  We present a detailed discussion of the structure of the low-lying
positive-parity energy spectrum of $^{12}$C from a no-core shell-model
perspective. The approach utilizes a fraction of the usual shell-model space
and extends its multi-shell reach via the symmetry-based no-core symplectic
shell model (NCSpM) with a simple, physically-informed effective interaction.
We focus on the ground-state rotational band, the Hoyle state and its $2^+$ and
$4^+$ excitations, as well as the giant monopole $0^+$ resonance, which is a
vibrational breathing mode of the ground state. This, in turn, allows us to
address the open question about the structure of the Hoyle state and its
rotational band. In particular, we find that the Hoyle state is best described
through deformed prolate collective modes rather than vibrational modes, while
we show that the higher-lying giant monopole $0^+$ resonance resembles the
oblate deformation of the $^{12}$C ground state. In addition, we identify the
giant monopole $0^+$ and quadrupole $2^+$ resonances of selected light and
intermediate-mass nuclei, along with other observables of $^{12}$C, including
matter rms radii, electric quadrupole moments, as well as $E2$ and $E0$
transition rates.
