This paper introduces Group Equivariant Convolutional Networks (G-CNNs), a generalization of convolutional neural networks (CNNs) that exploit symmetries to reduce sample complexity. G-CNNs use G-convolutions, a new type of layer that achieves higher weight sharing than regular convolutional layers, without increasing the number of parameters. G-convolutions are easy to implement and can be used with minimal computational overhead for discrete groups generated by translations, reflections, and rotations. G-CNNs achieve state-of-the-art results on CIFAR10 and rotated MNIST. The paper discusses the concept of equivariance, which is key to the generalization of CNNs to exploit larger symmetry groups. It shows that standard CNNs are equivariant to translations but may fail to be equivariant to more general transformations. Using a mathematical framework, the paper defines G-CNNs by analogy to standard CNNs, showing that G-convolutions and various other layers used in modern CNNs are all equivariant and thus compatible with G-CNNs. The paper also provides a detailed mathematical framework for G-CNNs, including the definition of symmetry groups, functions on groups, and the transformation properties of feature maps. It discusses the implementation of G-convolutions, showing that they can be implemented efficiently by leveraging recent advances in fast computation of planar convolutions. The paper presents experimental results on MNIST-rot and CIFAR10, where G-CNNs achieve state-of-the-art results. The results show that replacing planar convolutions with G-convolutions consistently improves results without additional tuning. The paper also discusses the implications of these results and considers several extensions of the method. In conclusion, the paper demonstrates that G-CNNs can be used as a drop-in replacement for standard convolutions in modern network architectures, improving their performance without further tuning. The results show that G-CNNs achieve state-of-the-art results on rotated MNIST and CIFAR10, and that the general theory of G-CNNs for discrete groups is applicable to these groups.This paper introduces Group Equivariant Convolutional Networks (G-CNNs), a generalization of convolutional neural networks (CNNs) that exploit symmetries to reduce sample complexity. G-CNNs use G-convolutions, a new type of layer that achieves higher weight sharing than regular convolutional layers, without increasing the number of parameters. G-convolutions are easy to implement and can be used with minimal computational overhead for discrete groups generated by translations, reflections, and rotations. G-CNNs achieve state-of-the-art results on CIFAR10 and rotated MNIST. The paper discusses the concept of equivariance, which is key to the generalization of CNNs to exploit larger symmetry groups. It shows that standard CNNs are equivariant to translations but may fail to be equivariant to more general transformations. Using a mathematical framework, the paper defines G-CNNs by analogy to standard CNNs, showing that G-convolutions and various other layers used in modern CNNs are all equivariant and thus compatible with G-CNNs. The paper also provides a detailed mathematical framework for G-CNNs, including the definition of symmetry groups, functions on groups, and the transformation properties of feature maps. It discusses the implementation of G-convolutions, showing that they can be implemented efficiently by leveraging recent advances in fast computation of planar convolutions. The paper presents experimental results on MNIST-rot and CIFAR10, where G-CNNs achieve state-of-the-art results. The results show that replacing planar convolutions with G-convolutions consistently improves results without additional tuning. The paper also discusses the implications of these results and considers several extensions of the method. In conclusion, the paper demonstrates that G-CNNs can be used as a drop-in replacement for standard convolutions in modern network architectures, improving their performance without further tuning. The results show that G-CNNs achieve state-of-the-art results on rotated MNIST and CIFAR10, and that the general theory of G-CNNs for discrete groups is applicable to these groups.