Table of Models

Model	Acting Group	Transitive?	Permutation Group Code	Orbits (Coset of …)?	Regular?
Model A (Classical)	$S ≅ D_{4} \times Z_{2}$	$✓$	$8 T_{9}$	$S ≅ D_{4} \times Z_{2}$	$\times$
Model A2	$S ≅ D_{4} \times Z_{2}$	$✓$	$16 T_{9}$	$S ≅ D_{4} \times Z_{2}$	$✓$
Model A (Presence Cube)	$S ≅ D_{4} \times Z_{2}$	$\times$	N/A	$Democratic/Aristocratic ≅ Z_{2}^{3}$	$\times$
Model A (SSS)	$S ≅ D_{4} \times Z_{2}$	$\times$	N/A	$Asking/Declaring ≅ D_{4}$	$\times$
Model A (SCS)	$S ≅ D_{4} \times Z_{2}$	$\times$	N/A	$Process/Result ≅ Z_{2} \times Z_{4}$	$\times$
Model G (SHS)	$S ≅ D_{4} \times Z_{2}$	$\times$	N/A	$Process/Result ≅ Z_{2} \times Z_{4}$	$\times$
Model L	$V_{15} ≅ Z_{2}^{4}$	$✓$	$16 T_{3}$	$V_{15} ≅ Z_{2}^{4}$	$✓$
Model W	$V_{14} ≅ Z_{2}^{4}$	$✓$	$16 T_{3}$	$V_{14} ≅ Z_{2}^{4}$	$✓$

Footnote

$8 T_{n}$ refers to the nth transitive subgroup of $S_{8}$ in the GAP classification scheme.
$16 T_{n}$ refers to the nth transitive subgroup of $S_{16}$ in the GAP classification scheme.
Technical Note: Systems of injections (charge models with only 8 functions) are best viewed as orbit models. Any compatible index-2 subgroup $H$ acts transitively on a single orbit of size 8 (hence an $8 T_{k}$ label applies). The full charge construction $F \times {A, B}$ is an intransitive $H$ -set with two orbits; it is not itself an $n T_{k}$ object unless extended by a sheet-swapping element, yielding a transitive $16 T_{l}$ action.