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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 =
[n1, ..., nr] of positive integers. The function constructs
the direct product of cyclic groups
Z(n1) x Z(n2) x ... x Z(nr).
AlternatingGroup(n) : RngIntElt -> GrpPerm
Alt(n) : RngIntElt -> GrpPerm
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.
CyclicGroup(n) : RngIntElt -> GrpPerm
Construct the cyclic group of order n with generator
(1, 2, ..., n).
DihedralGroup(n) : RngIntElt -> GrpPerm
Construct the dihedral group of degree n and order 2 * n on
generators (1, 2, ..., n) and (1, n)(2, n - 1) ... .
SymmetricGroup(GrpPerm, n) : Cat, RngIntElt -> GrpPerm
Sym(n) : RngIntElt -> GrpPerm
SymmetricGroup(n) : RngIntElt -> GrpPerm
Construct the symmetric group of degree n on generators
(1, 2, ..., n) and (1, 2).
ExtraSpecialGroup(p, n : parameters) : RngIntElt, RngIntElt -> GrpPerm
Given a small prime p and a small positive integer n, construct an
extra-special group G of order p2n + 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 p2n + 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 p2n + 1 and exponent p2.
Full: RngIntElt Default: false
Given a sequence L of positive integers, compute the Young subgroup
parameterized by L, i.e., the direct product of the
symmetric groups on Li points.
If the optional parameter Full is given, construct the group as a subgroup
of the symmetric group on Full elements.
- (1)
- The abelian group Z2 x Z2 x Z4:
> 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)
- (2)
- The alternating group of degree 12:
> 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)
- (3)
- The cyclic group Z24:
> 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)
- (4)
- The dihedral group of order 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)
- (5)
- The symmetric group of degree 8:
> 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]: Ki -> D(Ki) and P[i]: D -> Ki (where K1
= G, K2 = 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.
Given a block system B of some permutation group G,
compute the wreath-product corresponding to B.
Compute the smallest wreath product W to the block system B of G such that G
⊆W. Also return the complement as a subgroup of W. The third
parameter is a subgroup which is isomorphic to the action within a block.
We define G to be the symmetric group of degree 4 and
H to be the dihedral group of order 8. We then proceed to form the
direct, primitive-wreath and wreath products of G and H.
> 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)
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