# Transitive groups downloaded from the LMFDB on 12 May 2026.
# Search link: https://www.lmfdb.org/GaloisGroup/?n=5
# Query "{'n': 5}" returned 5 groups, sorted by degree.

# Each entry in the following data list has the form:
#    [Label, Name, Order, Parity, Solvable, $\#\Aut(F/K)$, Low Degree Siblings]
# For more details, see the definitions at the bottom of the file.



"5T1"	"$C_5$"	5	1	1	5	[[], 47]
"5T2"	"$D_{5}$"	10	1	1	1	[[[[10, 2], 1]], 47]
"5T3"	"$F_5$"	20	-1	1	1	[[[[10, 4], 1], [[20, 5], 1]], 47]
"5T4"	"$A_5$"	60	1	0	1	[[[[6, 12], 1], [[10, 7], 1], [[12, 33], 1], [[15, 5], 1], [[20, 15], 1], [[30, 9], 1]], 47]
"5T5"	"$S_5$"	120	-1	0	1	[[[[6, 14], 1], [[10, 12], 1], [[10, 13], 1], [[12, 74], 1], [[15, 10], 1], [[20, 30], 1], [[20, 32], 1], [[20, 35], 1], [[24, 202], 1], [[30, 22], 1], [[30, 25], 1], [[30, 27], 1], [[40, 62], 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:
#    <ul>
#     <li> $C_n$ denotes the cyclic group of order $n$
#     <li> $D_n$ denotes the dihedral group of order $2n$
#     <li> $A_n$ denotes the alternating group on $n$ letters
#     <li> $S_n$ denotes the symmetric group on $n$ letters
#    </ul>

#    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 $F$.


#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```.