Identification
Fano variety 223
 blowup of 116 in an intersection of $A\in\mathcal{O}_Q(1)$ and $B\in\mathcal{O}_Q(2)$ such that $A$ is smooth
 blowup of 116 in an intersection of $A\in\mathcal{O}_Q(1)$ and $B\in\mathcal{O}_Q(2)$ such that $A$ is singular
 Picard rank
 2 (others)
 $\mathrm{K}_X^3$
 30
 $\mathrm{h}^{1,2}(X)$
 1
Hodge diamond
1
0 0
0 2 0
0 1 1 0
0 2 0
0 0
1
0 0
0 2 0
0 1 1 0
0 2 0
0 0
1
1
0 0
0 2 11
0 0 0 18
0 0 0
0 0
0
0 0
0 2 11
0 0 0 18
0 0 0
0 0
0
Anticanonical bundle
 index
 1
 $\dim\mathrm{H}^0(X,\omega_X^\vee)$
 18
 $\mathrm{K}_X$ very ample?
 yes
 $\mathrm{K}_X$ basepoint free?
 yes
 hyperelliptic
 no
 trigonal
 no
Birational geometry
Deformation theory
 number of moduli

 2
 1
$\mathrm{Aut}^0(X)$  $\dim\mathrm{Aut}^0(X)$  number of moduli 

$0$  0  2 
Period sequence
Extremal contractions
Semiorthogonal decompositions
There exist interesting semiorthogonal decompositions, but this data is not yet added.
Structure of quantum cohomology
By Hertling–Manin–Teleman we have that quantum cohomology cannot be generically semisimple, as $\mathrm{h}^{1,2}\neq 0$.
Zero section description
Fano 3folds from homogeneous vector bundles over Grassmannians gives the following description(s):
 variety
 $\mathbb{P}^4 \times \mathbb{P}^5$
 bundle
 $\mathcal{Q}_{\mathbb{P}^4}(0,1) \oplus \mathcal{O}(2,0) \oplus \mathcal{O}(1,1)$
 variety
 $\operatorname{Fl}(1,2,6)$
 bundle
 $\mathcal{Q}_2 \oplus \mathcal{O}(0,2) \oplus \mathcal{O}(1,1)$
See the big table for more information.