Is there an authentic Hadith regarding the virtue of reciting the six phrases together: "Tabarak Allah, and Subhan Allah, and Al-hamdulillah, and Allahu Akbar, and La hawla wa la quwwata illa billah, and La ilaha illa Allah"?

No مرفوع (elevated) hadith has been found that collects all these wordings. Rather, a mawquf (stopped) hadith was found from Abdullah ibn Mas'ud, may Allah be pleased with him, in which he says: When a servant says: "SubhanAllah (Glory be to Allah), Al-hamdulillah (All praise is due to Allah), La ilaha illa Allah (There is no god but Allah), Allahu Akbar (Allah is the Greatest), and TabarakAllah (Blessed be Allah)," an angel ascends with them to Allah, referencing the verse: "To Him ascends good speech, and righteous work raises it."

Meanwhile, the four phrases (La ilaha illa Allah, SubhanAllah, Allahu Akbar, and Al-hamdulillah) along with the Hawqala (La hawla wa la quwwata illa billah - There is no might or power except with Allah) are confirmed in the hadith of Abu Sa'id al-Khudri as "the enduring good deeds" (Al-Baqiyat As-Salihat). It was also reported in another hadith from Abdullah ibn Amr ibn al-'As that whoever says these five words, "his sins will be forgiven, even if they are more than the foam of the sea."

As for the other words, they have numerous virtues. Takbir (saying Allahu Akbar), Tahmid (saying Al-hamdulillah), and Tasbih (saying SubhanAllah) are better than spending in the way of Allah, as stated in the hadith of Umm Hani'. The most beloved words to Allah are four: SubhanAllah, Al-hamdulillah, La ilaha illa Allah, and Allahu Akbar. They are more beloved to the Prophet, peace and blessings be upon him, than anything upon which the sun rises, and good deeds are recorded with them and sins are erased. They are "the enduring good deeds" and "your paradise from the Fire," as mentioned in other hadiths.