Question: Carloman of Bavaria was the eldest grandson of what descendant of Charlemagne?

full_context: Document 1: Gerold, Prefect of Bavaria: Gerold (died 799) was an Alamannian nobleman who served the Frankish King, Charlemagne, as Margrave of the Avarian March and Prefect of Bavaria in what is now South-Eastern Germany. Gerold played a significant role in the integration of Bavaria into the Frankish Kingdom during Carolingian expansion in the late 8th, and early 9th centuries. Gerold both aided the continuity of Agilofing rule of Bavaria, as well as took steps to integrate Bavarians into the wider scope of the Frankish Kingdom. Gerold was related both to the Agilofing family, the ruling class of Bavaria, as well as the Carolingian family. The Agilofings had ruled Bavaria since Duke Garibald I in 548. Gerold was born into the Agilofings, and his sister Hildegard was married to Charlemagne in 771.From these familial connections, he was appointed Prefect of Bavaria following the deposition of Duke Tassilo III in 788. Gerold was heralded as a superb military commander, giving rise to his promotion to Prefect as a defender of the eastern border of the Frankish Kingdom. In 799, Gerold is said to have fallen in battle against the Avars, shortly after the same Avars killed his ally, Erich, Duke of Friuli, through treachery.
Document 2: Charles I of Hungary: Charles I, also known as Charles Robert (Hungarian: "Károly Róbert" ; Croatian: "Karlo Robert" ; Slovak: "Karol Róbert" ; 128816 July 1342) was King of Hungary and Croatia from 1308 to his death. He was a member of the Capetian House of Anjou and the only son of Charles Martel, Prince of Salerno. His father was the eldest son of Charles II of Naples and Mary of Hungary. She laid claim to Hungary after her brother, Ladislaus IV of Hungary, died in 1290, but the Hungarian prelates and lords elected her cousin, Andrew III, king. Instead of abandoning her claim to Hungary, she transferred it to her son, Charles Martel, and after his death in 1295, to her grandson, Charles. On the other hand, her husband, Charles II of Naples, made their third son, Robert, heir to the Kingdom of Naples, thus disinheriting Charles.
Document 3: Drogo (mayor of the palace): Drogo was the eldest son of Carloman, mayor of the palace of Austrasia. He was born before Carloman came to power in 741. In 747 Carloman went on a pilgrimage to Rome, leaving his sons under the guardianship of his uncle Pepin the Short and the king, Childeric III. Drogo, the eldest, had probably already reached his majority and may have been left behind as mayor of the palace by his father. Drogo's position was strong at this point and he held a church synod in 747 or 748 competing with a simultaneous synod held by Pepin. The great reformer and apostle to the Germans, Boniface, was unsure of which one to attend.
Document 4: Stem duchy: A stem duchy (German: "Stammesherzogtum" , from "Stamm", meaning "tribe", in reference to the Germanic tribes of the Franks, Saxons, Bavarians and Swabians) was a constituent duchy of the kingdom of Germany at the time of the extinction of the Carolingian dynasty (the death of Louis the Child in 911) and the transitional period leading to the formation of the Holy Roman Empire later in the 10th century. The Carolingians had dissolved the original tribal duchies of the Frankish Empire in the 8th century. As the Carolingian Empire declined in the late 9th century, the old tribal areas assumed new identities as the subdivisions of the realm. These are the five stem duchies (sometimes also called "younger stem duchies" in reference to the pre-Carolingian tribal duchies):
Document 5: Conrad II of Italy: Conrad II (12 February 1074 – 27 July 1101) was the Duke of Lower Lorraine (1076–87), King of Germany (1087–98) and King of Italy (1093–98). He was the second son of Holy Roman Emperor Henry IV and Bertha of Savoy, and their eldest son to reach adulthood, his older brother Henry having been born and died in the same month of August 1071. Conrad's rule in Lorraine and Germany was nominal. He spent most of his life in Italy and there he was king in fact as well as in name.
Document 6: Pepin the Hunchback: Pepin, or Pippin, the Hunchback (French: Pépin le Bossu, German: Pippin der Buckelige; c. 769 – 811) was the eldest son of Charlemagne. Born to the Frankish noblewoman Himiltrude, Pepin probably developed kyphosis after birth, leading early medieval historians to give him the epithet "hunchback". He lived with his father's court even after Charlemagne dismissed his mother and took another wife, Hildegard. Around 781, Pepin's half brother Carloman was rechristened as "Pepin of Italy"—a step that may have signaled Charlemagne's decision to disinherit the elder Pepin, for a variety of possible reasons. In 792, Pepin the Hunchback revolted against his father with a group of leading Frankish nobles, but the plot was discovered and put down before the conspiracy could put it into action. Charlemagne commuted Pepin's death sentence, having him tonsured and exiled to the monastery of Prüm instead. Since his death in 811, Pepin has been the subject of numerous works of historical fiction.
