# Transitive groups downloaded from the LMFDB on 30 July 2026. # Search link: https://www.lmfdb.org/GaloisGroup/?n=38 # Query "{'n': 38}" returned 76 groups, sorted by degree. # Each entry in the following data list has the form: # [Label, Name, Order, Parity, Solvable, $\#\Aut(F/K)$, Subfields, Low Degree Siblings] # For more details, see the definitions at the bottom of the file. "38T1" "$C_{38}$" 38 -1 1 38 [[[2, 1], 1], [[19, 1], 1]] [[], 47] "38T2" NULL 38 -1 1 38 [[[2, 1], 1], [[19, 2], 1]] [[[[19, 2], 1]], 47] "38T3" "$D_{38}$" 76 -1 1 2 [[[2, 1], 1], [[19, 2], 1]] [[[[38, 3], 1]], 47] "38T4" NULL 114 -1 1 2 [[[2, 1], 1], [[19, 3], 1]] [[], 47] "38T5" NULL 114 -1 1 2 [[[2, 1], 1], [[19, 4], 1]] [[[[19, 4], 1]], 47] "38T6" NULL 228 -1 1 2 [[[2, 1], 1], [[19, 4], 1]] [[[[38, 6], 1]], 47] "38T7" NULL 342 -1 1 2 [[[2, 1], 1], [[19, 5], 1]] [[], 47] "38T8" NULL 342 -1 1 2 [[[2, 1], 1], [[19, 6], 1]] [[[[19, 6], 1]], 47] "38T9" NULL 684 -1 1 2 [[[2, 1], 1], [[19, 6], 1]] [[[[38, 9], 1]], 47] "38T10" NULL 722 -1 1 19 [[[2, 1], 1]] [[[[38, 10], 8]], 47] "38T11" NULL 1444 -1 1 1 [[[2, 1], 1]] [[[[38, 11], 8]], 47] "38T12" NULL 1444 1 1 1 [[[2, 1], 1]] [[[[38, 12], 9]], 47] "38T13" NULL 2166 -1 1 1 [[[2, 1], 1]] [[], 47] "38T14" NULL 2166 -1 1 1 [[[2, 1], 1]] [[[[38, 14], 8]], 47] "38T15" NULL 2888 -1 1 1 [[[2, 1], 1]] [[[[38, 15], 1]], 47] "38T16" NULL 4332 -1 1 1 [[[2, 1], 1]] [[], 47] "38T17" NULL 4332 1 1 1 [[[2, 1], 1]] [[], 47] "38T18" NULL 4332 1 1 1 [[[2, 1], 1]] [[[[38, 18], 9]], 47] "38T19" NULL 4332 -1 1 1 [[[2, 1], 1]] [[[[38, 19], 8]], 47] "38T20" NULL 6498 -1 1 1 [[[2, 1], 1]] [[], 47] "38T21" NULL 6498 -1 1 1 [[[2, 1], 1]] [[], 47] "38T22" NULL 6498 -1 1 1 [[[2, 1], 1]] [[[[38, 22], 8]], 47] "38T23" NULL 8664 -1 1 1 [[[2, 1], 1]] [[], 47] "38T24" NULL 8664 -1 1 1 [[[2, 1], 1]] [[[[38, 24], 1]], 47] "38T25" NULL 12996 -1 1 1 [[[2, 1], 1]] [[], 47] "38T26" NULL 12996 1 1 1 [[[2, 1], 1]] [[], 47] "38T27" NULL 12996 1 1 1 [[[2, 1], 1]] [[], 47] "38T28" NULL 12996 -1 1 1 [[[2, 1], 1]] [[], 47] "38T29" NULL 12996 1 1 1 [[[2, 1], 1]] [[[[38, 29], 9]], 47] "38T30" NULL 12996 -1 1 1 [[[2, 1], 1]] [[[[38, 30], 8]], 47] "38T31" NULL 19494 -1 1 1 [[[2, 1], 1]] [[], 47] "38T32" NULL 19494 -1 1 1 [[[2, 1], 1]] [[], 47] "38T33" "$\\PSL(2,37)$" 25308 1 0 1 [] [[], 47] "38T34" NULL 25992 -1 1 1 [[[2, 1], 1]] [[], 47] "38T35" NULL 25992 -1 1 1 [[[2, 1], 1]] [[], 47] "38T36" NULL 25992 -1 1 1 [[[2, 1], 1]] [[[[38, 36], 1]], 47] "38T37" NULL 38988 -1 1 1 [[[2, 1], 1]] [[], 47] "38T38" NULL 38988 1 1 1 [[[2, 1], 1]] [[], 47] "38T39" NULL 38988 1 1 1 [[[2, 1], 1]] [[], 47] "38T40" NULL 38988 -1 1 1 [[[2, 1], 1]] [[], 47] "38T41" NULL 50616 -1 0 1 [] [[], 47] "38T42" NULL 58482 -1 1 1 [[[2, 1], 1]] [[], 47] "38T43" NULL 