Fano threefolds with $\rho=4$
ID  $\mathrm{K}_X^3$  $\mathrm{h}^{1,2}$  description  blowups  blowdowns  rational  unirational  moduli  $\mathrm{Aut}^0$  

41  24  1 
divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of degree $(1,1,1,1)$ 
327  yes  yes  3  $0$  
42  28  1 
blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the disjoint union of the vertex and an elliptic curve on the quadric 
331  yes  yes  1  $\mathbb{G}_{\mathrm{m}}$  
43  30  0 
blowup of 327 in a curve of degree $(1,1,2)$ 
317, 327, 328  yes  yes  0  $\mathbb{G}_{\mathrm{m}}$  
44  32  0 
blowup of 319 in the proper transform of a conic through the points 
*  318, 319, 330  yes  yes  0  $\mathbb{G}_{\mathrm{m}}^2$  
45  32  0 
blowup of 234 in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$ 
321, 328, 331  yes  yes  0  $\mathbb{G}_{\mathrm{m}}^2$  
46  34  0 
blowup of 117 in the disjoint union of 3 lines

325, 327  yes  yes  0  $\mathrm{PGL}_2$  
47  36  0 
blowup of 232 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$ 
324, 328  yes  yes  0  $\mathrm{GL}_2$  
48  38  0 
blowup of 327 in a curve of degree $(0,1,1)$ 
327, 331  yes  yes  0  $\mathrm{B}\times\mathrm{PGL}_2$  
49  40  0 
blowup of 325 in an exceptional curve of the blowup 
*  325, 326, 328, 330  yes  yes  0  $\mathrm{PGL}_{(2,2);1}$  
410  42  0 
$\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$ 
*  327, 328  yes  yes  0  $\mathrm{PGL}_2\times\mathrm{B}^2$  
411  44  0 
blowup of 328 in $\{x\}\times E$, $x\in\mathbb{P}^1$ and $E$ the $(1)$curve 
*  328, 331  yes  yes  0  $\mathrm{B}\times\mathrm{PGL}_{3;1}$  
412  46  0 
blowup of 233 in the disjoint union of two exceptional lines of the blowup 
*  330  yes  yes  0  $\mathbb{G}_{\mathrm{a}}^4\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$  
413  26  0 
blowup of 327 in a curve of degree $(1,1,3)$ 
327, 331  yes  yes  1 
