### Re: [GAP Forum] semidirect products

Dear Forum, How to construct a group semidirect product of $Z_3$ and $Z_9$ where $Z_i$ is a cyclic group of order $i$. I nead the permutation representation of this group. Best, Moshtagh ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] semidirect products

Moshtagh: The presentation for the group Z_9 semi Z_3 is a^9=b^3=a^b*a^-4=1; More generally this class of groups [ C_(p^2}] Semi C_p is a^(p^2}=b^p=a^b*a^(-p-1) =1 Is this sufficient or did you need a permutation representation? Walter Becker Date: Sat, 9 Jun 2012 15:50:19 +0430 From: hs.mosht...@gmail.com To: fo...@gap-system.org Subject: Re: [GAP Forum] semidirect products Dear Forum, How to construct a group semidirect product of $Z_3$ and $Z_9$ where $Z_i$ is a cyclic group of order $i$. I nead the permutation representation of this group. Best, Moshtagh ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] semidirect products

From: w_bec...@hotmail.com To: hs.mosht...@gmail.com Subject: RE: [GAP Forum] semidirect products Date: Sat, 9 Jun 2012 10:00:19 -0800 Moshtagh: Sorry did not see the permutation part here it is Note most of the groups of low order can be found in older group theory literature in early 1900's namely G. A.Miller and others --groups of various degrees were done up to 12 at least. order 27 case: (r,s,t,u,v,w,x,y,z), (s,v,y)(t,z,w) Is this better. Date: Sat, 9 Jun 2012 15:50:19 +0430 From: hs.mosht...@gmail.com To: fo...@gap-system.org Subject: Re: [GAP Forum] semidirect products Dear Forum, How to construct a group semidirect product of $Z_3$ and $Z_9$ where $Z_i$ is a cyclic group of order $i$. I nead the permutation representation of this group. Best, Moshtagh ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] Semidirect products

Dear GAP-Forum, On Oct 4, 2011, at 10/4/11 8:57, Sandeep Murthy wrote: is there a quick way to directly access the factors of a semidirect product group? I have constructed a semidirect product G = N \rtimes_\theta P According to the manual, section 47.2 (Semidirect product): Embedding(G,1) returns the embedding P-G, Embedding(G,2) that of N. The subgroups of G you want then can be obtained as Image of these maps. For example: gap G:=SemidirectProduct(GL(3,2),GF(2)^3); matrix group of size 1344 with 3 generators gap hom1:=Embedding(G,1); CompositionMapping( [ (5,7)(6,8), (2,3,5)(4,7,6) ] - [ an immutable 4x4 matrix over GF2, an immutable 4x4 matrix over GF2 ], action isomorphism ) gap Pimg:=Image(hom1); matrix group of size 168 with 2 generators gap Size(Pimg); 168 gap hom2:=Embedding(G,2); MappingByFunction( ( GF(2)^3 ), matrix group with 3 generators, function( v ) ... end, function( a ) ... end ) gap Nimg:=Image(hom2); matrix group of size 8 with 3 generators gap Size(Nimg); 8 Regards, Alexander Hulpke -- Colorado State University, Department of Mathematics, Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA email: hul...@math.colostate.edu, Phone: ++1-970-4914288 http://www.math.colostate.edu/~hulpke ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] semidirect products

Dear GAP-forum, Dear Victor Bovdi, Let A and B be finite 2-groups. I am interested in the isomorphism classes of the semidirect (or/and central) products of these groups. Do you know about a package which can aid this sort of calculation ? There is no function that does these calculations directly -- you would have to go step-by-step in determining - Calculate Aut(N) (the package autpgrp will help) - Classify all homomorphisms from G to Aut(N) (there will be a function `AllHomomorphismClasses' in GAP4 and i can send you code for this privately if you are interested). - Form the semidirect products using `SemidirectProduct' - Test for isomorphism (the `anupq' package can help here, as the extensions are p-groups) Best wishes, Alexander Hulpke (with help from Bettina Eick) -- Colorado State University, Department of Mathematics, Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA email: hul...@math.colostate.edu, Phone: ++1-970-4914288 http://www.math.colostate.edu/~hulpke ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum

### Re: [GAP Forum] semidirect products

Dear Alexander and Bettina, Thank you very much. Best regards, Victor Bovdi Dear GAP-forum, Dear Victor Bovdi, Let A and B be finite 2-groups. I am interested in the isomorphism classes of the semidirect (or/and central) products of these groups. Do you know about a package which can aid this sort of calculation ? There is no function that does these calculations directly -- you would have to go step-by-step in determining - Calculate Aut(N) (the package autpgrp will help) - Classify all homomorphisms from G to Aut(N) (there will be a function `AllHomomorphismClasses' in GAP4 and i can send you code for this privately if you are interested). - Form the semidirect products using `SemidirectProduct' - Test for isomorphism (the `anupq' package can help here, as the extensions are p-groups) Best wishes, Alexander Hulpke (with help from Bettina Eick) -- Colorado State University, Department of Mathematics, Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA email: hul...@math.colostate.edu, Phone: ++1-970-4914288 http://www.math.colostate.edu/~hulpke ___ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum