAb: z4 to GrpAb: z6
Mapping from: GrpAb: z4 to GrpAb: z6
Mapping from: GrpAb: z6 to GrpAb: z4
A group of automorphisms of Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
4*z4.1 = 0
Generators:
Automorphism of Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
4*z4.1 = 0 which maps:
z4.1 |--> 3*z4.1
A group of automorphisms of Abelian Group isomorphic to Z/6
Defined on 1 generator
Relations:
6*z6.1 = 0
Generators:
Automorphism of Abelian Group isomorphic to Z/6
Defined on 1 generator
Relations:
6*z6.1 = 0 which maps:
3*z6.1 |--> 3*z6.1
2*z6.1 |--> 4*z6.1
Mapping from: GrpAb: z4 to GrpAb: z6
Mapping from: GrpAb: z4 to GrpAb: z6
Mapping from: GrpAb: z6 to GrpAb: z4
Mapping from: GrpAb: z6 to GrpAb: z4
Abelian Group isomorphic to Z/4
Defined on 1 generator in supergroup z4:
$.1 = z4.1
Relations:
4*$.1 = 0
Abelian Group isomorphic to Z/2
Defined on 1 generator in supergroup z4:
$.1 = 2*z4.1
Relations:
2*$.1 = 0
Abelian Group isomorphic to Z/6
Defined on 1 generator in supergroup z6:
$.1 = z6.1
Relations:
6*$.1 = 0
Abelian Group isomorphic to Z/3
Defined on 1 generator in supergroup z6:
$.1 = 2*z6.1
Relations:
3*$.1 = 0
Permutation group H1 acting on a set of cardinality 10
(3, 5)(4, 6)
(1, 3, 5)
(2, 4, 6)
(3, 4)
Permutation group H2 acting on a set of cardinality 10
(1, 5, 3)
(2, 4, 6)
Conjugacy classes of subgroups
------------------------------
[1] Order 1 Length 1
Permutation group acting on a set of cardinality 10
Order = 1
Id($)
[2] Order 360 Length 1
Permutation group acting on a set of cardinality 10
Order = 360 = 2^3 * 3^2 * 5
(2, 6)(3, 5)
(1, 5)(2, 4, 3, 6)
[3] Order 720 Length 1
Permutation group acting on a set of cardinality 10
Order = 720 = 2^4 * 3^2 * 5
(1, 4, 6, 2, 5)
(3, 5)
Conjugacy classes of subgroups
------------------------------
[1] Order 1 Length 1
Permutation group acting on a set of cardinality 10
Order = 1
[2] Order 3 Length 1
Permutation group acting on a set of cardinality 10
Order = 3
(1, 5, 3)(2, 6, 4)
[3] Order 3 Length 1
Permutation group acting on a set of cardinality 10
Order = 3
(2, 4, 6)
[4] Order 3 Length 1
Permutation group acting on a set of cardinality 10
Order = 3
(1, 5, 3)
[5] Order 3 Length 1
Permutation group acting on a set of cardinality 10
Order = 3
(1, 5, 3)(2, 4, 6)
[6] Order 9 Length 1
Permutation group acting on a set of cardinality 10
Order = 9 = 3^2
(1, 5, 3)
(2, 4, 6)
720
9
80
Mapping from: Cartesian Product<{ 1 .. 80 }, GrpPerm: H1, Degree 10, Order 2^4 *
3^2 * 5> to { 1 .. 80 }
$1 $2 $3 $4 -$2 -$3
1. 2 1 1 3 1 1
2. 1 2 2 4 2 2
3. 5 6 7 1 14 16
4. 8 9 10 2 21 23
5. 3 11 12 13 10 9
6. 10 14 8 15 3 19
7. 9 8 16 17 18 3
8. 4 18 19 20 7 6
9. 7 21 5 22 4 12
10. 6 5 23 24 11 4
Permutation group H1 acting on a set of cardinality 14
(3, 5, 4, 6, 7)
(1, 3, 5, 7)
(2, 4, 6, 7)
(3, 4, 5)
(2, 4)
(3, 11)
(3, 6)
Permutation group H2 acting on a set of cardinality 14
(1, 5, 6, 3, 7)
(4, 6, 7)
(3, 11)
Conjugacy classes of subgroups
------------------------------
[1] Order 1 Length 1
Permutation group acting on a set of cardinality 14
Order = 1
Id($)
[2] Order 20160 Length 1
Permutation group acting on a set of cardinality 14
Order = 20160 = 2^6 * 3^2 * 5 * 7
(1, 3)(2, 11)(4, 7)(5, 6)
