An interval is a pair of pitch classes $\{a,b\}$, and its quality is the quantity $q(a,b) = |a - b|$ mod 12. A minor third is an interval of quality 3, and a major third is an interval of quality 4. A triad is a triple of pitch classes $(a,b,c)$, where we always assume $a < b < c$ in the cyclic ordering. We are concerned with major and minor triads. A triad is major if $q(a,b) = 4$ and $q(b,c) = 3$, and a triad is minor if $q(a,b) = 3$ and $q(b,c) = 4$. (In other words, we always have $q(a,c) = 7$, and we call the triad major or minor depending on the quality of the interval $\{a,b\}$.) In music theory there are other triads, but for this answer I will use triad as a shorthand for major or minor triad.

The set of triads is a 24-element set, for a triad $(a,b,c)$ is completely characterized by its root note $a$ and its quality, major or minor.

The action of transposition and inversion is maybe easier to describe. Consider the group of bijection of $C_{12}$ generated by

$$\begin{cases}\iota\colon n \mapsto -n & \mod 12 \\ \tau\colon n \mapsto n+ 1 & \mod 12. \end{cases}$$ Note that $q(\iota(a),\iota(b) = q(a,b)$ and $q(\tau(a),\tau(b)) = q(a,b)$, and that this action is a faithful action of the dihedral group of order 24, which I am fond of calling $D_{12}$ for its natural action on a 12-element set, forgive me. Since this action preserves the quality of intervals, there is a diagonal action of $D_{12}$ on the set of triads. In fact, because we know $\tau$ preserves the cyclic ordering and $\iota$ reverses it, we can describe the action completely as $$\begin{cases} \iota\colon (a,b,c) \mapsto (\iota(c),\iota(b),\iota(a)) \\ \tau\colon (a,b,c) \mapsto (\tau(a),\tau(b),\tau(c)).\end{cases}$$

The “P/L/R” action is maybe slightly more annoying for someone with zero familiarity with Western harmony—indeed, I already see some confusion about it in the comments from people with plenty of familiarity! To properly explain where this comes from takes us a little further afield, so let me just give definitions. As we saw above, any triad is completely determined by its root note and its quality. Thus we can parametrize a triad as an element of $C_{12}\times C_2$. For some reason, I would like to think of $C_2$ multiplicatively as the group of units of the ring $\mathbb{Z}$, forgive me. Thus the triad $(a,b,c)$ is sent to $(a,+1)$ if $q(a,b) = 4$ and to $(a,-1)$ if $q(a,b)=3$.

The parallel major/minor of a triad $(a,t) \in C_{12}\times C_2$ is the triad $P(a,t) = (a,-t)$. Thus $P^2 = 1$, and $P(0,3,7) = (0,4,7)$.

The mediant of a triad is perhaps best described as follows. If $(a,b,c)$ is a triad, then its mediant $L(a,b,c)$ is the triad $(b,c,b+7)$. Thus if $(a,b,c)$ is a major triad, i.e. $(a,b,c) \leadsto (a,+1)$, then the mediant is $(b,-1)$. Conversely if $(a,b,c) \leadsto (a,-1)$, then the mediant is $(b,+1)$. From here you can easily reduce this to a map from $C_{12}\times C_2$ to itself, it just requires cases to state. Note that $L$ does not have order two.

The relative major/minor of a triad is again more easy to describe with music theory than directly as an action on a set, so I will content myself with describing it as $R = L^{-1}$.

Unfortunately there is a mistake. You can verify for yourself that the element $L$ has order $24$, and thus the group $\langle P,L,R\rangle$ cannot be the dihedral group of order $24$. Let me just show that $L$ does not have order $12$. Recall that $L$ inverts the quality of the triad and translates by $4$ or by $3$ according to the quality of the triad. We see that $L^2$ preserves the quality and has translated both by $4$ and by $3$ and thus $L^2$ satisfies $L^2 = \tau^7$. Thus $L^{12} = \tau^{6\cdot 7} = \tau^{42} = \tau^6$, since $\tau$ has order $12$.

An interval is a pair of pitch classes $\{a,b\}$, and its quality is the quantity $q(a,b) = |a - b|$ mod 12. A minor third is an interval of quality 3, and a major third is an interval of quality 4. A triad is an ordered triple of pitch classes $(a,b,c)$. We are concerned with major and minor triads. A triad is major if $q(a,b) = 4$ and $q(b,c) = 3$, and a triad is minor if $q(a,b) = 3$ and $q(b,c) = 4$. (In other words, we always have $q(a,c) = 7$, and we call the triad major or minor depending on the quality of the interval $\{a,b\}$.) In music theory there are other triads, but for this answer I will use triad as a shorthand for major or minor triad.

