# Questions tagged [gap]

GAP (Groups, Algorithms and Programming) is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. It provides a programming language, a library of thousands of functions implementing algebraic algorithms, and large data libraries of algebraic objects. Please note that GAP Forum or GAP Support may be more suitable places for questions about GAP: see http://www.gap-system.org/Contacts/Forum/forum.html

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### Which resources are available to self-study GAP?

Background: This year I'll do another Group Theory course ( Open University M336 ). In the past I have used Mathematica's AbstractAlgebra package but (although visually appealing ) this is no longer ...
7k views

### How to find all subgroups of a group in GAP

With the next group: ...
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### How do Gap generate the elements in permutation groups?

I understand that permutationgroups in Gap are represented by generators, which seems to be far more efficient than groups represented by all it's elements, but how could then for example ...
801 views

### finding code of a function in GAP packages [closed]

How can I find the codes related to a function in GAP? When I use "??" in front of the name of function there is no help, so I want to find the code of function among packages.
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### Proof that the Rubik’s Cube group is 2-generated

Singmaster (1981) writes, on page 32 of his Notes on Rubik’s Magic Cube: Frank Barnes observes that the group of the cube is generated by two moves: \begin{align*} \alpha &= L^2 B R D^{-1} L^{...
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### How to define own group action in GAP?

I am beginner in GAP. I have a group and a set. I wish to define an action the group on the set in my own way and wish to calculate its orbits and stabilizers. Is it possible? What is process?
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### Smart way to solve octics like $x^8+5992704x-304129728=0$

Question: How to quickly find explicit formula for roots of polynomials (solvable by radicals) like $f(x)=x^8+5992704x-304129728$? My current approach is just a brute force - not very fast and ...
2k views

### Easy way to get real irreducible characters (reps) from complex irreducible characters?

For plenty of groups, the real irreducible characters/representations aren't the same as the complex irreducible representations. I really enjoy James Montaldi's summary of real representations, for ...
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### GAP: how to work with elements of the group ring of the symmetric group $\mathbb{C}[S_k]$?

In GAP, working with elements of the symmetric group $S_k$ is straightforward. E.g. one can write (1,2)*(2,3); to obtain (1,3,2). Is there a similar ...
458 views

### Conjugacy classes and centralizers of a SmallGroup

What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)? Would someone be willing to provide the complete lists of conjugacy classes and centralizers of SmallGroup(...
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### Semidirect Products with GAP

I'm wondering how to specify to GAP which homomorphism to use when constructing a semidirect product. I'm trying to have it construct $\left(\mathbb{Z}_p\times\mathbb{Z}_p\right)\rtimes_\varphi S_3$. ...
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### GAP semidirect product algorithm

Can anybody guide me towards, or possibly even explain here, the algorithm that GAP uses to compute the semidirectproduct of two permutation groups which outputs another permutation group? EXAMPLE: ...
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### Factoring a polynomial in $\mathbb Q[t][x]$ over an extension of $\mathbb Q[t]$

I have found the following unit-distance embedding of the cubic symmetric graph $F_{42}A$ with $D_7$ symmetry. (Gerbracht's embedding on the MathWorld page for cubic symmetric graphs is only of $C_7$ ...
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### Differences between GAP and PARI/GP?

I was looking for free alternatives to Magma to perform number theory computations, such as modular, factorization, etc. and I've found GAP and Pari. What's the difference between them? Any other ...
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### Use GAP to define finite Clifford group

I am studying the computer algebra system GAP to do some calculations about Clifford group, which is defined (cf. Lawson and Michelsohn, Spin Geometry, Princeton 1989) as followings Definition. Let's ...
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### Computing Conjugacy Classes of Subgroups in GAP

GAP has the command ConjugacyClassesSubgroups which gives a list of the conjugacy classes of a finite group $G$. Is there a way I can specify further what types of subgroups GAP reports? For ...
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### A normal intermediate subgroup in $B_3$ lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups between $H$ and $G$. An intermediate subgroup $H \subset K \subset G$ is a ...
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### How can I calculate $gnu(17^3\times 2)=gnu(9826)$ with GAP?

I tried to calculate the number of groups of order $17^3\times 2=9826$ with GAP. Neither the NrSmallGroups-Command nor the ConstructAllGroups-Command work with GAP. The latter one because of the ...
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### Commutator of two elements in group algebra $\mathbb F_{5}D_{30}.$

I want to understand how to find the commutator of two elements in the group algebra $\mathbb{F}_{5}D_{30}$ using GAP. Additionally, I would like to determine the nilpotency class of the nilpotent ...
358 views

### Can graphs in GAP be obtained as a visual output

In GAP when we draw a cayley graph of a group using "CayleyGraph" command we get a list. Is there a way to visualize that cayley graph like a figure? Or else is there any other software that has the ...
344 views

### Subgroup lattice of UT(3,3)

I need to draw a subgroup lattice of $UT(3,\mathbb{Z_{3}})$, the group of upper triangular matrices with diagonal one. How to do it? And whether there is somebody ready image? (I know that it can ...
678 views

### Matrices' outputs in GAP

Does anyone know whether is a function in GAP via I would be able to change elements of matrices over GF(2) to the form 0 and 1. need to say, in GAP, outputs for elements of matrices defined over GF(2)...
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### Websites/Software for Group computation

Anyone knows a website or software that helps to do computations in a group? For example, by inputting generators and relations in the group, can we tell when two particular elements in the group ...
1 vote
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### How does one define a representation in GAP?

It seems that the reference manual for GAP never mentions how to construct a representation by sending generators to specified matrices. Yet I cannot help but feel that such a basic task must be ...
1 vote
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### How to make GAP understand that a homomorphism is a representation?

In a previous question I asked how one constructs an explicit representation in GAP. As a concrete example, I want to consider the $5$-dimensional $\mathbb{F}_3$-representation $\rho$ of $C_8$ sending ...
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### Sums of products of average character values on cosets

Consider a finite group $G$, a subgroup $H\leq G$, and a transversal $G/H = \{t_1H, t_2H,\ldots,t_rH\}$. Given three characters $\chi_1,\chi_2$ and $\chi_3$ of $G$, I'd like to compute:  \sum_{i}^r ...
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### Graphical interface in GAP

Is there any graphical interface in GAP? Something like RStudio for R or WxMaxima for Maxima. I'm using GAP under a Linux system. Thanks
For a finite group $G$, let $\mu(G)$ be the least positive integer $n$ such that $G$ is embedded as a subgroup of the symmetric group on $n$ points. In other words, $\mu(G)$ is the minimal permutation ...