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Properties

Label 14T37
Degree $14$
Order $1764$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^2:(C_6\times S_3)$

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Show commands: Magma

magma: G := TransitiveGroup(14, 37);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $37$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7^2:(C_6\times S_3)$
CHM label:   $[1/2.F_{42}(7)^{2}]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,4,6,8,10,12,14), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (2,4,8)(6,12,10), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

21T29 x 2, 28T170, 42T223 x 2, 42T224 x 2, 42T225 x 2, 42T252, 42T253, 42T254, 42T255

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $12$ $7$ $( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $18$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $18$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 3, 3, 3, 3, 1, 1 $ $98$ $3$ $( 3, 9, 5)( 4, 6,10)( 7,11,13)( 8,14,12)$
$ 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 14 $ $126$ $14$ $( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $3$ $( 4, 6,10)( 8,14,12)$
$ 7, 3, 3, 1 $ $84$ $21$ $( 1, 3, 5, 7, 9,11,13)( 4, 6,10)( 8,14,12)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 9, 5)( 4,10, 6)( 7,11,13)( 8,12,14)$
$ 6, 6, 2 $ $147$ $6$ $( 1, 8, 7,14, 5,12)( 2, 9)( 3,10,11, 4,13, 6)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $3$ $( 4,10, 6)( 8,12,14)$
$ 7, 3, 3, 1 $ $84$ $21$ $( 1, 3, 5, 7, 9,11,13)( 4,10, 6)( 8,12,14)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 5, 9)( 4, 6,10)( 7,13,11)( 8,14,12)$
$ 6, 6, 2 $ $147$ $6$ $( 1, 8, 5,12, 7,14)( 2, 9)( 3,10,13, 6,11, 4)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $49$ $2$ $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 6, 6, 1, 1 $ $98$ $6$ $( 3, 7, 5,13, 9,11)( 4,12,10,14, 6, 8)$
$ 14 $ $126$ $14$ $( 1, 8, 3, 6, 5, 4, 7, 2, 9,14,11,12,13,10)$
$ 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,14)(12,13)$
$ 6, 2, 2, 2, 1, 1 $ $98$ $6$ $( 3,13)( 4,12,10,14, 6, 8)( 5,11)( 7, 9)$
$ 6, 6, 1, 1 $ $49$ $6$ $( 3, 7, 5,13, 9,11)( 4, 8, 6,14,10,12)$
$ 6, 6, 2 $ $147$ $6$ $( 1, 8,11,12, 3, 6)( 2, 9,14,13,10, 7)( 4, 5)$
$ 6, 2, 2, 2, 1, 1 $ $98$ $6$ $( 3,13)( 4, 8, 6,14,10,12)( 5,11)( 7, 9)$
$ 6, 6, 1, 1 $ $49$ $6$ $( 3,11, 9,13, 5, 7)( 4,12,10,14, 6, 8)$
$ 6, 6, 2 $ $147$ $6$ $( 1, 8,13,10, 5, 4)( 2, 9,14, 3, 6, 7)(11,12)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1764=2^{2} \cdot 3^{2} \cdot 7^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1764.134
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 3B1 3B-1 3C 6A1 6A-1 6B 6C1 6C-1 6D1 6D-1 6E1 6E-1 7A 7B 7C 14A 14B 21A1 21A-1
Size 1 21 21 49 14 14 49 49 98 49 49 98 98 98 147 147 147 147 12 18 18 126 126 84 84
2 P 1A 1A 1A 1A 3A-1 3A1 3B-1 3B1 3C 3B1 3B-1 3C 3A1 3A-1 3B1 3B-1 3B-1 3B1 7A 7B 7C 7B 7C 21A-1 21A1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2C 2C 2C 2C 2C 2A 2A 2B 2B 7A 7B 7C 14A 14B 7A 7A
7 P 1A 2A 2B 2C 3A1 3A-1 3B1 3B-1 3C 6A1 6A-1 6B 6C1 6C-1 6D1 6D-1 6E1 6E-1 1A 1A 1A 2A 2B 3A1 3A-1
Type
1764.134.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1764.134.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1 1
1764.134.1c R 1 −1 1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1 1 −1 1 1 1
1764.134.1d R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 −1 −1 1 1 1 1 −1 1 1
1764.134.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 ζ3−1 ζ3
1764.134.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 ζ3 ζ3−1
1764.134.1f1 C 1 −1 −1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 1 −1 −1 ζ3−1 ζ3
1764.134.1f2 C 1 −1 −1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 1 −1 −1 ζ3 ζ3−1
1764.134.1g1 C 1 −1 1 −1 ζ3−1 ζ3 ζ3−1 ζ3 1 −ζ3 −ζ3−1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 1 1 1 −1 1 ζ3−1 ζ3
1764.134.1g2 C 1 −1 1 −1 ζ3 ζ3−1 ζ3 ζ3−1 1 −ζ3−1 −ζ3 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 1 1 1 −1 1 ζ3 ζ3−1
1764.134.1h1 C 1 1 −1 −1 ζ3−1 ζ3 ζ3−1 ζ3 1 −ζ3 −ζ3−1 −1 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1 −1 ζ3−1 ζ3
1764.134.1h2 C 1 1 −1 −1 ζ3 ζ3−1 ζ3 ζ3−1 1 −ζ3−1 −ζ3 −1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1 −1 ζ3 ζ3−1
1764.134.2a R 2 0 0 2 −1 −1 2 2 −1 2 2 −1 −1 −1 0 0 0 0 2 2 2 0 0 −1 −1
1764.134.2b R 2 0 0 −2 −1 −1 2 2 −1 −2 −2 1 1 1 0 0 0 0 2 2 2 0 0 −1 −1
1764.134.2c1 C 2 0 0 2 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −1 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 0 0 0 2 2 2 0 0 −ζ3 −ζ3−1
1764.134.2c2 C 2 0 0 2 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −1 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 0 0 0 2 2 2 0 0 −ζ3−1 −ζ3
1764.134.2d1 C 2 0 0 −2 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −1 −2ζ3−1 −2ζ3 1 ζ3−1 ζ3 0 0 0 0 2 2 2 0 0 −ζ3 −ζ3−1
1764.134.2d2 C 2 0 0 −2 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −1 −2ζ3 −2ζ3−1 1 ζ3 ζ3−1 0 0 0 0 2 2 2 0 0 −ζ3−1 −ζ3
1764.134.12a R 12 0 0 0 6 6 0 0 0 0 0 0 0 0 0 0 0 0 5 −2 −2 0 0 −1 −1
1764.134.12b1 C 12 0 0 0 6ζ3−1 6ζ3 0 0 0 0 0 0 0 0 0 0 0 0 5 −2 −2 0 0 −ζ3−1 −ζ3
1764.134.12b2 C 12 0 0 0 6ζ3 6ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 5 −2 −2 0 0 −ζ3 −ζ3−1
1764.134.18a R 18 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 4 −3 0 −1 0 0
1764.134.18b R 18 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 −3 4 −1 0 0 0
1764.134.18c R 18 −6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 −3 4 1 0 0 0
1764.134.18d R 18 0 −6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 4 −3 0 1 0 0

magma: CharacterTable(G);