[oxofrygy] Finite simple groups of the 24-cell

SOME REMARKS ON THE ALGEBRAIC STRUCTURE OF THE FINITE COXETER GROUP F4, by MUHAMMAD A. ALBAR and NORAH AL-SALEH

The Coxeter group F4 (order 1152) is the split extension of the Coxeter group D4 by S3 (order 3! = 6). D4 is the split extension of Z₂³ (order 2³ = 8) by S4 (order 4! = 24).

Thus 1152 = 6 * 8 * 24.

(Notation made searching for this complicated. There exist finite simple groups called F4(q), and ²F4(2²ⁿ⁺¹). "The 24 vertices of the 24-cell represent the root vectors of the simple Lie group D4." and there also exist finite simple groups named Dn(q), ²Dn(q²), ³D4(q³).)