6ef2f6c4a0fa7e52da9f4abff6d899d0250eeabde1a601d53c40052a841e7039
The multiplication trace form is tr(Lx Ly)=(7/30)<x,y>. The 45 axes are exactly the nonzero idempotents of minimum squared norm 480/49. Therefore the full unmarked automorphism group is O⁻₆(2) ≅ U₄(2):2, order 51,840.
M2(Q) + Q + M2(Q) + Q + M3(Q), with explicit irreducible modules of dimensions 1, 15a, 15b, 24, and 81. The 200-point module is 1² + 15a² + 15b + 24³ + 81.
The degree-480 character has norm 23 and known part 1 + 2·81 + 2·15a + 15b + 3·24. The orthogonal residual has degree 200 and norm four.
The former calculation overflowed signed 64-bit integers before modular reduction. The replacement is stable over six primes.
| Dimension | Weighted generating-set count | Subalgebra orbit size | Contained axes |
|---|---|---|---|
| 4 | 135 | 27 | 5 |
| 5 | 720 | 720 | 4 |
| 6 | 1,080 | 120 | 6 |
| 10 | 16,740 | 540 | 7 |
| 12 | 5,040 | 40 | 9 |
| 14 | 27,000 | 36 | 15 |
| 16 | 14,040 | 45 | 13 |
| 24 | 84,240 | 1 | 45 |
All eight species are simple and nonunital, with full multiplication envelope.
| Mechanism | Center | Distinguishing data |
|---|---|---|
| W(D₄) | 2 | derived order 96; census 1,43,32,84,32 |
| D₈ × S₄ | 2 | derived order 24; order-12 elements |
| Octonion axis-line stabilizer | 1 | 48 order-8 elements; free 192-torsor |
| Exceptional tomotope 2⁴:D₁₂ | 1 | normal C₂⁴; split outer extension |
Boundary. No Monster words or class fusion, complete residual decomposition, complete tensor table, remote CI/PDF result, hardware result, laboratory result, or physical mechanism is asserted.