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Properties

Label 14T36
Degree $14$
Order $1764$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^2:C_3:C_{12}$

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Show commands: Magma

magma: G := TransitiveGroup(14, 36);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $36$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7^2:C_3:C_{12}$
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), (2,4,8)(6,12,10), (1,6,13,8)(2,9,12,5)(3,4,11,10)(7,14), (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)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $S_3$, $C_6$
$12$:  $C_{12}$, $C_3 : C_4$
$18$:  $S_3\times C_3$
$36$:  $C_3\times (C_3 : C_4)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

28T169, 42T248, 42T249, 42T250, 42T251, 42T257

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 $ $36$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 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, 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 $ $49$ $3$ $( 3, 5, 9)( 4, 6,10)( 7,13,11)( 8,14,12)$
$ 3, 3, 3, 3, 1, 1 $ $98$ $3$ $( 3, 5, 9)( 4,10, 6)( 7,13,11)( 8,12,14)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 9, 5)( 4,10, 6)( 7,11,13)( 8,12,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $49$ $2$ $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 6, 2, 2, 2, 1, 1 $ $98$ $6$ $( 3,13)( 4,12,10,14, 6, 8)( 5,11)( 7, 9)$
$ 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, 1, 1 $ $98$ $6$ $( 3,11, 9,13, 5, 7)( 4, 8, 6,14,10,12)$
$ 6, 6, 1, 1 $ $49$ $6$ $( 3, 7, 5,13, 9,11)( 4, 8, 6,14,10,12)$
$ 4, 4, 4, 2 $ $147$ $4$ $( 1, 6,13, 8)( 2, 9,12, 5)( 3, 4,11,10)( 7,14)$
$ 12, 2 $ $147$ $12$ $( 1,10, 3, 6,13,14, 7,12, 5, 2, 9, 8)( 4,11)$
$ 12, 2 $ $147$ $12$ $( 1, 4,11, 6,13,12, 5, 2, 9,14, 7, 8)( 3,10)$
$ 4, 4, 4, 2 $ $147$ $4$ $( 1,12,11, 8)( 2, 7, 4, 5)( 3,14, 9, 6)(10,13)$
$ 12, 2 $ $147$ $12$ $( 1, 8)( 2, 7, 6, 3,12,11,14, 9,10,13, 4, 5)$
$ 12, 2 $ $147$ $12$ $( 1,14, 9, 4, 5, 2, 7,10,13, 6, 3, 8)(11,12)$

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:  1764.133
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B1 3B-1 3C 4A1 4A-1 6A1 6A-1 6B 6C1 6C-1 7A 7B 12A1 12A-1 12A5 12A-5 21A1 21A-1
Size 1 49 14 14 49 49 98 147 147 49 49 98 98 98 12 36 147 147 147 147 84 84
2 P 1A 1A 3A-1 3A1 3B-1 3B1 3C 2A 2A 3B1 3B-1 3A1 3A-1 3C 7A 7B 6A-1 6A1 6A-1 6A1 21A-1 21A1
3 P 1A 2A 1A 1A 1A 1A 1A 4A-1 4A1 2A 2A 2A 2A 2A 7A 7B 4A-1 4A-1 4A1 4A1 7A 7A
7 P 1A 2A 3A1 3A-1 3B1 3B-1 3C 4A-1 4A1 6A1 6A-1 6C1 6C-1 6B 1A 1A 12A5 12A1 12A-1 12A-5 3A1 3A-1
Type
1764.133.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1764.133.1b R 1 1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1
1764.133.1c1 C 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
1764.133.1c2 C 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
1764.133.1d1 C 1 −1 1 1 1 1 1 −i i −1 −1 −1 −1 −1 1 1 i −i i −i 1 1
1764.133.1d2 C 1 −1 1 1 1 1 1 i −i −1 −1 −1 −1 −1 1 1 −i i −i i 1 1
1764.133.1e1 C 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 −1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3
1764.133.1e2 C 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 −1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1
1764.133.1f1 C 1 −1 −ζ122 ζ124 −ζ122 ζ124 1 −ζ123 ζ123 −ζ124 ζ122 −1 −ζ124 ζ122 1 1 −ζ125 ζ12 −ζ12 ζ125 −ζ122 ζ124
1764.133.1f2 C 1 −1 ζ124 −ζ122 ζ124 −ζ122 1 ζ123 −ζ123 ζ122 −ζ124 −1 ζ122 −ζ124 1 1 ζ12 −ζ125 ζ125 −ζ12 ζ124 −ζ122
1764.133.1f3 C 1 −1 −ζ122 ζ124 −ζ122 ζ124 1 ζ123 −ζ123 −ζ124 ζ122 −1 −ζ124 ζ122 1 1 ζ125 −ζ12 ζ12 −ζ125 −ζ122 ζ124
1764.133.1f4 C 1 −1 ζ124 −ζ122 ζ124 −ζ122 1 −ζ123 ζ123 ζ122 −ζ124 −1 ζ122 −ζ124 1 1 −ζ12 ζ125 −ζ125 ζ12 ζ124 −ζ122
1764.133.2a R 2 2 −1 −1 2 2 −1 0 0 2 2 −1 −1 −1 2 2 0 0 0 0 −1 −1
1764.133.2b S 2 −2 −1 −1 2 2 −1 0 0 −2 −2 1 1 1 2 2 0 0 0 0 −1 −1
1764.133.2c1 C 2 2 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −1 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 2 2 0 0 0 0 −ζ3 −ζ3−1
1764.133.2c2 C 2 2 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −1 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 2 2 0 0 0 0 −ζ3−1 −ζ3
1764.133.2d1 C 2 −2 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −1 0 0 −2ζ3−1 −2ζ3 1 ζ3−1 ζ3 2 2 0 0 0 0 −ζ3 −ζ3−1
1764.133.2d2 C 2 −2 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −1 0 0 −2ζ3 −2ζ3−1 1 ζ3 ζ3−1 2 2 0 0 0 0 −ζ3−1 −ζ3
1764.133.12a R 12 0 6 6 0 0 0 0 0 0 0 0 0 0 5 −2 0 0 0 0 −1 −1
1764.133.12b1 C 12 0 6ζ3−1 6ζ3 0 0 0 0 0 0 0 0 0 0 5 −2 0 0 0 0 −ζ3−1 −ζ3
1764.133.12b2 C 12 0 6ζ3 6ζ3−1 0 0 0 0 0 0 0 0 0 0 5 −2 0 0 0 0 −ζ3 −ζ3−1
1764.133.36a R 36 0 0 0 0 0 0 0 0 0 0 0 0 0 −6 1 0 0 0 0 0 0

magma: CharacterTable(G);