# Transitive groups downloaded from the LMFDB on 27 July 2026.
# Search link: https://www.lmfdb.org/GaloisGroup/?n=33
# Query "{'n': 33}" returned 162 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.



"33T1"	"$C_{33}$"	33	1	1	33	[[[3, 1], 1], [[11, 1], 1]]	[[], 47]
"33T2"	NULL	66	-1	1	11	[[[3, 2], 1], [[11, 1], 1]]	[[], 47]
"33T3"	"$D_{33}$"	66	1	1	1	[[[3, 2], 1], [[11, 2], 1]]	[[], 47]
"33T4"	NULL	66	-1	1	3	[[[3, 1], 1], [[11, 2], 1]]	[[], 47]
"33T5"	NULL	132	-1	1	1	[[[3, 2], 1], [[11, 2], 1]]	[[], 47]
"33T6"	NULL	165	1	1	3	[[[3, 1], 1], [[11, 3], 1]]	[[], 47]
"33T7"	NULL	330	-1	1	1	[[[3, 2], 1], [[11, 3], 1]]	[[], 47]
"33T8"	NULL	330	-1	1	3	[[[3, 1], 1], [[11, 4], 1]]	[[], 47]
"33T9"	NULL	330	1	1	1	[[[3, 2], 1], [[11, 4], 1]]	[[], 47]
"33T10"	NULL	363	1	1	11	[[[3, 1], 1]]	[[[[33, 10], 3]], 47]
"33T11"	NULL	660	-1	1	1	[[[3, 2], 1], [[11, 4], 1]]	[[], 47]
"33T12"	NULL	726	-1	1	1	[[[3, 1], 1]]	[[[[33, 12], 3]], 47]
"33T13"	NULL	726	-1	1	11	[[[3, 2], 1]]	[[[[33, 14], 1]], 47]
"33T14"	NULL	726	1	1	1	[[[3, 2], 1]]	[[[[33, 13], 1]], 47]
"33T15"	NULL	1452	-1	1	1	[[[3, 2], 1]]	[[[[33, 15], 1]], 47]
"33T16"	NULL	1815	1	1	1	[[[3, 1], 1]]	[[[[33, 16], 3]], 47]
"33T17"	NULL	1980	1	0	3	[[[3, 1], 1], [[11, 5], 1]]	[[[[33, 17], 1], [[36, 3021], 1]], 47]
"33T18"	NULL	2673	1	1	3	[[[11, 1], 1]]	[[[[33, 18], 10]], 47]
"33T19"	NULL	3630	-1	1	1	[[[3, 1], 1]]	[[[[33, 19], 3]], 47]
"33T20"	NULL	3630	1	1	1	[[[3, 2], 1]]	[[[[33, 21], 1]], 47]
"33T21"	NULL	3630	-1	1	1	[[[3, 2], 1]]	[[[[33, 20], 1]], 47]
"33T22"	NULL	3960	-1	0	1	[[[3, 2], 1], [[11, 5], 1]]	[[[[33, 22], 1], [[36, 4977], 1]], 47]
"33T23"	NULL	3993	1	1	11	[[[3, 1], 1]]	[[[[33, 23], 39]], 47]
"33T24"	NULL	5346	-1	1	1	[[[11, 1], 1]]	[[[[33, 24], 10]], 47]
"33T25"	NULL	7260	-1	1	1	[[[3, 2], 1]]	[[[[33, 25], 1]], 47]
"33T26"	NULL	7986	-1	1	1	[[[3, 1], 1]]	[[[[33, 26], 39]], 47]
"33T27"	NULL	7986	-1	1	11	[[[3, 2], 1]]	[[[[33, 27], 9]], 47]
"33T28"	NULL	7986	1	1	1	[[[3, 2], 1]]	[[[[33, 28], 9]], 47]
"33T29"	NULL	8019	1	1	3	[[[11, 1], 1]]	[[[[33, 29], 21]], 47]
"33T30"	NULL	13365	1	1	3	[[[11, 3], 1]]	[[], 47]
