# A5 in A6

From Groupprops

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) alternating group:A5 and the group is (up to isomorphism) alternating group:A6 (see subgroup structure of alternating group:A6).VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

This article describes a subgroup in a group , where is alternating group:A6, the alternating group on the set , and is the subgroup of comprising those permutations that fix . can be identified with alternating group:A5 via its action on .

has five *other* conjugate subgroups:

- is the subgroup of fixing
- is the subgroup of fixing
- is the subgroup of fixing
- is the subgroup of fixing
- is the subgroup of fixing

## Arithmetic functions

Function | Value | Explanation | Comment |
---|---|---|---|

order of the group | 360 | ||

order of the subgroup | 60 | ||

index of the subgroup | 6 | ||

size of conjugacy class | 6 | ||

number of conjugacy classes in automorphism class | 2 |

## GAP implementation

### Construction of group-subgroup pair

The group-subgroup pair can be constructed using the AlternatingGroup function:

`G := AlternatingGroup(6); H := AlternatingGroup(5);`