# 73.52 ConjugationCat1

`ConjugationCat1( R, S )`

When S is a normal subgroup of a group R form the semi-direct product G = R semidirect S to R and take this as the source, with R as the range. The tail and head homomorphisms are defined by t(r,s) = r(partial s), ; h(r,s) = r. In the example `h20` is the range, rather than the source.

```    gap> c5 := Subgroup( h20, [(1,2,3,4,5)] );;
gap> c5.name := "c5";;
gap> CC := ConjugationCat1( h20, c5 );
cat1-group [Perm(h20 ```
`|`
```X c5) ==> h20]
gap> Cat1Print( CC );

cat1-group [Perm(h20 ```
`|`
```X c5) ==> h20] :-
: source group has generators:
[ ( 6, 7, 8, 9,10), ( 2, 3, 5, 4)( 7, 8,10, 9), (1,2,3,4,5) ]
:  range group has generators:
[ (1,2,3,4,5), (2,3,5,4) ]
: tail homomorphism maps source generators to:
[ ( 1, 2, 3, 4, 5), ( 2, 3, 5, 4), () ]
: head homomorphism maps source generators to:
[ ( 1, 2, 3, 4, 5), ( 2, 3, 5, 4), ( 1, 2, 3, 4, 5) ]
: range embedding maps range generators to:
[ ( 6, 7, 8, 9,10), ( 2, 3, 5, 4)( 7, 8,10, 9) ]
: kernel has generators:
[ (1,2,3,4,5) ]
: boundary homomorphism maps generators of kernel to:
[ ( 1, 2, 3, 4, 5) ]
: kernel embedding maps generators of kernel to:
[ ( 1, 2, 3, 4, 5) ]
: associated crossed module is Crossed module [c5->h20]

gap> ct := CC.tail;;
gap> ch := CC.head;;
gap> CG := CC.source;;
gap> genCG := CG.generators;;
gap> x := genCG[2] * genCG[3];
( 1, 2, 4, 3 )( 7, 8,10, 9 )
gap> tx := Image( ct, x );
( 2, 3, 5, 4)
gap> hx := Image( ch, x );
( 1, 2, 4, 3)

gap> RecFields( CC );
[ "source", "range", "tail", "head", "embedRange", "kernel",
"boundary", "embedKernel", "isDomain", "operations", "isCat1",
"name", "xmod" ]```

GAP 3.4.4
April 1997