# Endomorphism structure of M16

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$\langle a,x \mid a^8 = x^2 = e, xax = a^5 \rangle$

where $e$ denotes the identity element.

## Summary of information

Construct Value Order Second part of GAP ID (if group) Comment
endomorphism monoid  ?  ? --
automorphism group direct product of D8 and Z2 16 11
inner automorphism group Klein four-group 4 2
outer automorphism group Klein four-group 4 2
group of class-preserving automorphisms Klein four-group 4 2 same as inner automorphism group
group of IA-automorphisms Klein four-group 4 2 same as inner automorphism group
quotient of class-preserving automorphism group by inner automorphism group trivial group 1 1
quotient of IA-automorphism group by inner automorphism group trivial group 1 1
group of center-fixing automorphisms dihedral group:D8 8 3
extended automorphism group direct product of D8 and V4 32 46
quasiautomorphism group  ?  ?  ?
1-automorphism group
holomorph
inner holomorph

## Description of automorphism group

### Inner automorphisms

For the actual description of automorphisms, we use the left action convention, so conjugation by $g$ is the map $h \mapsto ghg^{-1}$. With the right action convention, what we call conjugation by $g$ becomes conjugation by $g^{-1}$. However, the data in the table below is the same for both left and right action conventions because, since every square is in the center, every element and its inverse are in the same coset of the center.

For every inner automorphisms, there are four candidate elements that give rise to that inner automorphism via the action by conjugation. These elements together form a single coset of the center, which is the cyclic subgroup of order four generated by $a^2$.

The columns below have been arranged so that elements of a conjugacy class are in adjacent columns to each other. From this, you can notice that the inner automorphisms permute elements within each conjugacy class.

Candidate elements by which conjugation describes the inner automorphism Image of $e$ Image of $a^4$ Image of $a^2$ Image of $a^6$ Image of $a$ Image of $a^5$ Image of $a^3$ Image of $a^7$ Image of $ax$ Image of $a^5x$ Image of $a^3x$ Image of $a^7x$ Image of $x$ Image of $a^4x$ Image of $a^2x$ Image of $a^6x$
$\{ e,a^2,a^4,a^6 \}$ $e$ $a^4$ $a^2$ $a^6$ $a$ $a^5$ $a^3$ $a^7$ $ax$ $a^5x$ $a^3x$ $a^7x$ $x$ $a^4x$ $a^2x$ $a^6x$
$\{ a,a^3,a^5,a^7 \}$ $e$ $a^4$ $a^2$ $a^6$ $a$ $a^5$ $a^3$ $a^7$ $a^5x$ $ax$ $a^7x$ $a^3x$ $a^4x$ $x$ $a^6x$ $a^2x$
$\{ ax, a^3x, a^5x, a^7x \}$ $e$ $a^4$ $a^2$ $a^6$ $a^5$ $a$ $a^7$ $a^3$ $ax$ $a^5x$ $a^3x$ $a^7x$ $a^4x$ $x$ $a^6x$ $a^2x$
$\{ x, a^2x, a^4x, a^6x \}$ $e$ $a^4$ $a^2$ $a^6$ $a^5$ $a$ $a^7$ $a^3$ $a^5x$ $ax$ $a^7x$ $a^3x$ $x$ $a^4x$ $a^2x$ $a^6x$

The inner automorphism group is itself isomorphic to a Klein four-group. The multiplication table, viewed as cosets of the center, is as follows:

Element/element $\{e, a^2, a^4, a^6 \}$ $\{ a, a^3, a^5, a^7 \}$ $\{ ax, a^3x, a^5x, a^7x \}$ $\{ x, a^2x, a^4x, a^6x \}$
$\{e, a^2, a^4, a^6 \}$ $\{e, a^2, a^4, a^6 \}$ $\{ a, a^3, a^5, a^7 \}$ $\{ ax, a^3x, a^5x, a^7x \}$ $\{ x, a^2x, a^4x, a^6x \}$
$\{ a, a^3, a^5, a^7 \}$ $\{ a, a^3, a^5, a^7 \}$ $\{e, a^2, a^4, a^6 \}$ $\{ x, a^2x, a^4x, a^6x \}$ $\{ ax, a^3x, a^5x, a^7x \}$
$\{ ax, a^3x, a^5x, a^7x \}$ $\{ ax, a^3x, a^5x, a^7x \}$ $\{ x, a^2x, a^4x, a^6x \}$ $\{e, a^2, a^4, a^6 \}$ $\{ a, a^3, a^5, a^7 \}$
$\{ x, a^2x, a^4x, a^6x \}$ $\{ x, a^2x, a^4x, a^6x \}$ $\{ ax, a^3x, a^5x, a^7x \}$ $\{ a, a^3, a^5, a^7 \}$ $\{e, a^2, a^4, a^6 \}$

### Outer automorphisms and outer automorphism classes

The outer automorphism group is a Klein four-group, i.e., there are four cosets of the inner automorphism group in the automorphism group. Excluding the identity coset (which comprises the inner automorphisms) we thus get three outer automorphism classes. Each of these has size four (the size of the inner automorphism group) so there is a total of 12 outer automorphisms.

The automorphism group as a whole is isomorphic to direct product of D8 and Z2.

