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Properties

Label 14T21
Degree $14$
Order $448$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^3:F_8$

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Show commands: Magma

magma: G := TransitiveGroup(14, 21);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $21$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^3:F_8$
CHM label:   $[2^{6}]7$
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:  (2,9)(7,14), (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)
$7$:  $C_7$
$56$:  $C_2^3:C_7$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7$

Low degree siblings

14T21 x 6, 28T62 x 21, 28T63 x 14, 28T64 x 42, 28T65 x 7, 28T66 x 7

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 6,13)( 7,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 5,12)( 7,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 4,11)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 4,11)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 4,11)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 4,11)( 5,12)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 2, 9)( 4,11)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $7$ $2$ $( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 7, 7 $ $64$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)$
$ 7, 7 $ $64$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)$
$ 7, 7 $ $64$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,11,14,10,13, 9,12)$
$ 7, 7 $ $64$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)$
$ 7, 7 $ $64$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,13,11, 9,14,12,10)$
$ 7, 7 $ $64$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8,14,13,12,11,10, 9)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $448=2^{6} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  448.1394
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 7A1 7A-1 7A2 7A-2 7A3 7A-3
Size 1 7 7 7 7 7 7 7 7 7 64 64 64 64 64 64
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 7A-1 7A1 7A-3 7A3 7A2 7A-2
7 P 1A 2H 2F 2G 2B 2I 2C 2D 2A 2E 7A2 7A-2 7A-1 7A1 7A3 7A-3
Type
448.1394.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
448.1394.1b1 C 1 1 1 1 1 1 1 1 1 1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72
448.1394.1b2 C 1 1 1 1 1 1 1 1 1 1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2
448.1394.1b3 C 1 1 1 1 1 1 1 1 1 1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1
448.1394.1b4 C 1 1 1 1 1 1 1 1 1 1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7
448.1394.1b5 C 1 1 1 1 1 1 1 1 1 1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73
448.1394.1b6 C 1 1 1 1 1 1 1 1 1 1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3
448.1394.7a R 7 −1 −1 −1 −1 −1 −1 −1 7 −1 0 0 0 0 0 0
448.1394.7b R 7 −1 −1 −1 −1 7 −1 −1 −1 −1 0 0 0 0 0 0
448.1394.7c R 7 −5 −1 3 3 −1 3 −1 −1 −1 0 0 0 0 0 0
448.1394.7d R 7 −1 −1 −5 3 −1 −1 3 −1 3 0 0 0 0 0 0
448.1394.7e R 7 −1 3 −1 −5 −1 3 3 −1 −1 0 0 0 0 0 0
448.1394.7f R 7 −1 3 3 −1 −1 −5 −1 −1 3 0 0 0 0 0 0
448.1394.7g R 7 3 −5 3 −1 −1 −1 3 −1 −1 0 0 0 0 0 0
448.1394.7h R 7 3 −1 −1 −1 −1 3 −5 −1 3 0 0 0 0 0 0
448.1394.7i R 7 3 3 −1 3 −1 −1 −1 −1 −5 0 0 0 0 0 0

magma: CharacterTable(G);