"33T31"	NULL	15972	1	1	1	[[[3, 1], 1]]	[[[[44, 142], 1]], 47]
"33T32"	NULL	15972	-1	1	1	[[[3, 2], 1]]	[[[[33, 32], 9]], 47]
"33T33"	NULL	16038	-1	1	1	[[[11, 1], 1]]	[[[[33, 33], 21]], 47]
"33T34"	NULL	19965	1	1	1	[[[3, 1], 1]]	[[[[33, 34], 39]], 47]
"33T35"	NULL	23760	1	0	3	[[[3, 1], 1], [[11, 6], 1]]	[[[[36, 12768], 1]], 47]
"33T36"	NULL	26730	-1	1	1	[[[11, 3], 1]]	[[], 47]
"33T37"	NULL	31944	-1	1	1	[[[3, 1], 1]]	[[[[44, 229], 1]], 47]
"33T38"	NULL	31944	-1	1	1	[[[3, 2], 1]]	[[[[44, 231], 1]], 47]
"33T39"	NULL	31944	1	1	1	[[[3, 2], 1]]	[[[[44, 230], 1]], 47]
"33T40"	"$\\PSL(2,32)$"	32736	1	0	1	[]	[[], 47]
"33T41"	NULL	39930	-1	1	1	[[[3, 1], 1]]	[[[[33, 41], 39]], 47]
"33T42"	NULL	39930	-1	1	1	[[[3, 2], 1]]	[[[[33, 42], 9]], 47]
"33T43"	NULL	39930	1	1	1	[[[3, 2], 1]]	[[[[33, 43], 9]], 47]
"33T44"	NULL	40095	1	1	3	[[[11, 3], 1]]	[[[[33, 44], 1]], 47]
"33T45"	NULL	47520	-1	0	1	[[[3, 2], 1], [[11, 6], 1]]	[[[[36, 16207], 1]], 47]
"33T46"	NULL	63888	-1	1	1	[[[3, 2], 1]]	[[[[44, 293], 1]], 47]
"33T47"	NULL	79860	1	1	1	[[[3, 1], 1]]	[[[[44, 296], 1]], 47]
"33T48"	NULL	79860	-1	1	1	[[[3, 2], 1]]	[[[[33, 48], 9]], 47]
"33T49"	NULL	80190	-1	1	1	[[[11, 3], 1]]	[[[[33, 49], 1]], 47]
"33T50"	NULL	99825	1	1	1	[[[3, 1], 1]]	[[], 47]
"33T51"	NULL	159720	-1	1	1	[[[3, 1], 1]]	[[[[44, 326], 1]], 47]
"33T52"	NULL	159720	-1	1	1	[[[3, 2], 1]]	[[[[44, 327], 1]], 47]
"33T53"	NULL	159720	1	1	1	[[[3, 2], 1]]	[[[[44, 328], 1]], 47]
"33T54"	NULL	160380	1	0	3	[[[11, 5], 1]]	[[], 47]
"33T55"	NULL	163680	1	0	1	[]	[[], 47]
"33T56"	NULL	199650	-1	1	1	[[[3, 1], 1]]	[[], 47]
"33T57"	NULL	199650	1	1	1	[[[3, 2], 1]]	[[], 47]
"33T58"	NULL	199650	-1	1	1	[[[3, 2], 1]]	[[], 47]
"33T59"	NULL	319440	-1	1	1	[[[3, 2], 1]]	[[[[44, 358], 1]], 47]
"33T60"	NULL	320760	-1	0	1	[[[11, 5], 1]]	[[], 47]
"33T61"	NULL	399300	1	1	1	[[[3, 1], 1]]	[[], 47]
"33T62"	NULL	399300	-1	1	1	[[[3, 2], 1]]	[[], 47]
"33T63"	NULL	481140	1	0	3	[[[11, 5], 1]]	[[[[33, 63], 1]], 47]
"33T64"	NULL	499125	1	1	1	[[[3, 1], 1]]	[[], 47]
"33T65"	NULL	649539	1	1	3	[[[11, 1], 1]]	[[[[33, 65], 2661]], 47]
