Real mutually unbiased bases and representations of groups of odd order by real scaled Hadamard matrices of 2-power size

We prove the following two results relating real mutually unbiased bases and representations of finite groups of odd order. Let q be a power of 2 and r a positive integer. Then we can find a q^2rtimes q^2r real orthogonal matrix D, say, of multiplicative order q^2r-1+1, whose q^2r-1+1 powers D, \dots, D^{q^2r-1+1}=I define q^2r-1+1 mutually unbiased bases in mathbb{R}^{q^2r}. Thus the scaled matrices q^rD, \dots, q^rD^{q^2r-1} are q^2r-1 different Hadamard matrices. When we take q=2, we achieve the maximum number of real mutually unbiased bases in dimension 2^2r using the elements of a cyclic group. We also prove the following. Let G be an arbitrary finite group of odd order 2k+1, where kgeq 3. Then G has a real representation R, say, of degree 2^{2^k-1} such that the elements R(σ), σin G, define |G| mutually unbiased bases in mathbb{R}^d, where d= 2^{2^k-1}. In addition, a group of order 5 defines five real mutually unbiased bases in mathbb{R}¹⁶ and a group of order 3 defines three real mutually unbiased bases in mathbb{R}⁴. Thus, an arbitrary group of odd order has a faithful representation by real scaled Hadamard matrices of 2-power size.