# Transitive groups downloaded from the LMFDB on 22 September 2026.
# Search link: https://www.lmfdb.org/GaloisGroup/?n=34
# Query "{'n': 34}" returned 115 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.



"34T1"	"$C_{34}$"	34	-1	1	34	[[[2, 1], 1], [[17, 1], 1]]	[[], 47]
"34T2"	NULL	34	-1	1	34	[[[2, 1], 1], [[17, 2], 1]]	[[[[17, 2], 1]], 47]
"34T3"	"$D_{34}$"	68	-1	1	2	[[[2, 1], 1], [[17, 2], 1]]	[[[[34, 3], 1]], 47]
"34T4"	NULL	68	-1	1	2	[[[2, 1], 1], [[17, 3], 1]]	[[[[17, 3], 1]], 47]
"34T5"	NULL	136	-1	1	2	[[[2, 1], 1], [[17, 3], 1]]	[[[[34, 5], 1]], 47]
"34T6"	NULL	136	-1	1	2	[[[2, 1], 1], [[17, 4], 1]]	[[[[17, 4], 1]], 47]
"34T7"	NULL	272	-1	1	2	[[[2, 1], 1], [[17, 4], 1]]	[[[[34, 7], 1]], 47]
"34T8"	NULL	272	-1	1	2	[[[2, 1], 1], [[17, 5], 1]]	[[[[17, 5], 1]], 47]
"34T9"	NULL	544	-1	1	2	[[[2, 1], 1], [[17, 5], 1]]	[[[[34, 9], 1]], 47]
"34T10"	NULL	578	-1	1	17	[[[2, 1], 1]]	[[[[34, 10], 7]], 47]
"34T11"	NULL	1156	-1	1	1	[[[2, 1], 1]]	[[[[34, 11], 7]], 47]
"34T12"	NULL	1156	-1	1	1	[[[2, 1], 1]]	[[[[34, 12], 7]], 47]
"34T13"	NULL	2312	-1	1	1	[[[2, 1], 1]]	[[[[34, 15], 2]], 47]
"34T14"	NULL	2312	-1	1	1	[[[2, 1], 1]]	[[[[34, 14], 2]], 47]
"34T15"	NULL	2312	-1	1	1	[[[2, 1], 1]]	[[[[34, 13], 1], [[34, 15], 1]], 47]
"34T16"	NULL	2312	-1	1	1	[[[2, 1], 1]]	[[[[34, 16], 7]], 47]
"34T17"	NULL	2312	-1	1	1	[[[2, 1], 1]]	[[[[34, 17], 7]], 47]
"34T18"	NULL	4352	1	1	2	[[[17, 1], 1]]	[[[[34, 18], 14]], 47]
"34T19"	NULL	4624	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T20"	NULL	4624	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T21"	NULL	4624	-1	1	1	[[[2, 1], 1]]	[[[[34, 21], 7]], 47]
"34T22"	NULL	4624	-1	1	1	[[[2, 1], 1]]	[[[[34, 22], 7]], 47]
"34T23"	NULL	4624	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T24"	NULL	4624	-1	1	1	[[[2, 1], 1]]	[[[[34, 24], 2]], 47]
"34T25"	NULL	4624	-1	1	1	[[[2, 1], 1]]	[[[[34, 26], 2]], 47]
"34T26"	NULL	4624	-1	1	1	[[[2, 1], 1]]	[[[[34, 25], 1], [[34, 26], 1]], 47]
"34T27"	NULL	8160	-1	0	2	[[[2, 1], 1], [[17, 6], 1]]	[[], 47]
"34T28"	NULL	8160	-1	0	2	[[[2, 1], 1], [[17, 7], 1]]	[[[[17, 7], 1]], 47]
"34T29"	NULL	8704	-1	1	2	[[[17, 1], 1]]	[[[[34, 29], 14]], 47]
"34T30"	NULL	8704	1	1	2	[[[17, 2], 1]]	[[[[34, 30], 14], [[34, 31], 15]], 47]
"34T31"	NULL	8704	-1	1	2	[[[17, 2], 1]]	[[[[34, 30], 15], [[34, 31], 14]], 47]
"34T32"	NULL	9248	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T33"	NULL	9248	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T34"	NULL	9248	-1	1	1	[[[2, 1], 1]]	[[[[34, 34], 1], [[34, 36], 1]], 47]
"34T35"	NULL	9248	-1	1	1	[[[2, 1], 1]]	[[[[34, 35], 2]], 47]
"34T36"	NULL	9248	-1	1	1	[[[2, 1], 1]]	[[[[34, 34], 2]], 47]