Document 7: Carloman (mayor of the palace): Carloman (between 706 and 716 – 17 August 754) was the eldest son of Charles Martel, "majordomo" or mayor of the palace and duke of the Franks, and his wife Chrotrud of Treves. On Charles's death (741), Carloman and his brother Pepin the Short succeeded to their father's legal positions, Carloman in Austrasia, and Pepin in Neustria. He was a member of the family later called the Carolingians and it can be argued that he was instrumental in consolidating their power at the expense of the ruling Merovingian kings of the Franks. He withdrew from public life in 747 to take up the monastic habit, "the first of a new type of saintly king," according to Norman Cantor, "more interested in religious devotion than royal power, who frequently appeared in the following three centuries and who was an indication of the growing impact of Christian piety on Germanic society".
Document 8: Louis the German: Louis (also Ludwig or Lewis) "the German" (c. 802-876), also known as Louis II, was the first king of East Francia. Grandson of emperor Charlemagne and the third son of emperor of Francia, Louis the Pious and his first wife, Ermengarde of Hesbaye, he received the appellation "Germanicus" shortly after his death in recognition of the fact that the bulk of his kingdom was in the former Germania.
Document 9: Conrad II, Holy Roman Emperor: Conrad II (c. 990 4 June 1039), also known as Conrad the Elder and Conrad the Salic , was Emperor of the Holy Roman Empire from 1027 until his death in 1039. The founder of the Salian dynasty of emperors, Conrad also served as King of Germany from 1024, King of Italy from 1026, and King of Burgundy from 1033.
Document 10: Duke of the Franks: The title Duke [and Prince] of the Franks ("dux" ["et princeps"] "Francorum") has been used for three different offices, always with "duke" implying military command and "prince", on those occasions when it was used either with or in preference to "duke", implying something approaching sovereign or regalian rights. The term "Franks" may refer to an ethnic group or the inhabitants of a territory called "Francia", named after the original Franks. The first office was that of the mayors of the palace of the Merovingian kings of the Franks, whose powers increased as those of the kings declined. The second was that of the second-in-command to the early kings of France, the last incumbent of which succeeded to the throne in 987. This title was sometimes rendered Duke of France ("dux Franciae"). The third instance was that of the rulers of the Frankish-inhabited lands of Germany, the so-called "tribal" duchy of Franconia.
Document 11: Berengar I of Italy: Berengar I (Latin: "Berengarius, Perngarius" ; Italian: "Berengario" ; 845 – 7 April 924) was the King of Italy from 887, and Holy Roman Emperor after 915, until his death. He is usually known as Berengar of Friuli, since he ruled the March of Friuli from 874 until at least 890, but he had lost control of the region by 896.
Document 12: Carloman I: Carloman I, also "Karlmann" (28 June 751 – 4 December 771) was king of the Franks from 768 until his death in 771. He was the second surviving son of Pepin the Short and Bertrada of Laon and was a younger brother of Charlemagne. No one knows who killed him. Carloman is mainly regarded by historians as Charlemagne's first steps towards acquiring the Holy Roman Empire, as his death allowed Charlemagne to take all of Francia and begin his expansion into other kingdoms.
Document 13: Carloman of Bavaria: Carloman (German: "Karlmann" , Latin: "Karlomannus" ; c. 830 – 22 March 880), was a Frankish king of the Carolingian dynasty. He was the eldest son of Louis the German, king of East Francia, and Hemma, daughter of a Bavarian count. His father appointed him margrave of Pannonia in 856, and upon his father's death in 876 he became King of Bavaria. He was appointed by King Louis II of Italy as his successor, but the Kingdom of Italy was taken by his uncle Charles the Bald in 875. Carloman only conquered it in 877. In 879 he was incapacitated, perhaps by a stroke, and abdicated Bavaria to Louis the Younger and Italy to Charles the Fat.
Document 14: Louis II, Duke of Bavaria: Ludwig I or Louis I of Upper Bavaria (German: "Ludwig II der Strenge, Herzog von Bayern, Pfalzgraf bei Rhein" ) (13 April 1229 – 2 February 1294) was Duke of Upper Bavaria and Count Palatine of the Rhine from 1253. He is known as Ludwig II or Louis II as Duke of Bavaria, and also as Louis the Strict. Born in Heidelberg, he was a son of duke Otto II and Agnes of the Palatinate. She was a daughter of the Welf Henry V, Count Palatine of the Rhine, her grandfathers were Henry XII the Lion and Conrad of Hohenstaufen.
Document 15: Ludwig II of Bavaria: Ludwig II (German: "Ludwig Otto Friedrich Wilhelm" ; English: Louis Otto Frederick William ; 25 August 1845 – 13 June 1886) was King of Bavaria from 1864 until his death in 1886. He is sometimes called the Swan King, Mad King Ludwig or der Märchenkönig (the 'Fairy Tale King'). He also held the titles of Count Palatine of the Rhine, Duke of Bavaria, Duke of Franconia, and Duke in Swabia.
ground_truth: Louis II