Hanani's original 1960 proof of the existence of Steiner quadruple systems $SQS(n)$ for all $n\equiv2,4\bmod6$ involves explicitly constructing an $SQS(14)$ and an $SQS(38)$ in what is otherwise a purely inductive proof. However, the listed $SQS(14)$ is devoid of any noticeable pattern and would certainly not be memorisable by a human – its automorphism group has order only $6$.
0125 038D 1236 157C 24AC 3579 479C
013B 039A 1247 1589 24BD 358B 5678
0146 0459 128B 15BD 257B 35AC 569B
0178 047B 129A 1679 258A 367B 56CD
019D 048A 12CD 168D 259C 3689 59AD
01AC 04CD 1345 16BC 267C 36AD 68AC
0234 057D 137D 17AB 269D 3BCD 789A
0268 058C 138A 235D 26AB 457A 78BC
0279 05AB 139C 237A 278D 458D 79BD
02AD 067A 148C 238C 346C 45BC 7ACD
02BC 069C 149B 239B 3478 467D 89CD
0356 06BD 14AD 2456 349D 468B 8ABD
037C 089B 156A 2489 34AB 469A 9ABC
Thus I want to find the $SQS(14)$ whose automorphism group is of the highest possible order, just like how two non-isomorphic solutions to Kirkman's schoolgirl problem have the largest possible automorphism group of $\operatorname{PSL}(2,7)$. The most symmetric $SQS(14)$ I have found so far has $C_7\rtimes C_6$ as its automorphism group, with its blocks falling into five orbits under the actions of $(1234567)(ABCDEFG)$ and $(1A)(2F5G3D)(4B6E7C)$. The orbit representatives are $$1235,126C,12AB,12DE,12FG$$ and the blocks are
1235 2346 3457 4561 5672 6713 7124
AGFD GFEC FEDB EDCA DCBG CBAF BAGE
126C 237D 341E 452F 563G 674A 715B
AGC6 GFB5 FEA4 EDG3 DCF2 CBE1 BAD7
12AB 23BC 34CD 45DE 56EF 67FG 71GA
15AE 52EB 26BF 63FC 37CG 74GD 41DA
13AC 35CE 57EG 72GB 24BD 46DF 61FA
12DE 23EF 34FG 45GA 56AB 67BC 71CD
15FC 52CG 26GD 63DA 37AE 74EB 41BF
13GB 35BD 57DF 72FA 24AC 46CE 61EG
12FG 23GA 34AB 45BC 56CD 67DE 71EF
15GD 52DA 26AE 63EB 37BF 74FC 41CG
13DF 35FA 57AC 72CE 24EG 46GB 61BD
Is there an $SQS(14)$ whose automorphism group has more than $42$ elements, or whose blocks fall into fewer than $5$ orbits?
The SQS(14) that you have found is the most symmetric example.
The SQS(14) have been completely classified in [1] by a computer search, by extending the two nonisomorphic STS(13) examples in all possible ways. [To do this, a new point is added to every block of the STS(13), and then a computer is used to construct a collection of new blocks in all possible ways which will complete this to an SQS(14).]
While I cannot find an original copy of the paper classifying these examples online, the results are summarized in the survey paper [2], which mentions that the four examples have automorphism groups of order 42, 14, and 6 (with two nonisomorphic examples having automorphism groups of order 6).
[1]: Mendelsohn, N. S.; Hung, Stephen H. Y., On the Steiner systems S(3, 4, 14) and S(4, 5, 15), Util. Math. 1, 5-95 (1972). ZBL0258.05017.
[2]: Lindner, Charles C.; Rosa, Alexander, Steiner quadruple systems - a survey, Discrete Math. 22, 147-181 (1978). ZBL0398.05015.