The set of triads is a 24-element set, for a triad $(a,b,c)$ is completely characterized by its root note $a$ and its quality, major or minor. Thus it is convenient to parametrize triads as elements of the set $C_{12}\times C_2$. For some reason I would like to think of $C_2$ as the group of units of the ring $\mathbb{Z}$, forgive me. If $(a,b,c)$ is a major triad, it corresponds to the element $(a,+1) \in C_{12}\times C_2$. Likewise if $(a,b,c)$ is a minor triad, it corresponds to $(a,-1)$.

The action of transposition and inversion is simple to describe: consider the group of bijection of $C_{12}\times C_2$ generated by

$$\begin{cases}\iota\colon (a,\pm 1) \mapsto (-c,\mp 1) = (-a-7,\mp 1) \\ \tau\colon (a,\pm 1) \mapsto (a+1,\pm 1). \end{cases}$$ (Here and throughout addition and subtraction are mod 12.)

Note that $\iota$ has order $2$, $\tau$ has order $12$, and that $$\iota\tau\iota (a,\pm 1) = \iota\tau(-a-7,\mp 1) = \iota(-a-6,\mp 1) = (a-1,\pm 1) = \tau^{-1}(a,\pm 1),$$ so this defines an action of the dihedral group of order 24—which I am fond of writing as $D_{12}$ because of its action on a $12$-element set, forgive me—on the set of triads.

The “P/L/R” action is maybe slightly more involved.

The parallel major/minor of a triad $(a,\pm 1) \in C_{12}\times C_2$ is the triad $P(a,t) = (a,\mp 1)$. Thus $P^2 = 1$, and $P(0,3,7) = (0,4,7)$.

The leading-tone exchange of a major triad $(a,b,c)$ is the triad $L(a,b,c) = (b,c,a-1)$. Thus if $(a,b,c)$ is a major triad, $L(a,+1) = (b,-1)$. The leading-tone exchange of a minor triad $(a,b,c)$ is the triad $L(a,b,c) = (c+1,a,b)$. So if $(a,b,c)$ is a minor triad, then $L(a,-1) = (c+1,+1)$. One checks that $L^2 =1$.

The relative major of a minor triad $(a,b,c)$ is the triad $R(a,b,c) = (b,c,b+7)$—that is, if $(a,b,c)$ is a minor triad, $R(a,-1) = (b,+1)$. The relative minor of a major triad is the inverse operation: $R(a,b,c) = (b-7,a,b)$, so if $(a,b,c)$ is a major triad, $R(a,+1) = (b-7,-1)$. Similarly one checks that $R^2 = 1$.

I claim that the action of $RL$ on major triads is given by $RL = \tau^7$. (We are acting on the left.) Indeed, suppose $(a,b,c)$ is a major triad. Then $L(a,+1) = (b,-1)$, which is the minor triad $(b,c,b+7)$, so $RL(a,+1) = R(b,-1) = (c,+1)$.

On the other hand, the action of $LR$ on minor triads is given by $LR = \tau^7$ (so on minor triads we have $RL = (LR)^{-1} = \tau^{-7}$). Indeed, suppose $(a,b,c)$ is a minor triad. Then $R(a,-1) = (b,+1)$, which is the major triad $(b,c,b+7)$, so $LR(a,-1) = L(b,+1) = (c,-1)$. Thus in both cases, the quality is preserved, and the root note has shifted up $7$ pitch classes.

Since $7$ is relatively prime to $12$, $\langle R,L \rangle$ is a quotient of a group with presentation $\langle x,y \mid x^2 = y^2 = (xy)^{12} = 1\rangle$, and the latter is a presentation of $D_{12}$. I suppose one has to show that the action is faithful; it is, but I’ll leave that to you.

I claim further that $L$ and $R$ are sufficient to recover $P$. I don’t immediately see a slick way of doing this, so I’ll leave it to you.

To show that each centralizes the other, it suffices to show that $\tau R = R\tau$, $\tau L = L\tau$, $\iota R = R\iota$, and finally $\iota L = L\iota$. Let me maybe just talk through a tiny piece.

Suppose first $(a,b,c)$ is a minor triad. Then

$$\tau R (a,-1) = \tau(b,+1) = (b+1,+1) = R(a+1,-1) = R\tau(a,-1).$$ The argument for a major triad is analogous.

On the other hand, note that $L(a,-1) = (a-4,+1)$ and $L(a,+1) = (a+4,-1)$. Thus we have $$\iota L(a,-1) = \iota(a-4,+1) = (-a-3,-1) = L(-a-7,+1) = \iota(a,-1).$$ The argument for a major triad is analogous.

Of course, even buying the gaps I’ve left, this doesn’t show that these copies of $D_{12}$ are the full centralizer of each other in $S_{24}$, but hopefully you have some ideas about how to prove it.