"33T66"	NULL	798600	-1	1	1	[[[3, 1], 1]]	[[], 47]
"33T67"	NULL	798600	-1	1	1	[[[3, 2], 1]]	[[], 47]
"33T68"	NULL	798600	1	1	1	[[[3, 2], 1]]	[[], 47]
"33T69"	NULL	962280	-1	0	1	[[[11, 5], 1]]	[[[[33, 69], 1]], 47]
"33T70"	NULL	998250	-1	1	1	[[[3, 1], 1]]	[[], 47]
"33T71"	NULL	998250	1	1	1	[[[3, 2], 1]]	[[], 47]
"33T72"	NULL	998250	-1	1	1	[[[3, 2], 1]]	[[], 47]
"33T73"	NULL	1299078	-1	1	1	[[[11, 1], 1]]	[[[[33, 73], 2661]], 47]
"33T74"	NULL	1299078	1	1	1	[[[11, 2], 1]]	[[[[33, 74], 120], [[33, 75], 121]], 47]
"33T75"	NULL	1299078	-1	1	3	[[[11, 2], 1]]	[[[[33, 74], 121], [[33, 75], 120]], 47]
"33T76"	NULL	1597200	-1	1	1	[[[3, 2], 1]]	[[], 47]
"33T77"	NULL	1924560	1	0	1	[[[11, 6], 1]]	[[], 47]
"33T78"	NULL	1948617	1	1	3	[[[11, 1], 1]]	[[[[33, 78], 5323]], 47]
"33T79"	NULL	1996500	1	1	1	[[[3, 1], 1]]	[[], 47]
"33T80"	NULL	1996500	-1	1	1	[[[3, 2], 1]]	[[], 47]
"33T81"	NULL	2598156	-1	1	1	[[[11, 2], 1]]	[[[[33, 81], 241]], 47]
"33T82"	NULL	3247695	1	1	3	[[[11, 3], 1]]	[[[[33, 82], 1]], 47]
"33T83"	NULL	3849120	-1	0	1	[[[11, 6], 1]]	[[], 47]
"33T84"	NULL	3897234	-1	1	1	[[[11, 1], 1]]	[[[[33, 84], 5323]], 47]
"33T85"	NULL	3897234	1	1	1	[[[11, 2], 1]]	[[[[33, 85], 241]], 47]
"33T86"	NULL	3897234	-1	1	3	[[[11, 2], 1]]	[[[[33, 86], 241]], 47]
"33T87"	NULL	3993000	-1	1	1	[[[3, 1], 1]]	[[], 47]
"33T88"	NULL	3993000	1	1	1	[[[3, 2], 1]]	[[], 47]
"33T89"	NULL	3993000	-1	1	1	[[[3, 2], 1]]	[[], 47]
"33T90"	NULL	5773680	1	0	1	[[[11, 6], 1]]	[[], 47]
"33T91"	NULL	6495390	-1	1	1	[[[11, 3], 1]]	[[[[33, 91], 1]], 47]
"33T92"	NULL	6495390	-1	1	3	[[[11, 4], 1]]	[[[[33, 93], 1]], 47]
"33T93"	NULL	6495390	1	1	1	[[[11, 4], 1]]	[[[[33, 92], 1]], 47]
"33T94"	NULL	7794468	-1	1	1	[[[11, 2], 1]]	[[[[33, 94], 241]], 47]
"33T95"	NULL	7986000	-1	1	1	[[[3, 2], 1]]	[[], 47]
"33T96"	NULL	9743085	1	1	3	[[[11, 3], 1]]	[[[[33, 96], 3]], 47]
"33T97"	NULL	11547360	-1	0	1	[[[11, 6], 1]]	[[], 47]
"33T98"	NULL	12990780	-1	1	1	[[[11, 4], 1]]	[[[[33, 98], 1]], 47]
"33T99"	NULL	19486170	-1	1	1	[[[11, 3], 1]]	[[[[33, 99], 3]], 47]
"33T100"	NULL	19486170	-1	1	3	[[[11, 4], 1]]	[[[[33, 100], 1]], 47]
