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Properties

Label 14T32
Degree $14$
Order $1176$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_7^2:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(14, 32);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $32$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_7^2:C_6$
CHM label:   $[D(7)^{2}:3]2$
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), (2,12)(4,10)(6,8), (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$8$:  $D_{4}$
$12$:  $C_6\times C_2$
$24$:  $D_4 \times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T32, 28T136 x 2, 28T137 x 2, 28T138 x 2, 28T143, 42T195 x 2

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 $ $12$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $24$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $2$ $( 4,14)( 6,12)( 8,10)$
$ 7, 2, 2, 2, 1 $ $84$ $14$ $( 1, 3, 5, 7, 9,11,13)( 4,14)( 6,12)( 8,10)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $49$ $2$ $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 5, 9)( 4, 6,10)( 7,13,11)( 8,14,12)$
$ 6, 3, 3, 1, 1 $ $98$ $6$ $( 3, 5, 9)( 4,12,10,14, 6, 8)( 7,13,11)$
$ 6, 6, 1, 1 $ $49$ $6$ $( 3,11, 9,13, 5, 7)( 4,12,10,14, 6, 8)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 9, 5)( 4,10, 6)( 7,11,13)( 8,12,14)$
$ 6, 3, 3, 1, 1 $ $98$ $6$ $( 3, 9, 5)( 4, 8, 6,14,10,12)( 7,11,13)$
$ 6, 6, 1, 1 $ $49$ $6$ $( 3, 7, 5,13, 9,11)( 4, 8, 6,14,10,12)$
$ 2, 2, 2, 2, 2, 2, 2 $ $14$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 14 $ $84$ $14$ $( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$
$ 4, 4, 4, 2 $ $98$ $4$ $( 1,10, 3, 8)( 2, 9)( 4,11,14, 7)( 5, 6,13,12)$
$ 6, 6, 2 $ $98$ $6$ $( 1,14,13,10, 5, 8)( 2, 3, 4, 7,12, 9)( 6,11)$
$ 12, 2 $ $98$ $12$ $( 1, 4, 7, 6,11,12, 9, 2, 3,14,13, 8)( 5,10)$
$ 6, 6, 2 $ $98$ $6$ $( 1,12, 3, 6, 7, 8)( 2, 5,14,11,10, 9)( 4,13)$
$ 12, 2 $ $98$ $12$ $( 1, 6, 7,10, 9, 2, 5, 4,13,14,11, 8)( 3,12)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1176=2^{3} \cdot 3 \cdot 7^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1176.217
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 4A 6A1 6A-1 6B1 6B-1 6C1 6C-1 7A 7B 7C 12A1 12A-1 14A 14B
Size 1 14 14 49 49 49 98 49 49 98 98 98 98 12 12 24 98 98 84 84
2 P 1A 1A 1A 1A 3A-1 3A1 2C 3A1 3A-1 3A1 3A1 3A-1 3A-1 7A 7B 7C 6A1 6A-1 7A 7B
3 P 1A 2A 2B 2C 1A 1A 4A 2C 2C 2A 2B 2A 2B 7A 7B 7C 4A 4A 14A 14B
7 P 1A 2A 2B 2C 3A1 3A-1 4A 6A1 6A-1 6B1 6C-1 6B-1 6C1 1A 1A 1A 12A1 12A-1 2B 2A
Type
1176.217.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1176.217.1b R 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 −1
1176.217.1c R 1 −1 1 1 1 1 −1 1 1 −1 −1 1 1 1 1 1 −1 −1 1 −1
1176.217.1d R 1 1 −1 1 1 1 −1 1 1 1 1 −1 −1 1 1 1 −1 −1 −1 1
1176.217.1e1 C 1 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 ζ3−1 ζ3 1 1
1176.217.1e2 C 1 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 ζ3 ζ3−1 1 1
1176.217.1f1 C 1 −1 −1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 1 ζ3−1 ζ3 −1 −1
1176.217.1f2 C 1 −1 −1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 1 ζ3 ζ3−1 −1 −1
1176.217.1g1 C 1 −1 1 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 1 1 1 −ζ3−1 −ζ3 1 −1
1176.217.1g2 C 1 −1 1 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 1 1 1 −ζ3 −ζ3−1 1 −1
1176.217.1h1 C 1 1 −1 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 −ζ3−1 −ζ3 −1 1
1176.217.1h2 C 1 1 −1 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 −ζ3 −ζ3−1 −1 1
1176.217.2a R 2 0 0 −2 2 2 0 −2 −2 0 0 0 0 2 2 2 0 0 0 0
1176.217.2b1 C 2 0 0 −2 2ζ3−1 2ζ3 0 −2ζ3 −2ζ3−1 0 0 0 0 2 2 2 0 0 0 0
1176.217.2b2 C 2 0 0 −2 2ζ3 2ζ3−1 0 −2ζ3−1 −2ζ3 0 0 0 0 2 2 2 0 0 0 0
1176.217.12a R 12 0 6 0 0 0 0 0 0 0 0 0 0 5 −2 −2 0 0 −1 0
1176.217.12b R 12 6 0 0 0 0 0 0 0 0 0 0 0 −2 5 −2 0 0 0 −1
1176.217.12c R 12 −6 0 0 0 0 0 0 0 0 0 0 0 −2 5 −2 0 0 0 1
1176.217.12d R 12 0 −6 0 0 0 0 0 0 0 0 0 0 5 −2 −2 0 0 1 0
1176.217.24a R 24 0 0 0 0 0 0 0 0 0 0 0 0 −4 −4 3 0 0 0 0

magma: CharacterTable(G);