Examples of permutation groups are routinely constructed by taking one or more standard groups and applying some extension procedure to construct a group having the given groups as subgroups or quotient groups. In the first subsection we describe functions which construct some well-known groups and in the following subsection we give functions for constructing direct and wreath products.
A number of functions are provided which construct various standard groups. The effect of these functions is to construct the group on some standard set of generating permutations.
Construct the abelian group defined by the sequence Q = [n_1, ..., n_r] of positive integers. The function constructs the direct product of cyclic groups Z(n_1) x Z(n_2) x ... x Z(n_r).
Construct the alternating group of degree n on generators (3, 4, ..., n) and (1, 2, 3), if n is odd, or (1, 2)(3, 4, ..., n) and (1, 2, 3), if n is even.
Construct the cyclic group of order n with generator (1, 2, ..., n).
Construct the dihedral group of degree n and order 2 * n on generators (1, 2, ..., n) and (1, n)(2, n - 1) ... .
Construct the symmetric group of degree n on generators (1, 2, ..., n) and (1, 2).
Given a small prime p and a small positive integer n, construct an extra-special group G of order p^(2n + 1) in the category GrpPerm. The isomorphism type of G can be selected using the parameter Type.
Type: MonStgElt Default: "+"Possible values for this parameter are "+" (default) and "-".
If Type is set to "+", the function returns for p = 2 the central product of n copies of the dihedral group of order 8, and for p > 2 it returns the unique extra-special group of order p^(2n + 1) and exponent p.
If Type is set to "-", the function returns for p = 2 the central product of a quaternion group of order 8 and n - 1 copies of the dihedral group of order 8, and for p > 2 it returns the unique extra-special group of order p^(2n + 1) and exponent p^2.
> A := AbelianGroup(GrpPerm, [2, 2, 4] );
> A;
Permutation group A acting on a set of cardinality 8
Order = 16 = 2^4
(1, 2)
(3, 4)
(5, 6, 7, 8)
> A12 := AlternatingGroup(GrpPerm, 12);
> A12;
Permutation group A12 acting on a set of cardinality 12
Order = 239500800 = 2^9 * 3^5 * 5^2 * 7 * 11
(1, 2)(3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
(1, 2, 3)
> Z24 := CyclicGroup(GrpPerm, 24);
> Z24;
Permutation group Z24 on a set of cardinality 24
Order = 24 = 2^3 * 3
(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24)
> D12 := DihedralGroup(GrpPerm, 12);
> D12;
Permutation group D12 acting on a set of cardinality 12
Order = 24 = 2^3 * 3
(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
(1, 12)(2, 11)(3, 10)(4, 9)(5, 8)(6, 7)
> S8 := SymmetricGroup(GrpPerm, 8); > S8; Symmetric group S8 acting on a set of cardinality 8 Order = 40320 = 2^7 * 3^2 * 5 * 7
Given two permutation groups G and H, construct the direct product D of G and H as an intransitive group having degree equal to the sum of the degrees of G and H. In addition, the sequences I of inclusions and P of projections are returned, satisfying I[i]: K_i -> D(K_i) and P[i]: D -> K_i (where K_1 = G, K_2 = H and D(K) is the group K represented naturally as a subgroup of D).
Given a sequence Q of n permutation groups, construct the direct product Q[1] x Q[2] x ... x Q[n] as an intransitive group of degree equal to the sum of the degrees of the groups Q[i], (i = 1, ..., n). In addition, the sequences I of inclusion and P of projections are returned, satisfying I[i]: Q[i] -> D(Q[i]) and P[i]: D -> Q[i] (where D(K) is the group K represented naturally as a subgroup of D).
Given permutation groups G and H, construct the wreath product G wreath H of G and H, where G wreath H has product action.
Given a sequence Q of n permutation groups, construct the iterated wreath product T = ( ... (Q[1] wreath Q[2]) wreath ... wreath Q[n]), where T has product action.
Given permutation groups G and H, construct the wreath product W = G wreath H of G and H, where G wreath H has imprimitive action. The function also returns the sequence of Degree(H) inclusions of G into W, the inclusion of H into W and the projection of W onto H.
Given a sequence Q of n permutation groups, construct the iterated wreath product W = ( ... (Q[1] wreath Q[2]) wreath ... wreath Q[n]), where W has imprimitive action.
> G := SymmetricGroup(GrpPerm, 4);
> H := DihedralGroup(GrpPerm, 3);
> D := DirectProduct(G, H);
> D;
Permutation group D acting on a set of cardinality 7
Order = 144 = 2^4 * 3^2
(1, 2, 3, 4)
(1, 2)
(5, 6, 7)
(5, 6)
> T := PrimitiveWreathProduct(G, H);
> T;
Permutation group T acting on a set of cardinality 64
Order = 82944 = 2^10 * 3^4
(2, 5, 17)(3, 9, 33)(4, 13, 49)(6, 21, 18)(7, 25, 34)(8, 29, 50)
(10, 37, 19) (11, 41, 35)(12, 45, 51) (14, 53, 20)(15, 57, 36)
(16, 61, 52)(23, 26, 38) (24, 30, 54)(27, 42, 39)(28, 46, 55)
(31, 58, 40) (32, 62, 56)(44, 47, 59)(48, 63, 60)
(2, 5)(3, 9)(4, 13)(7, 10)(8, 14)(12, 15)(18, 21)(19 , 25)(20, 29)
(23, 26)(24, 30)(28, 31)(34, 37)(35 , 41)(36, 45)(39, 42)(40, 46)
(44, 47)(50, 53)(51 , 57)(52, 61)(55, 58)(56, 62)(60, 63)
(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)(33, 34, 35, 36)
(37, 38, 39, 40)(41, 42, 43, 44)(45, 46, 47, 48)(49, 50, 51, 52)
(53, 54, 55, 56)(57, 58, 59, 60)(61, 62, 63, 64)
(1, 2)(5, 6)(9, 10)(13, 14)(17, 18)(21, 22)(25, 26)( 29, 30)(33, 34)
(37, 38)(41, 42)(45, 46)(49, 50)( 53, 54)(57, 58)(61, 62)
> W := WreathProduct(G, H);
> W;
Permutation group W acting on a set of cardinality 12
Order = 82944 = 2^10 * 3^4
(1, 5, 9)(2, 6, 10)(3, 7, 11)(4, 8, 12)
(1, 5)(2, 6)(3, 7)(4, 8)
(1, 2, 3, 4)
(1, 2)