  We present the analysis of four first overtone RR Lyrae stars observed with
the Kepler space telescope, based on data obtained over nearly 2.5yr. All four
stars are found to be multiperiodic. The strongest secondary mode with
frequency f_2 has an amplitude of a few mmag, 20 - 45 times lower than the main
radial mode with frequency f_1. The two oscillations have a period ratio of
P_2/P_1 = 0.612 - 0.632 that cannot be reproduced by any two radial modes.
Thus, the secondary mode is nonradial. Modes yielding similar period ratios
have also recently been discovered in other variables of the RRc and RRd types.
These objects form a homogenous group and constitute a new class of multimode
RR Lyrae pulsators, analogous to a similar class of multimode classical
Cepheids in the Magellanic Clouds. Because a secondary mode with P_2/P_1 ~ 0.61
is found in almost every RRc and RRd star observed from space, this form of
multiperiodicity must be common. In all four Kepler RRc stars studied, we find
subharmonics of f_2 at ~1/2 f_2 and at ~3/2 f_2. This is a signature of period
doubling of the secondary oscillation, and is the first detection of period
doubling in RRc stars. The amplitudes and phases of f_2 and its subharmonics
are variable on a timescale of 10 - 200d. The dominant radial mode also shows
variations on the same timescale, but with much smaller amplitude. In three
Kepler RRc stars we detect additional periodicities, with amplitudes below
1mmag, that must correspond to nonradial g-modes. Such modes never before have
been observed in RR Lyrae variables.