77976 -1 1 1 [[[2, 1], 1]] [[], 47] "38T44" NULL 77976 -1 1 1 [[[2, 1], 1]] [[], 47] "38T45" NULL 116964 1 1 1 [[[2, 1], 1]] [[], 47] "38T46" NULL 116964 -1 1 1 [[[2, 1], 1]] [[], 47] "38T47" NULL 233928 -1 1 1 [[[2, 1], 1]] [[], 47] "38T48" NULL 4980736 1 1 2 [[[19, 1], 1]] [[[[38, 48], 13796]], 47] "38T49" NULL 9961472 -1 1 2 [[[19, 1], 1]] [[[[38, 49], 13796]], 47] "38T50" NULL 9961472 1 1 2 [[[19, 2], 1]] [[[[38, 50], 510], [[38, 51], 511]], 47] "38T51" NULL 9961472 -1 1 2 [[[19, 2], 1]] [[[[38, 50], 511], [[38, 51], 510]], 47] "38T52" NULL 14942208 1 1 2 [[[19, 3], 1]] [[[[38, 52], 62]], 47] "38T53" NULL 19922944 -1 1 2 [[[19, 2], 1]] [[[[38, 53], 1021]], 47] "38T54" NULL 29884416 -1 1 2 [[[19, 3], 1]] [[[[38, 54], 62]], 47] "38T55" NULL 29884416 -1 1 2 [[[19, 4], 1]] [[[[38, 55], 6], [[38, 56], 7]], 47] "38T56" NULL 29884416 1 1 2 [[[19, 4], 1]] [[[[38, 55], 7], [[38, 56], 6]], 47] "38T57" NULL 44826624 1 1 2 [[[19, 5], 1]] [[[[38, 57], 2]], 47] "38T58" NULL 59768832 -1 1 2 [[[19, 4], 1]] [[[[38, 58], 13]], 47] "38T59" NULL 89653248 -1 1 2 [[[19, 5], 1]] [[[[38, 59], 2]], 47] "38T60" NULL 89653248 -1 1 2 [[[19, 6], 1]] [[[[38, 61], 1]], 47] "38T61" NULL 89653248 1 1 2 [[[19, 6], 1]] [[[[38, 60], 1]], 47] "38T62" NULL 179306496 -1 1 2 [[[19, 6], 1]] [[[[38, 62], 1]], 47] "38T63" NULL 121645100408832000 -1 0 2 [[[2, 1], 1], [[19, 7], 1]] [[], 47] "38T64" NULL 121645100408832000 -1 0 2 [[[2, 1], 1], [[19, 8], 1]] [[[[19, 8], 1]], 47] "38T65" NULL 243290200817664000 -1 0 2 [[[2, 1], 1], [[19, 8], 1]] [[[[38, 65], 1]], 47] "38T66" NULL 15944266600786427904000 1 0 2 [[[19, 7], 1]] [[], 47] "38T67" NULL 31888533201572855808000 -1 0 2 [[[19, 7], 1]] [[], 47] "38T68" NULL 31888533201572855808000 -1 0 2 [[[19, 8], 1]] [[[[38, 69], 1]], 47] "38T69" NULL 31888533201572855808000 1 0 2 [[[19, 8], 1]] [[[[38, 68], 1]], 47] "38T70" NULL 63777066403145711616000 -1 0 2 [[[19, 8], 1]] [[[[38, 70], 1]], 47] "38T71" NULL 7398765226737409606771802112000000 -1 0 1 [[[2, 1], 1]] [[], 47] "38T72" NULL 14797530453474819213543604224000000 1 0 1 [[[2, 1], 1]] [[], 47] "38T73" NULL 14797530453474819213543604224000000 -1 0 1 [[[2, 1], 1]] [[], 47] "38T74" NULL 29595060906949638427087208448000000 -1 0 1 [[[2, 1], 1]] [[], 47] "38T75" "$A_{38}$" 261511308733300555880003612050037145600000000 1 0 1 [] [[], 47] "38T76" "$S_{38}$" 523022617466601111760007224100074291200000000 -1 0 1 [] [[], 47] # Label -- # Labels for Galois groups are of the form $\tt{nTt}$ where $n$ is the degree and $t$ is the $T$-number. #Name (pretty) -- # We describe abstract groups using standard building blocks: # # Groups $A$ and $B$ may be used to construct a larger group: # - $A\times B$ for the direct product of $A$ and $B$ # - $A:B$ for the semidirect