"34T37"	NULL	9248	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T38"	NULL	9248	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T39"	NULL	9248	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T40"	NULL	9248	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T41"	NULL	9248	-1	1	1	[[[2, 1], 1]]	[[[[34, 41], 7]], 47]
"34T42"	NULL	9248	1	1	1	[[[2, 1], 1]]	[[[[34, 42], 8]], 47]
"34T43"	NULL	16320	-1	0	2	[[[2, 1], 1], [[17, 7], 1]]	[[[[34, 43], 1]], 47]
"34T44"	NULL	16320	-1	0	2	[[[2, 1], 1], [[17, 8], 1]]	[[[[17, 8], 1]], 47]
"34T45"	NULL	17408	-1	1	2	[[[17, 2], 1]]	[[[[34, 45], 29]], 47]
"34T46"	NULL	17408	1	1	2	[[[17, 3], 1]]	[[[[34, 46], 2], [[34, 47], 3]], 47]
"34T47"	NULL	17408	-1	1	2	[[[17, 3], 1]]	[[[[34, 46], 3], [[34, 47], 2]], 47]
"34T48"	NULL	18496	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T49"	NULL	18496	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T50"	NULL	18496	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T51"	NULL	18496	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T52"	NULL	18496	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T53"	NULL	18496	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T54"	NULL	18496	-1	1	1	[[[2, 1], 1]]	[[[[34, 54], 2]], 47]
"34T55"	NULL	18496	1	1	1	[[[2, 1], 1]]	[[], 47]
"34T56"	NULL	32640	-1	0	2	[[[2, 1], 1], [[17, 8], 1]]	[[[[34, 56], 1]], 47]
"34T57"	NULL	34816	-1	1	2	[[[17, 3], 1]]	[[[[34, 57], 5]], 47]
"34T58"	NULL	34816	-1	1	2	[[[17, 4], 1]]	[[[[34, 59], 1]], 47]
"34T59"	NULL	34816	1	1	2	[[[17, 4], 1]]	[[[[34, 58], 1]], 47]
"34T60"	NULL	36992	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T61"	NULL	36992	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T62"	NULL	36992	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T63"	NULL	36992	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T64"	NULL	36992	1	1	1	[[[2, 1], 1]]	[[], 47]
"34T65"	NULL	69632	-1	1	2	[[[17, 4], 1]]	[[[[34, 65], 1]], 47]
"34T66"	NULL	73984	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T67"	NULL	73984	1	1	1	[[[2, 1], 1]]	[[], 47]
"34T68"	NULL	147968	-1	1	1	[[[2, 1], 1]]	[[], 47]
"34T69"	NULL	1114112	1	1	2	[[[17, 1], 1]]	[[[[34, 69], 3824]], 47]
"34T70"	NULL	2228224	-1	1	2	[[[17, 1], 1]]	[[[[34, 70], 3824]], 47]
"34T71"	NULL	2228224	-1	1	2	[[[17, 2], 1]]	[[[[34, 71], 224], [[34, 72], 225]], 47]
"34T72"	NULL	2228224	1	1	2	[[[17, 2], 1]]	[[[[34, 71], 225], [[34, 72], 224]], 47]
"34T73"	NULL	4456448	-1	1	2	[[[17, 2], 1]]	[[[[34, 73], 449]], 47]
"34T74"	NULL	4456448	1	1	2	[[[17, 3], 1]]	[[[[34, 74], 8], [[34, 75], 9]], 47]
"34T75"	NULL	4456448	-1	1	2	[[[17, 3], 1]]	[[[[34, 74], 9], [[34, 75], 8]], 47]
"34T76"	NULL	8912896	-1	1	2	[[[17, 3], 1]]	[[[[34, 76], 17]], 47]
"34T77"	NULL	8912896	-1	1	2	[[[17, 4], 1]]	[[[[34, 78], 1]], 47]
"34T78"	NULL	8912896	1	1	2	[[[17, 4], 1]]	[[[[34, 77], 1]], 47]
