  The electric quadrupole transitions between $0^+$, $2^+$, and $4^+$ states in
$^{12}$C are investigated in a $3\alpha$ model. The three-body wave functions
are obtained by means of the hyperspherical adiabatic expansion method, and the
continuum is discretized by imposing a box boundary condition. Corresponding
expressions for the continuum three-body ($3\alpha$) bremsstrahlung and photon
dissociation cross sections are derived and computed for two different
$\alpha-\alpha$ potentials. The available experimental energy dependence is
reproduced and a series of other cross sections are predicted. The transition
strengths are defined and derived from the cross sections, and compared to
schematic rotational model predictions. The computed properties of the $^{12}$C
resonances suggest that the two lowest bands are made, respectively, by the
states $\{0^+_1, 2^+_1, 4^+_2\}$ and $\{0^+_2, 2^+_2, 4^+_1\}$. The transitions
between the states in the first band are consistent with the rotational pattern
corresponding to three alphas in an equal sided triangular structure. For the
second band, the transitions are also consistent with a rotational pattern, but
with the three alphas in an aligned distribution.