"33T101"	NULL	19486170	1	1	1	[[[11, 4], 1]]	[[[[33, 101], 1]], 47]
"33T102"	NULL	38972340	-1	1	1	[[[11, 4], 1]]	[[[[33, 102], 1]], 47]
"33T103"	NULL	38972340	1	0	3	[[[11, 5], 1]]	[[], 47]
"33T104"	NULL	59875200	1	0	3	[[[3, 1], 1], [[11, 7], 1]]	[[], 47]
"33T105"	NULL	77944680	-1	0	1	[[[11, 5], 1]]	[[], 47]
"33T106"	NULL	116917020	1	0	3	[[[11, 5], 1]]	[[[[33, 106], 1]], 47]
"33T107"	NULL	119750400	-1	0	1	[[[3, 2], 1], [[11, 7], 1]]	[[], 47]
"33T108"	NULL	119750400	-1	0	3	[[[3, 1], 1], [[11, 8], 1]]	[[], 47]
"33T109"	NULL	119750400	1	0	1	[[[3, 2], 1], [[11, 8], 1]]	[[], 47]
"33T110"	NULL	233834040	-1	0	1	[[[11, 5], 1]]	[[[[33, 110], 1]], 47]
"33T111"	NULL	239500800	-1	0	1	[[[3, 2], 1], [[11, 8], 1]]	[[], 47]
"33T112"	NULL	467668080	1	0	3	[[[11, 6], 1]]	[[], 47]
"33T113"	NULL	862488000	1	0	1	[[[3, 1], 1]]	[[[[33, 113], 1], [[36, 92835], 1]], 47]
"33T114"	NULL	935336160	-1	0	1	[[[11, 6], 1]]	[[], 47]
"33T115"	NULL	1403004240	1	0	3	[[[11, 6], 1]]	[[[[33, 115], 1]], 47]
"33T116"	NULL	1403004240	1	0	1	[[[11, 6], 1]]	[[], 47]
"33T117"	NULL	1724976000	-1	0	1	[[[3, 2], 1]]	[[[[33, 117], 1], [[36, 97582], 1]], 47]
"33T118"	NULL	1995383808	1	1	1	[[[11, 1], 1]]	[[], 47]
"33T119"	NULL	2806008480	-1	0	1	[[[11, 6], 1]]	[[[[33, 119], 1]], 47]
"33T120"	NULL	2806008480	-1	0	1	[[[11, 6], 1]]	[[], 47]
"33T121"	NULL	3990767616	-1	1	1	[[[11, 1], 1]]	[[], 47]
"33T122"	NULL	3990767616	-1	1	1	[[[11, 2], 1]]	[[], 47]
"33T123"	NULL	3990767616	1	1	1	[[[11, 2], 1]]	[[], 47]
"33T124"	NULL	7981535232	-1	1	1	[[[11, 2], 1]]	[[], 47]
"33T125"	NULL	9976919040	1	1	1	[[[11, 3], 1]]	[[], 47]
"33T126"	NULL	19953838080	-1	1	1	[[[11, 3], 1]]	[[], 47]
"33T127"	NULL	19953838080	1	1	1	[[[11, 4], 1]]	[[], 47]
"33T128"	NULL	19953838080	-1	1	1	[[[11, 4], 1]]	[[], 47]
"33T129"	NULL	39907676160	-1	1	1	[[[11, 4], 1]]	[[], 47]
"33T130"	NULL	119723028480	1	0	1	[[[11, 5], 1]]	[[], 47]
"33T131"	NULL	239446056960	-1	0	1	[[[11, 5], 1]]	[[], 47]
"33T132"	NULL	1178523561600	1	0	3	[[[11, 7], 1]]	[[], 47]
"33T133"	NULL	1436676341760	1	0	1	[[[11, 6], 1]]	[[], 47]