"34T79"	NULL	17825792	-1	1	2	[[[17, 4], 1]]	[[[[34, 79], 1]], 47]
"34T80"	NULL	17825792	1	1	2	[[[17, 5], 1]]	[[[[34, 81], 1]], 47]
"34T81"	NULL	17825792	-1	1	2	[[[17, 5], 1]]	[[[[34, 80], 1]], 47]
"34T82"	NULL	33292800	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T83"	NULL	35651584	-1	1	2	[[[17, 5], 1]]	[[[[34, 83], 1]], 47]
"34T84"	NULL	66585600	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T85"	NULL	66585600	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T86"	NULL	133171200	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T87"	NULL	133171200	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T88"	NULL	133171200	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T89"	NULL	133171200	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T90"	NULL	133171200	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T91"	NULL	266342400	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T92"	NULL	266342400	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T93"	NULL	267386880	1	0	2	[[[17, 6], 1]]	[[], 47]
"34T94"	NULL	532684800	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T95"	NULL	534773760	-1	0	2	[[[17, 6], 1]]	[[], 47]
"34T96"	NULL	534773760	-1	0	2	[[[17, 7], 1]]	[[[[34, 97], 1]], 47]
"34T97"	NULL	534773760	1	0	2	[[[17, 7], 1]]	[[[[34, 96], 1]], 47]
"34T98"	NULL	1069547520	-1	0	2	[[[17, 7], 1]]	[[[[34, 98], 1]], 47]
"34T99"	NULL	1069547520	-1	0	2	[[[17, 8], 1]]	[[[[34, 100], 1]], 47]
"34T100"	NULL	1069547520	1	0	2	[[[17, 8], 1]]	[[[[34, 99], 1]], 47]
"34T101"	NULL	2139095040	-1	0	2	[[[17, 8], 1]]	[[[[34, 101], 1]], 47]
"34T102"	NULL	355687428096000	-1	0	2	[[[2, 1], 1], [[17, 9], 1]]	[[], 47]
"34T103"	NULL	355687428096000	-1	0	2	[[[2, 1], 1], [[17, 10], 1]]	[[[[17, 10], 1]], 47]
"34T104"	NULL	711374856192000	-1	0	2	[[[2, 1], 1], [[17, 10], 1]]	[[[[34, 104], 1]], 47]
"34T105"	NULL	11655165643849728000	1	0	2	[[[17, 9], 1]]	[[], 47]
"34T106"	NULL	23310331287699456000	-1	0	2	[[[17, 9], 1]]	[[], 47]
"34T107"	NULL	23310331287699456000	1	0	2	[[[17, 10], 1]]	[[[[34, 108], 1]], 47]
"34T108"	NULL	23310331287699456000	-1	0	2	[[[17, 10], 1]]	[[[[34, 107], 1]], 47]
"34T109"	NULL	46620662575398912000	-1	0	2	[[[17, 10], 1]]	[[[[34, 109], 1]], 47]
"34T110"	NULL	63256773252773585092608000000	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T111"	NULL	126513546505547170185216000000	1	0	1	[[[2, 1], 1]]	[[], 47]
"34T112"	NULL	126513546505547170185216000000	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T113"	NULL	253027093011094340370432000000	-1	0	1	[[[2, 1], 1]]	[[], 47]
"34T114"	"$A_{34}$"	147616399519802070423809304821760000000	1	0	1	[]	[[], 47]
"34T115"	"$S_{34}$"	295232799039604140847618609643520000000	-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```.