(2, 3, 7, 11, 5)
[3] Order 40320 Length 1
Permutation group acting on a set of cardinality 14
Order = 40320 = 2^7 * 3^2 * 5 * 7
(1, 2)
(1, 4, 3)(2, 6, 11, 5, 7)
Conjugacy classes of subgroups
------------------------------
[1] Order 1 Length 1
Permutation group acting on a set of cardinality 14
Order = 1
Id($)
[2] Order 2520 Length 1
Permutation group acting on a set of cardinality 14
Order = 2520 = 2^3 * 3^2 * 5 * 7
(1, 3, 11)
(1, 4, 5)(3, 7)(6, 11)
[3] Order 5040 Length 1
Permutation group acting on a set of cardinality 14
Order = 5040 = 2^4 * 3^2 * 5 * 7
(1, 3)
(1, 4, 5, 6)(3, 7, 11)
40320
5040
8
Mapping from: Cartesian Product<{ 1 .. 8 }, GrpPerm: H1, Degree 14, Order 2^7 *
3^2 * 5 * 7> to { 1 .. 8 }
$1 $2 $3 $4 $5 $6 $7 -$1 -$2 -$3 -$4
1. 1 1 2 1 2 1 1 1 1 3 1
2. 4 2 4 5 1 2 2 5 2 1 6
3. 6 7 1 3 3 3 3 4 5 4 3
4. 3 4 3 4 4 4 6 2 4 2 4
5. 2 3 5 6 5 5 5 6 6 5 2
6. 5 5 6 2 6 8 4 3 7 6 5
7. 7 6 7 7 7 7 7 7 3 7 7
8. 8 8 8 8 8 6 8 8 8 8 8
Homomorphism of GrpPerm: H2, Degree 14, Order 2^4 * 3^2 * 5 * 7 into GrpPerm:
H1, Degree 14, Order 2^7 * 3^2 * 5 * 7 induced by
(1, 5, 6, 3, 7) |--> (3, 5, 4, 6, 7)
Homomorphism of GrpPerm: H1, Degree 14, Order 2^7 * 3^2 * 5 * 7 into GrpPerm:
H2, Degree 14, Order 2^4 * 3^2 * 5 * 7 induced by
(3, 5, 4, 6, 7) |--> (1, 5, 6, 3, 7)
Permutation group H1 acting on a set of cardinality 13
(3, 5)(4, 6)
(1, 3, 5)
(2, 4, 6)
(3, 4)
Permutation group H2 acting on a set of cardinality 13
(1, 5)(3, 4)
(1, 2, 3, 5)(4, 6)
Conjugacy classes of subgroups
------------------------------
[1] Order 1 Length 1
Permutation group acting on a set of cardinality 13
Order = 1
Id($)
[2] Order 360 Length 1
Permutation group acting on a set of cardinality 13
Order = 360 = 2^3 * 3^2 * 5
(2, 3)(4, 6)
(1, 3, 4, 6)(2, 5)
[3] Order 720 Length 1
Permutation group acting on a set of cardinality 13
Order = 720 = 2^4 * 3^2 * 5
(1, 4, 6, 2, 5)
(2, 3)
Permutation group acting on a set of cardinality 2
Id($)
Id($)
Id($)
(1, 2)
小群表：
Size Construction Notes
1 SymmetricGroup(1) Trivial
2 SymmetricGroup(2) Also CyclicPermutationGroup(2)
3 CyclicPermutationGroup(3) Prime order
4 CyclicPermutationGroup(4) Cyclic
4 KleinFourGroup() Abelian, non-cyclic
5 CyclicPermutationGroup(5) Prime order
6 CyclicPermutationGroup(6) Cyclic
6 SymmetricGroup(3) Non-abelian, also DihedralGroup(3)
7 CyclicPermutationGroup(7) Prime order
8 CyclicPermutationGroup(8) Cyclic
8 D1=CyclicPermutationGroup(4)
D2=CyclicPermutationGroup(2)
G=direct product permgroups([D1,D2])
Abelian, non-cyclic
8 D1=CyclicPermutationGroup(2)
D2=CyclicPermutationGroup(2)
D3=CyclicPermutationGroup(2)
G=direct product permgroups([D1,D2,D3])
Abelian, non-cyclic
8 DihedralGroup(4) Non-abelian
8 PermutationGroup(["(1,2,5,6)(3,4,7,8)",
"(1,3,5,7)(2,8,6,4)" ])
Quaternions
The two generators are I and J
9 CyclicPermutationGroup(9) Cyclic
9 D1=CyclicPermutationGroup(3)
D2=CyclicPermutationGroup(3)
G=direct product permgroups([D1,D2])
Abelian, non-cyclic
10 CyclicPermutationGroup(10) Cyclic
10 DihedralGroup(5) Non-abelian
11 CyclicPermutationGroup(11) Prime order
12 CyclicPermutationGroup(12) Cyclic
12 D1=CyclicPermutationGroup(6)
D2=CyclicPermutationGroup(2)
G=direct product permgroups([D1,D2])
