SO24×SO6

SO(24) × SO(6) denotes the direct product of the special orthogonal groups in 24 and 6 dimensions. An element is a pair (A, B) with A ∈ SO(24) and B ∈ SO(6), and multiplication is performed componentwise: (A, B)(A′, B′) = (AA′, BB′). The group is a compact, connected Lie group and is semisimple, as it is the direct product of two semisimple factors.

The Lie algebra of SO(24) × SO(6) is so(24) ⊕ so(6). Its dimension is 276 + 15 = 291, and its rank is 12 + 3 = 15. The center of the group is Z2 × Z2, generated by the elements (-I24, I6) and (I24, -I6), since -I is in SO(n) when n is even.

The fundamental group is π1(SO(24) × SO(6)) ≅ π1(SO(24)) × π1(SO(6)) ≅ Z2 × Z2, reflecting the well-known fact that π1(SO(n)) ≅ Z2 for n ≥ 3.

Representations: there is a natural faithful representation on R24 ⊕ R6 given by (A, B) · (v, w) = (Av, Bw). This yields a faithful 30-dimensional representation, combining the standard actions of SO(24) on R24 and SO(6) on R6.

Related structures: Spin(24) × Spin(6) is the universal cover of SO(24) × SO(6), with a kernel isomorphic to Z2 × Z2. The group is a compact semisimple Lie group and commonly arises in contexts involving products of orthogonal symmetries and their geometric or physical applications.