The outer automorphism group has a natural action as the normal V4 in S4 with its non-identity elements acting as double transpositions on the set of conjugacy classes $\{ \{ a,a^5 \}, \{ a^3, a^7 \}, \{ ax, a^5x \}, \{a^3x, a^7x \}\}$. More explicitly:

1. One outer automorphism class interchanges the conjugacy class $\{ a, a^5 \}$ and $\{ a^3, a^7 \}$ and simultaneously also interchanges the conjugacy classes $\{ax, a^5x \}$ and $\{ a^4x, a^7x \}$.
2. One outer automorphism class interchanges the conjugacy classes $\{ a, a^5 \}$ and $\{ ax, a^5x \}$ and simultaneously also interchanges the conjugacy classes $\{ a^3, a^7 \}$ and $\{a^3x, a^7x \}$.
3. One outer automorphism class interchanges the conjugacy classes $\{ a, a^5 \}$ and $\{ a^3x, a^7x \}$ and simultaneously also interchanges the conjugacy classes $\{ a^3, a^7 \}$ and $\{ ax, a^5x \}$.

Here now is a detailed description of all the 12 outer automorphisms in terms of their action on every element:

Automorphism class no. in above list Order of automorphism Quick description of automorphism Image of $e$ Image of $a^2$ Image of $a^4$ Image of $a^6$ Image of $a$ Image of $a^5$ Image of $a^3$ Image of $a^7$ Image of $ax$ Image of $a^5x$ Image of $a^3x$ Image of $a^7x$ Image of $x$ Image of $a^4x$ Image of $a^2x$ Image of $a^6x$
1 2 $a \mapsto a^3, x \mapsto x$ $e$ $a^4$ $a^6$ $a^2$ $a^3$ $a^7$ $a$ $a^5$ $a^3x$ $a^7x$ $ax$ $a^5x$ $x$ $a^4x$ $a^2x$ $a^6x$
1 2 $a \mapsto a^7, x \mapsto x$ $e$ $a^4$ $a^6$ $a^2$ $a^7$ $a^3$ $a^5$ $a$ $a^7x$ $a^3x$ $a^5x$ $ax$ $x$ $a^4x$ $a^2x$ $a^6x$
1 2 $a \mapsto a^3, x \mapsto a^4x$ $e$ $a^4$ $a^6$ $a^2$ $a^3$ $a^7$ $a$ $a^5$ $a^7x$ $a^3x$ $a^5x$ $ax$ $a^4x$ $x$ $a^6x$ $a^2x$
1 2 $a \mapsto a^7, x \mapsto a^4x$ $e$ $a^4$ $a^6$ $a^2$ $a^7$ $a^3$ $a^5$ $a$ $a^3x$ $a^7x$ $ax$ $a^5x$ $a^4x$ $x$ $a^6x$ $a^2x$
2 2 $a \mapsto ax, x \mapsto x$ $e$ $a^4$ $a^6$ $a^2$ $ax$ $a^5x$ $a^7x$ $a^3x$ $a$ $a^5$ $a^7$ $a^3$ $x$ $a^4x$ $a^2x$ $a^6x$
2 2 $a \mapsto a^5x, x \mapsto x$ $e$ $a^4$ $a^6$ $a^2$ $a^5x$ $ax$ $a^3x$ $a^7x$ $a^5$ $a$ $a^3$ $a^7$ $x$ $a^4x$ $a^2x$ $a^6x$
2 4 $a \mapsto ax, x \mapsto a^4x$ $e$ $a^4$ $a^6$ $a^2$ $ax$ $a^5x$ $a^7x$ $a^3x$ $a^5$ $a$ $a^3$ $a^7$ $a^4x$ $x$ $a^6x$ $a^2x$
2 4 $a \mapsto a^5x, x ,\mapsto a^4x$ $e$ $a^4$ $a^2$ $a^6$ $a^5x$ $ax$ $a^3x$ $a^7x$ $a$ $a^5$ $a^7$ $a^3$ $x$ $a^4x$ $a^2x$ $a^6x$
3 4 $a \mapsto a^3x, x \mapsto x$ $e$ $a^4$ $a^2$ $a^6$ $a^3x$ $a^7x$ $a^5x$ $ax$ $a^3$ $a^7$ $a^5$ $a$ $x$ $a^4x$ $a^2x$ $a^6x$
3 4 $a \mapsto a^7x, x \mapsto x$ $e$ $a^4$ $a^2$ $a^6$ $a^7x$ $a^3x$ $ax$ $a^5x$ $a^7$ $a^3$ $a$ $a^5$ $x$ $a^4x$ $a^2x$ $a^6x$
3 2 $a \mapsto a^3x, x \mapsto a^4x$ $e$ $a^4$ $a^2$ $a^6$ $a^3x$ $a^7x$ $a^5x$ $ax$ $a^7$ $a^3$ $a$ $a^5$ $a^4x$ $x$ $a^6x$ $a^2x$
3 2 $a \mapsto a^7x, x \mapsto a^4x$ $e$ $a^4$ $a^2$ $a^6$ $a^7x$ $a^3x$ $ax$ $a^5x$ $a^3$ $a^7$ $a^5$ $a$ $x$ $a^4x$ $a^2x$ $a^6x$