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Mentioned Ito's Theorem
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Geoff Robinson
Geoff Robinson
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Later edit: Another general fact which is often useful, is a result of N. Ito, which states that if the finite group $G$ has an Abelian normal subgroup $A$, then the degree of any complex irreducible representation of $G$ is a divisor of the index $[G:A].$

Later edit: Another general fact which is often useful, is a result of N. Ito, which states that if the finite group $G$ has an Abelian normal subgroup $A$, then the degree of any complex irreducible representation of $G$ is a divisor of the index $[G:A].$

typos
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Geoff Robinson
Geoff Robinson
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If $\chi$ is a (complex) irreducible character of a finite group $G$, the Frobenius-Schur indicator of $\chi$ is denoted by $\nu(\chi)$ defined to be $0$ if $\chi$ is not real-valued, to be $-1$ if $\chi$ is real-valued, but $\chi$ may NOT be afforded by a representation over $\mathbb{R}$, and to be $1$ if $\chi$ is afforded by a representation over $\mathbb{R}.$ For example, the unique irreducible complex character of degree $2$ of the quaternion group of order $8$ has frobeniusFrobenius-Schur indicator $-1$, and the unique irreducible character of degree $2$ of the dihedral group of order $8$ ( I mean the one with $8$ elements) has Frobenius-Schur indicator $1$.

As to the question of what you term degeneracy, there is some ambiguity (related to the Frobenius-Schur indicator and also the ScurSchur index). For example, the quaternion group of order $8$ has a $4$-dimensional representation which is irreducible as a real representation, but which is equivalent to the sum of two equivalent $2$-dimensional complex irreducible representations. An absolutely irreducible real representation is a real irreducible representation which remains irreducible as a complex representation. This is a representation whose character $\chi$ is irreducible as a complex character and has $\nu(\chi) = 1.$

If $\chi$ is a (complex) irreducible character of a finite group $G$, the Frobenius-Schur indicator of $\chi$ is denoted by $\nu(\chi)$ defined to be $0$ if $\chi$ is not real-valued, to be $-1$ if $\chi$ is real-valued, but $\chi$ may NOT be afforded by a representation over $\mathbb{R}$, and to be $1$ if $\chi$ is afforded by a representation over $\mathbb{R}.$ For example, the unique irreducible complex character of degree $2$ of the quaternion group of order $8$ has frobenius-Schur indicator $-1$, and the unique irreducible character of degree $2$ of the dihedral group of order $8$ ( I mean the one with $8$ elements) has Frobenius-Schur indicator $1$.

As to the question of what you term degeneracy, there is some ambiguity (related to the Frobenius-Schur indicator and also the Scur index). For example, the quaternion group of order $8$ has a $4$-dimensional representation which is irreducible as a real representation, but which is equivalent to the sum of two equivalent $2$-dimensional complex irreducible representations. An absolutely irreducible real representation is a real irreducible representation which remains irreducible as a complex representation. This is a representation whose character $\chi$ is irreducible as a complex character and has $\nu(\chi) = 1.$

If $\chi$ is a (complex) irreducible character of a finite group $G$, the Frobenius-Schur indicator of $\chi$ is denoted by $\nu(\chi)$ defined to be $0$ if $\chi$ is not real-valued, to be $-1$ if $\chi$ is real-valued, but $\chi$ may NOT be afforded by a representation over $\mathbb{R}$, and to be $1$ if $\chi$ is afforded by a representation over $\mathbb{R}.$ For example, the unique irreducible complex character of degree $2$ of the quaternion group of order $8$ has Frobenius-Schur indicator $-1$, and the unique irreducible character of degree $2$ of the dihedral group of order $8$ ( I mean the one with $8$ elements) has Frobenius-Schur indicator $1$.

As to the question of what you term degeneracy, there is some ambiguity (related to the Frobenius-Schur indicator and also the Schur index). For example, the quaternion group of order $8$ has a $4$-dimensional representation which is irreducible as a real representation, but which is equivalent to the sum of two equivalent $2$-dimensional complex irreducible representations. An absolutely irreducible real representation is a real irreducible representation which remains irreducible as a complex representation. This is a representation whose character $\chi$ is irreducible as a complex character and has $\nu(\chi) = 1.$

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Geoff Robinson
Geoff Robinson
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Your question touches on many issues in group representation theory, and I can only give a few general remarks which may point you in interesting directions for further reading.

