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The results below are complete, since the LMFDB contains all transitive groups of degree at most 47 (except 32)

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Results (displaying both matches)

  displayed columns for results
Label Name Order Parity Solvable Subfields Low Degree Siblings
36T46557 $A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$ $2654208$ $-1$ $C_2$, $S_3$, $D_{6}$, $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$, $C_2\times C_3^3:S_4$ 36T46557 x 3, 36T46638 x 4
36T46638 $A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$ $2654208$ $-1$ $S_3$, $S_4$, $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$, $(C_3\times C_6^2):S_4$ 36T46557 x 4, 36T46638 x 3
  displayed columns for results

Results are complete for degrees $\leq 23$.