product of $A$ and $B$ (with normal subgroup $A$) # - $A.B$ an extension with normal subgroup $A$ and quotient isomorphic to $B$ # - $A\wr B$ for the wreath product of A and B # Order -- # The **order** of a group is its cardinality as a set. # Parity -- # A Galois group $G\leq S_n$ has **parity** $1$ if $G\leq A_n$, and $-1$ otherwise. #Solvable (solv) -- # A group $G$ is **solvable** if there exists a chain of subgroups # \[ \langle e\rangle =H_0\leq H_1 \leq H_2 \leq \cdots \leq H_n=G\] # such that for all $i< n$, $H_i$ is a normal subgroup of $H_{i+1}$ (i.e., it is a subnormal series) and each quotient $H_{i+1}/H_i$ is abelian. #$\#\Aut(F/K)$ (auts) -- # Let $G$ be a transitive subgroup of $S_n$ and $G_1=\{\sigma\in G \subset S_n \mid \sigma(1)=1\}$. The group $G \subset S_n$ can be realized as the Galois group of a field extension $L/K$. Let $F \subset L$ be the fixed field of $G_1$ defined by the Galois action of $G$, i.e., $F = \{ a \in L \mid \forall g \in G_1,\, g(a) = a \}$. Then, $F/K$ is a field extension of degree $n$ whose Galois closure is $L/K$. # Although $F$ might not be Galois over $K$, one can consider the group $\Aut(F/K)$. # The group $\Aut(F/K)$ can be computed via the Galois correspondence. Group-theoretically, the order $\# \Aut(F/K)$ equals the order of the centralizer of $G$ in $S_n$. # If $F/K$ is realized as the extension $F = K[x]/(f(x))$ for some polynomial $f(x) \in K[x]$ then, the order $\#\Aut(F/K)$ is the number of roots of $f(x)$ in $F$. # Subfields -- # In terms of the Galois correspondence, a transitive group $G\leq S_n$ is the Galois group of an irreducible degree $n$ polynomial $f(x)\in K[x]$. Let $F=K(\alpha)$ where $\alpha$ is a root of $f(x)$. Then the **subfields** are intermediate fields for the extension $F/K$. #Low Degree Siblings (siblings) -- # An abstract group $G$ may be a transitive subgroup of $S_n$ for different $n$ and even in different (non-conjugate) ways for a given $n$. Each action is identified by a label of the form ```nTt``` where $n$ is the degree and ```t``` is the T-number classifying the action. # In terms of the Galois correspondence, the group $G$ corresponds to a degree $n$ extension $F/K$ where $F=K(\alpha)$, and $G$ is the Galois group of the splitting field of the monic irreducible polynomial for $\alpha$. The **siblings** correspond to sibling fields, which are not isomorphic to $F$, yet have the same normal closure. # There can be more than one action with the same transitive classification. This corresponds to non-isomorphic fields with the same degree, Galois group, and Galois closure. We indicate multiplicity using the notation "````nTt x k````" where there are ```k``` non-conjugate subgroups such that the action is ```nTt```.