Abstract We study dark matter (DM) abundance in the framework of the extension of the Standard Model (SM) with an additional $$U(1)_X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>U</mml:mi><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mi>X</mml:mi></mml:msub></mml:mrow></mml:math> gauge symmetry. One complex singlet is included to break the $$U(1)_X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>U</mml:mi><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mi>X</mml:mi></mml:msub></mml:mrow></mml:math> gauge symmetry, meanwhile one of the doublets is considered inert to introduce a DM candidate. The stability of the DM candidate is analyzed with a continuous $$U(1)_X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>U</mml:mi><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mi>X</mml:mi></mml:msub></mml:mrow></mml:math> gauge symmetry as well as discrete $$Z_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> symmetry. We find allowed regions for the free model parameters which are in agreement with the most up-to-date experimental results reported by CMS and ATLAS Collaborations, the upper limit on WIMP-nucleon cross section imposed by XENON1T Collaboration and the upper limit on the production cross-section of a $$Z^{\prime }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>Z</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math> gauge boson times the branching ratio of the $$Z^{\prime }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>Z</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math> boson decaying into $$\ell ^-\ell ^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>ℓ</mml:mi><mml:mo>-</mml:mo></mml:msup><mml:msup><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math> . We also obtain allowed regions for the DM candidate mass from the relic density reported by the PLANCK Collaboration including light, intermediate and heavy masses; depending mainly on two parameters of the scalar potential, $$\lambda _{2x}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>λ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:math> and $$\lambda _{345}=\lambda _3+\lambda _4+2\lambda _5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mn>345</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>λ</mml:mi><mml:mn>5</mml:mn></mml:msub></mml:mrow></mml:math> . We find that trough $$pp\rightarrow \chi \chi \gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mi>p</mml:mi><mml:mo>→</mml:mo><mml:mi>χ</mml:mi><mml:mi>χ</mml:mi><mml:mi>γ</mml:mi></mml:mrow></mml:math> production, it may only be possible for a future hadron–hadron circular collider (FCC-hh) to be able to detect a DM candidate within the range of masses 10–60 GeV.