As to your question regarding the maximal real irreducible representation of a finite group, there is an interesting connection with the Frobenius Schur indicator.

If $\chi$ is a (complex) irreducible character of a finite group $G$, the Frobenius-Schur indicator of $\chi$ is denoted by $\nu(\chi)$ defined to be $0$ if $\chi$ is not real-valued, to be $-1$ if $\chi$ is real-valued, but $\chi$ may NOT be afforded by a representation over $\mathbb{R}$, and to be $1$ if $\chi$ is afforded by a representation over $\mathbb{R}.$ For example, the unique irreducible complex character of degree $2$ of the quaternion group of order $8$ has frobenius-Schur indicator $-1$, and the unique irreducible character of degree $2$ of the dihedral group of order $8$ ( I mean the one with $8$ elements) has Frobenius-Schur indicator $1$.

The number of solutions of $x^{2}=1 $ in the finite group $G$ is equal to $\sum_{\chi} \nu(\chi) \chi(1)$, where $\chi$ runs over the complex irreducible characters of $G$.

This is especially useful if all irreducible characters $\chi$ of $G$ have $\nu(\chi) = 1$, which is always the case for $G = S_{n}$ (the symmetric group).

The FS-indicator may (in principle at least) be calculated via the formula $\nu(\chi) = \frac{1}{|G|} \sum_{g \in G} \chi(g^{2}).$

In the case of the alternating group of degree $5$, for example, all irreducible characters $\chi$ have $\nu(\chi) = 1$, the irreducible characters have degree $1,3,3,4,5$. Hence we get $\sum_{\chi} \nu(\chi)\chi(1) = 16$, and there are indeed $16$ solutions of $x^{2} = 1$ in $G$ (the identity and fifteen elements of order $2$).

As to the question of what you term degeneracy, there is some ambiguity (related to the Frobenius-Schur indicator and also the Scur index). For example, the quaternion group of order $8$ has a $4$-dimensional representation which is irreducible as a real representation, but which is equivalent to the sum of two equivalent $2$-dimensional complex irreducible representations. An absolutely irreducible real representation is a real irreducible representation which remains irreducible as a complex representation. This is a representation whose character $\chi$ is irreducible as a complex character and has $\nu(\chi) = 1.$

A real irreducible representation which is not absolutely irreducible is one which is not irreducible as a complex representation. Such a representation may afford a character of the form $2\chi$ where $\chi$ is a complex irreducible character with $\nu(\chi) = -1$, or it may afford a character of the form $\chi + \overline{\chi}$, where $\chi$ is a complex irreducible character with $\nu(\chi) = 0$ (ie $\chi$ is not real-valued).

In terms of complex irreducible representations, it is one of the earliest theorems in group theory (due to C. Jordan) that if a finite group $G$ has a complex representation of degree $n$ (irreducible or not), then $G$ has an Abelian normal subgroup whose index is bounded in terms of $n$. This also applies to real irreducible representations.

If we restrict to complex irreducible representations which are primitive (that is, can not be induced from a representation of a proper subgroup), this tells us that if $G$ has a primitive complex irreducible representation of degree $n$, then the number of possibilities for $G/Z(G)$ is bounded in terms of $n$.

On the other hand, the symmetric group $S_{n+1}$ always has a real irreducible representation of degree $n$, and has order $(n+1)!$, yet has no non-identity Abelian normal subgroup if $n >3.$ This is related to the "generic" worst case bound for Jordan's Theorem, and is genuinely an upper bound for that Theorem if $n$ is large enough.

I think that in general, it is very difficult to relate the order of generators of a finite group $G$ with the largest degree of its real (or complex) irreducible representations. For example, there are arbitrarily large finite simple groups $G$ which may be generated by an element of order $2$ and an element of order $3$, and there is therefore no upper bound on the dimensions of the real irreducible representations of finte groups which may be generated by an element of order $2$ and an element of order $3$.