2-transitive permutation groups of small degree

Question

I've recently been learning about 2-transitive permutation groups. To better my understanding of the classification of 2-transitive groups I'm trying to learn about 2-transitive permutation groups of small degree, say $ n \leq 30 $. To make this slightly more manageable I've confined myself to odd degree.

Is this the full list of 2-transitive permutation groups for odd degree $ n \leq 30 $ (excluding $ A_n,S_n $)? $ n $ is the degree $ k $ is the transitivity

$n$	$k$	Group
5	2	$AGL(1,5)$
7	2	$AGL(1,7)$
7	2	$PSL(3,2)$
9	2	$AGL(1,9)=3^2:8$
9	2	$AGL(2,3)=3^2:2S_4$
9	2	$ASL(2,3)=3^2:2A_4$
9	2	$3^2:Q_8$
9	2	$3^2:2D_8$
9	3	$PSL(2,8)$
9	3	$P\Gamma L(2,8)$
11	2	$AGL(1,11)$
11	2	$PSL(2,11)$
11	4	$M_{11}$
13	2	$AGL(1,13)$
13	2	$PSL(3,3)$
15	2	$PSL(4,2)\cong A_8$
15	2	$A_7$
17	2	$AGL(1,17)$
17	3	$PSL(2,16)\leq G \leq P\Gamma L(2,16)$ (3 total)
19	2	$AGL(1,19)$
21	2	$PSL(3,4)\leq G \leq P\Gamma L(3,4)$(4 total)
23	2	$AGL(1,23)$
23	4	$M_{23}$
25	2	$ASL(2,5)\leq G \leq A \Gamma L(2,5)$ (7 total)
25	2	$AGL(1,25)$
25	2	$A\Gamma L(1,25)$
27	2	$AGL(3,3)$
27	2	$ASL(3,3)$
27	2	$AGL(1,27)$
27	2	$A\Gamma L(1,27)$
29	2	$AGL(1,29)$

Edit: I updated the list using the answer/comments/GAP code from comments

Edit 2: reader beware I think I fixed the degree 27 groups but seems like my degree 25 groups still aren't right

Answer 1

No, these are not all. You are missing:

• Subgroups of $AGL(k,p)$ that contain the full $p^k$, and are semidirect product with a transitive complement (e.g. a Singer Cycle, or automorphism of the SL): $3^2:Q_8$, $3^2:2D_8$, $AGL(2,3)$. Groups containing $PSL(2,16)$, but with Galois automorphisms on top. 8 in degree 25 (amongst them $ASL(2,5):2$ and $AGL(2,5)$, $AGL(1,25)$, $A\Gamma L(1,25)$, in Degree 27 $AGL(3,3)$ and $A\Gamma L(1,27)$.

(The groups $\Gamma L$ arise by taking Galois automorphisms of SL in addition to GL.)

• $PSL(3,4)$ (and 3 supergroups) in degree $21=\frac{4^3-1}{4-1}$.