Why do higher symmetry GCNNs have more parameters?

[snehjp2]

It's known that equivariance is implemented by employing a higher degree of weight sharing in convolution, corresponding to the endowed symmetry. More-symmetric networks (higher order symmetries) should therefore have more weight sharing and therefore less trainable parameters.

I'm noticing in my own implementation of E(2) equivariant networks that GCNNs equivariant to $D_1$ have significantly less parameters than an identically constructed $D_{16}$ network. I'm wondering why this is? Naively, it should be the case that the $D_{16}$ has less parameters than $D_1$, because there is significantly more parameter sharing. This is concerning to me cause then it could be argued that the performance bump of a higher-order network such as $D_{16}$ can be attributed to more trainable parameters, rather than exploiting more symmetry.

Is there something in the backend of escnn / e2cnn that I'm not aware of? Is counting trainable parameters in PyTorch the usual way not appropriate for these GCNNs? Is my theoretical understanding of group convolution incorrect?