# Transitive groups downloaded from the LMFDB on 11 December 2023.
# Search link: https://www.lmfdb.org/GaloisGroup/
# Query "{'n': 31}" returned 12 groups, sorted by degree.
# Each entry in the following data list has the form:
# [Label, Name, Order, Parity, Solvable, Subfields, Low Degree Siblings]
# For more details, see the definitions at the bottom of the file.
# To create a list of groups, type "groups = make_data()"
columns = ["label", "pretty", "order", "parity", "solv", "subfields", "siblings"]
data = [
["31T1", "$C_{31}$", 31, 1, 1, [], [[], 47]],
["31T2", "$D_{31}$", 62, -1, 1, [], [[], 47]],
["31T3", "$C_{31}:C_{3}$", 93, 1, 1, [], [[], 47]],
["31T4", "$C_{31}:C_{5}$", 155, 1, 1, [], [[], 47]],
["31T5", "$C_{31}:C_{6}$", 186, -1, 1, [], [[], 47]],
["31T6", "$C_{31}:C_{10}$", 310, -1, 1, [], [[], 47]],
["31T7", "$C_{31}:C_{15}$", 465, 1, 1, [], [[], 47]],
["31T8", "$F_{31}$", 930, -1, 1, [], [[], 47]],
["31T9", "$\\PSL(3,5)$", 372000, 1, 0, [], [[[[31, 9], 1]], 47]],
["31T10", "$\\PSL(5,2)$", 9999360, 1, 0, [], [[[[31, 10], 1]], 47]],
["31T11", "$A_{31}$", 4111419327088961408862781440000000, 1, 0, [], [[], 47]],
["31T12", "$S_{31}$", 8222838654177922817725562880000000, -1, 0, [], [[], 47]]
]
def create_record(row):
out = {col: val for col, val in zip(columns, row)}
TransitiveGroup(*[int(c) for c in out["label"].split("T")])
out["group"] = group
return out
def make_data():
return [create_record(row) for row in data]
# 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:
#
# - $C_n$ denotes the cyclic group of order $n$
#
- $D_n$ denotes the dihedral group of order $2n$
#
- $A_n$ denotes the alternating group on $n$ letters
#
- $S_n$ denotes the symmetric group on $n$ letters
#

# 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