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Properties

Label 14T9
Degree $14$
Order $112$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times F_8$

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Show commands: Magma

magma: G := TransitiveGroup(14, 9);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times F_8$
CHM label:   $[2^{4}]7$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8)(2,9)(4,11), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$7$:  $C_7$
$14$:  $C_{14}$
$56$:  $C_2^3:C_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7$

Low degree siblings

16T196, 28T19 x 3, 28T20

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $112=2^{4} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  112.41
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 7A1 7A-1 7A2 7A-2 7A3 7A-3 14A1 14A-1 14A3 14A-3 14A5 14A-5
Size 1 1 7 7 8 8 8 8 8 8 8 8 8 8 8 8
2 P 1A 1A 1A 1A 7A2 7A-2 7A-3 7A3 7A-1 7A1 7A1 7A3 7A2 7A-3 7A-2 7A-1
7 P 1A 2A 2B 2C 7A3 7A-3 7A-1 7A1 7A2 7A-2 14A3 14A-5 14A-1 14A5 14A1 14A-3
Type
112.41.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
112.41.1b R 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
112.41.1c1 C 1 1 1 1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3
112.41.1c2 C 1 1 1 1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73
112.41.1c3 C 1 1 1 1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2
112.41.1c4 C 1 1 1 1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72
112.41.1c5 C 1 1 1 1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1
112.41.1c6 C 1 1 1 1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7
112.41.1d1 C 1 −1 1 −1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3
112.41.1d2 C 1 −1 1 −1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73
112.41.1d3 C 1 −1 1 −1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2
112.41.1d4 C 1 −1 1 −1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72
112.41.1d5 C 1 −1 1 −1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1
112.41.1d6 C 1 −1 1 −1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7
112.41.7a R 7 7 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0
112.41.7b R 7 −7 −1 1 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);