The imperial period of Rome lasted approximately 1,500 years compared to the 500 years of the Republican era. The first two centuries of the empire's existence were a period of unprecedented political stability and prosperity known as the Pax Romana, or ``Roman Peace ''. Following Octavian's victory, the size of the empire was dramatically increased. After the assassination of Caligula in AD 41, the Senate briefly considered restoring the republic, but the Praetorian Guard proclaimed Claudius emperor instead. Under Claudius, the empire invaded Britannia, its first major expansion since Augustus. After Claudius' successor, Nero, committed suicide in AD 68, the empire suffered a series of brief civil wars, as well as a concurrent major rebellion in Judea, during which four different legionary generals were proclaimed emperor. Vespasian emerged triumphant in AD 69, establishing the Flavian dynasty, before being succeeded by his son Titus, who opened the Colosseum shortly after the eruption of Mount Vesuvius. His short reign was followed by the long reign of his brother Domitian, who was eventually assassinated. The Senate then appointed the first of the Five Good Emperors. The empire reached its greatest extent under Trajan, the second in this line.
Marcus Ulpius Traianus was born on 18 September 53AD in the Roman province of Hispania Baetica (in what is now Andalusia in modern Spain), in the city of Italica (now in the municipal area of Santiponce, in the outskirts of Seville). Although frequently designated the first provincial emperor, and dismissed by later writers such as Cassius Dio (himself of provincial origin) as "an Iberian, and neither an Italian nor even an Italiot", Trajan appears to have hailed on his father's side from the area of Tuder (modern Todi) in Umbria, at the border with Etruria, and on his mother's side from the Gens Marcia, of an Italic family of Sabine origin. Trajan's birthplace of Italica was founded as a Roman military colony of "Italian" settlers in 206BC, though it is unknown when the Ulpii arrived there. It is possible, but cannot be substantiated, that Trajan's ancestors married local women and lost their citizenship at some point, but they certainly recovered their status when the city became a municipium with Latin citizenship in the mid-1st century BC.
The precise start and end of the Industrial Revolution is still debated among historians, as is the pace of economic and social changes. Eric Hobsbawm held that the Industrial Revolution began in Britain in the 1780s and was not fully felt until the 1830s or 1840s, while T.S. Ashton held that it occurred roughly between 1760 and 1830. Rapid industrialization first began in Britain, starting with mechanized spinning in the 1780s, with high rates of growth in steam power and iron production occurring after 1800. Mechanized textile production spread from Great Britain to continental Europe and the United States in the early 19th century, with important centres of textiles, iron and coal emerging in Belgium and the United States and later textiles in France.
Hiberni (Ireland), Pictish (northern Britain) and Britons (southern Britain) tribes, all speaking Insular Celtic, inhabited the islands at the beginning of the 1st millennium AD. Much of Brittonic-controlled Britain was conquered by the Roman Empire from AD 43. The first Anglo-Saxons arrived as Roman power waned in the 5th century and eventually dominated the bulk of what is now England. Viking invasions began in the 9th century, followed by more permanent settlements and political change—particularly in England. The subsequent Norman conquest of England in 1066 and the later Angevin partial conquest of Ireland from 1169 led to the imposition of a new Norman ruling elite across much of Britain and parts of Ireland. By the Late Middle Ages, Great Britain was separated into the Kingdoms of England and Scotland, while control in Ireland fluxed between Gaelic kingdoms, Hiberno-Norman lords and the English-dominated Lordship of Ireland, soon restricted only to The Pale. The 1603 Union of the Crowns, Acts of Union 1707 and Acts of Union 1800 attempted to consolidate Britain and Ireland into a single political unit, the United Kingdom of Great Britain and Ireland, with the Isle of Man and the Channel Islands remaining as Crown Dependencies. The expansion of the British Empire and migrations following the Irish Famine and Highland Clearances resulted in the distribution of the islands' population and culture throughout the world and a rapid de-population of Ireland in the second half of the 19th century. Most of Ireland seceded from the United Kingdom after the Irish War of Independence and the subsequent Anglo-Irish Treaty (1919–1922), with six counties remaining in the UK as Northern Ireland.

Question: Who was the mother of the emperor when the foreign group that conquered the nation where the industrial revolution began reach its greatest extent?
Answer: Marcia