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Bhargava proved a formula for counting, with certain weights, degree n etaleextensions of a local field, or equivalently, local Galois representations toS_n. This formula is motivation for his conjectures about the density ofdiscriminants of S_n-number fields. We prove there are analogous ``massformulas'' that count local Galois representations to any group that can beformed from symmetric groups by wreath products and cross products,corresponding to counting towers and direct sums of etale extensions. We obtainas a corollary that the above mentioned groups have rational character tables.Our result implies that D_4 has a mass formula for certain weights, but we showthat D_4 does not have a mass formula when the local Galois representations toD_4 are weighted in the same way as representations to S_4 are weighted inBhargava's mass formula.

Mass formulas for local Galois representations to wreath products and cross products