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half-fourth-cousin 

1- Half-great-great-grandpibling's great-great-grandchild.
2- Someone who, in relation to others, has zero parents, zero grandparents, zero great-grandparents, zero great-great-grandparents and one great-great-great-grandparent in common.

irregular-double-fourth-cousin

Someone who, in relation to others, has zero parents, zero grandparents, zero great-grandparents, zero great-great-grandparents and three great-great-great-grandparents in common.

double-fourth-cousin 

Someone who, in relation to others, has zero parents, zero grandparents, zero great-grandparents, zero great-great-grandparents and four great-great-great-grandparents in common.

irregular-triple-fourth-cousin 

Someone who, in relation to others, has zero parents, zero grandparents, zero great-grandparents, zero great-great-grandparents and five great-great-great-grandparents in common.

triple-fourth-cousin 

Someone who, in relation to others, has zero parents, zero grandparents, zero great-grandparents, zero great-great-grandparents and six great-great-great-grandparents in common.

irregular-quadruple-fourth-cousin

Someone who, in relation to others, has zero parents, zero grandparents, zero great-grandparents, zero great-great-grandparents and seven great-great-great-grandparents in common.