Different Dimension Gate is a Continuous Spell Card.
Spell Cards are usually activated from the hand on your turn unless their type gives another timing window.
On the current TCG and OCG Forbidden & Limited lists tracked here, Different Dimension Gate is Unlimited.
YGOPRODeck lists Different Dimension Gate for these formats: GOAT, OCG GOAT, Duel Links, Common Charity, Edison, Speed Duel, TCG, OCG, Master Duel.
A listed print of Different Dimension Gate appears in Battle Pack: Epic Dawn (BP01-EN077) at Common rarity.
Among markets shown on this page, the lowest listed snapshot for Different Dimension Gate is about $0.04 on TCGPlayer (prices change frequently).