Abelian, non-cyclic
12 DihedralGroup(6) Non-abelian
12 AlternatingGroup(4) Non-abelian, symmetries of tetrahedron
12 PermutationGroup(["(1,2,3)(4,6)(5,7)",
"(1,2)(4,5,6,7)"])
Non-abelian
Semi-direct product Z3 o Z4
13 CyclicPermutationGroup(13) Prime order
14 CyclicPermutationGroup(14) Cyclic
14 DihedralGroup(7) Non-abelian
15 CyclicPermutationGroup(15) Cyclic
Conjugacy classes of subgroups
------------------------------
[1] Order 1 Length 1
Permutation group acting on a set of cardinality 4
Order = 1
[2] Order 4 Length 1
Permutation group acting on a set of cardinality 4
Order = 4 = 2^2
(1, 4)(2, 3)
(1, 3)(2, 4)
[3] Order 12 Length 1
Permutation group acting on a set of cardinality 4
Order = 12 = 2^2 * 3
(2, 3, 4)
(1, 4)(2, 3)
(1, 3)(2, 4)
[4] Order 24 Length 1
Permutation group acting on a set of cardinality 4
Order = 24 = 2^3 * 3
(3, 4)
(2, 3, 4)
(1, 4)(2, 3)
(1, 3)(2, 4)
Permutation group ss acting on a set of cardinality 4
(2, 3, 4)
(1, 4)(2, 3)
(1, 3)(2, 4)
12
2
Permutation group acting on a set of cardinality 2
(1, 2)
(1, 2)
Mapping from: Cartesian Product<{ 1 .. 2 }, GrpPerm: s4, Degree 4, Order 2^3 *
3> to { 1 .. 2 }
$1 $2 -$1
1. 2 2 2
2. 1 1 1
Symmetric group s3 acting on a set of cardinality 3
Order = 6 = 2 * 3
Symmetric group quotientgroup acting on a set of cardinality 2
Order = 2
(1, 2)
(1, 2)
Symmetric group s5 acting on a set of cardinality 5
Order = 120 = 2^3 * 3 * 5
Permutation group fatherquotientgroup acting on a set of cardinality 5
(1, 2)
(1, 2)
(1, 2)
Conjugacy classes of subgroups
------------------------------
[1] Order 1 Length 1
Permutation group acting on a set of cardinality 5
Order = 1
[2] Order 2 Length 1
Permutation group acting on a set of cardinality 5
Order = 2
(1, 2)
Conjugacy classes of subgroups
------------------------------
[1] Order 1 Length 1
Permutation group acting on a set of cardinality 5
Order = 1
[2] Order 2 Length 1
Permutation group acting on a set of cardinality 5
Order = 2
(1, 2)
Homomorphism of GrpPerm: s4, Degree 4, Order 2^3 * 3 into GrpPerm:
fatherquotientgroup, Degree 5, Order 2 induced by
(1, 2, 3, 4) |--> (1, 2)
Mapping from: GrpAb: z4 to GrpAb: z6
Mapping from: GrpAb: z4 to GrpAb: z6
Mapping from: GrpAb: z6 to GrpAb: z4
Mapping from: GrpAb: z6 to GrpAb: z4
Abelian Group of order 1
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
2*$.1 = 0
Abelian Group of order 1
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
2*$.1 = 0
Conjugacy classes of subgroups
------------------------------
[1] Order 3 Length 1
Abelian Group isomorphic to Z/3
Defined on 1 generator in supergroup z6:
$.1 = 2*z6.1
Relations:
3*$.1 = 0
[2] Order 2 Length 1
Abelian Group isomorphic to Z/2
Defined on 1 generator in supergroup z6:
$.1 = 3*z6.1
Relations:
2*$.1 = 0
Conjugacy classes of subgroups
------------------------------
[1] Order 2 Length 1
Abelian Group isomorphic to Z/2
Defined on 1 generator in supergroup z4:
$.1 = 2*z4.1
Relations:
2*$.1 = 0
>> IsIsomorphic(s6, m4sub) ;
^
Runtime error in 'IsIsomorphic': Bad argument types
Argument types given: GrpAb, SeqEnum[Rec]
>> IsIsomorphic(Q62, m4sub) ;
^
User error: Identifier 'Q62' has not been declared or assigned