"33T134"	NULL	1490379264000	1	0	1	[[[3, 1], 1]]	[[[[36, 119673], 1]], 47]
"33T135"	NULL	2357047123200	-1	0	1	[[[11, 7], 1]]	[[], 47]
"33T136"	NULL	2357047123200	-1	0	3	[[[11, 8], 1]]	[[], 47]
"33T137"	NULL	2357047123200	1	0	1	[[[11, 8], 1]]	[[], 47]
"33T138"	NULL	2873352683520	-1	0	1	[[[11, 6], 1]]	[[], 47]
"33T139"	NULL	2980758528000	-1	0	1	[[[3, 2], 1]]	[[[[36, 120048], 1]], 47]
"33T140"	NULL	3535570684800	1	0	3	[[[11, 7], 1]]	[[[[33, 140], 1]], 47]
"33T141"	NULL	4714094246400	-1	0	1	[[[11, 8], 1]]	[[], 47]
"33T142"	NULL	7071141369600	-1	0	1	[[[11, 7], 1]]	[[[[33, 142], 1]], 47]
"33T143"	NULL	7071141369600	-1	0	3	[[[11, 8], 1]]	[[[[33, 143], 1]], 47]
"33T144"	NULL	7071141369600	1	0	1	[[[11, 8], 1]]	[[[[33, 144], 1]], 47]
"33T145"	NULL	14142282739200	-1	0	1	[[[11, 8], 1]]	[[[[33, 145], 1]], 47]
"33T146"	NULL	3620424381235200	1	0	1	[[[11, 7], 1]]	[[], 47]
"33T147"	NULL	7240848762470400	-1	0	1	[[[11, 7], 1]]	[[], 47]
"33T148"	NULL	7240848762470400	1	0	1	[[[11, 8], 1]]	[[], 47]
"33T149"	NULL	7240848762470400	-1	0	1	[[[11, 8], 1]]	[[], 47]
"33T150"	NULL	14481697524940800	-1	0	1	[[[11, 8], 1]]	[[], 47]
"33T151"	NULL	23850551284826112000000	1	0	1	[[[3, 1], 1]]	[[], 47]
"33T152"	NULL	47701102569652224000000	-1	0	1	[[[3, 1], 1]]	[[], 47]
"33T153"	NULL	47701102569652224000000	1	0	1	[[[3, 2], 1]]	[[], 47]
"33T154"	NULL	47701102569652224000000	-1	0	1	[[[3, 2], 1]]	[[], 47]
"33T155"	NULL	95402205139304448000000	1	0	1	[[[3, 1], 1]]	[[], 47]
"33T156"	NULL	95402205139304448000000	-1	0	1	[[[3, 2], 1]]	[[], 47]
"33T157"	NULL	190804410278608896000000	-1	0	1	[[[3, 1], 1]]	[[], 47]
"33T158"	NULL	190804410278608896000000	-1	0	1	[[[3, 2], 1]]	[[], 47]
"33T159"	NULL	190804410278608896000000	1	0	1	[[[3, 2], 1]]	[[], 47]
"33T160"	NULL	381608820557217792000000	-1	0	1	[[[3, 2], 1]]	[[], 47]
"33T161"	"$A_{33}$"	4341658809405943247759097200640000000	1	0	1	[]	[[], 47]
"33T162"	"$S_{33}$"	8683317618811886495518194401280000000	-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:
#    <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 $F$.


# 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```.