Is it possible to determine the universality class of phase transitions by just analysing symmetry? Since phase transition is closely connected with symmetry, I am wondering whether it is possible to determine the universality class of phase transitions just by symmetry? 
Actually, I found it is quite boring to calculate critical exponent numerically
and I want to find a new method. 
 A: In the simplest cases, yes, if you know the microscopic symmetries of the system, you know the universality class of the transition (if of course it's a second order phase transition). However, it does not have to be the case. Two examples: 1- emergent symmetries; 2- unexpected first order phase transition. Let me explain :
1- It might be the case that the effective (low energy / long distance) description of the system is described by an action/Hamiltonian that has different (emergent) symmetries. For example, in the Bose-Hubbard model (describing bosons on a lattice with on-site interaction), there is a phase transition between a superfluid and a Mott insulator. The transition is described by two different universality classes depending on the value of the parameters. This is because at some specific points in the phase diagram, a new (emergent) `Lorentzian' symmetry is present, with different critical exponents.
2- You can always imagine to fine tune the parameters of a system in order to transform a second order transition into a first order transition (no scaling, no critical exponents, etc.). For example, imagine a classical spin model, Ising-like, with first-, second-, third-, ...-neighbour interactions. By choosing carefully the parameters, there is no reason that the transition must always be second order.
A: At least for the case of the three "conventional" symmetry classes (Wigner-Dyson classes), the symmetry is directly visible in the structure of your Hamiltonian or, if you do numerics, in the matrix elements of your Hamiltonian matrix. For example, if your tight binding matrix contains only real matrix elements and is symmetric, the corresponding Hamiltonian belongs to the orthogonal symmetry class. Same is true for hermitian matrices with complex matrix elements (unitary symmetry class) and for the symplectic symmetry class, the symmetry relation is $H = \sigma_y H^T \sigma_y$, where $\sigma_y$ is the second Pauli matrix.
From what I know, the universality class also directly classifies the kind of phase transition. Maybe somebody else can shed more light on this topic.
