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Properties

Label 28T4
Degree $28$
Order $28$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{14}$

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Show commands: Magma

magma: G := TransitiveGroup(28, 4);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{14}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $28$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,25)(2,26)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,28), (1,4,6,8,10,12,14,16,18,20,22,24,26,27)(2,3,5,7,9,11,13,15,17,19,21,23,25,28)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$14$:  $D_{7}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: $D_{7}$

Degree 14: $D_{7}$, $D_{14}$ x 2

Low degree siblings

14T3 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1, 2)( 3,27)( 4,28)( 5,26)( 6,25)( 7,24)( 8,23)( 9,22)(10,21)(11,20)(12,19) (13,18)(14,17)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1, 3)( 2, 4)( 5,27)( 6,28)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21)(13,20) (14,19)(15,18)(16,17)$
$ 14, 14 $ $2$ $14$ $( 1, 4, 6, 8,10,12,14,16,18,20,22,24,26,27)( 2, 3, 5, 7, 9,11,13,15,17,19,21, 23,25,28)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 6,10,14,18,22,26)( 2, 5, 9,13,17,21,25)( 3, 7,11,15,19,23,28) ( 4, 8,12,16,20,24,27)$
$ 14, 14 $ $2$ $14$ $( 1, 8,14,20,26, 4,10,16,22,27, 6,12,18,24)( 2, 7,13,19,25, 3, 9,15,21,28, 5, 11,17,23)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1,10,18,26, 6,14,22)( 2, 9,17,25, 5,13,21)( 3,11,19,28, 7,15,23) ( 4,12,20,27, 8,16,24)$
$ 14, 14 $ $2$ $14$ $( 1,12,22, 4,14,24, 6,16,26, 8,18,27,10,20)( 2,11,21, 3,13,23, 5,15,25, 7,17, 28, 9,19)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1,14,26,10,22, 6,18)( 2,13,25, 9,21, 5,17)( 3,15,28,11,23, 7,19) ( 4,16,27,12,24, 8,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,16)( 2,15)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)( 8,22)( 9,23)(10,24)(11,25) (12,26)(13,28)(14,27)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $28=2^{2} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  28.3
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 7A1 7A2 7A3 14A1 14A3 14A5
Size 1 1 7 7 2 2 2 2 2 2
2 P 1A 1A 1A 1A 7A1 7A2 7A3 7A3 7A2 7A1
7 P 1A 2A 2B 2C 7A2 7A3 7A1 14A5 14A1 14A3
Type
28.3.1a R 1 1 1 1 1 1 1 1 1 1
28.3.1b R 1 −1 −1 1 1 1 1 −1 −1 −1
28.3.1c R 1 −1 1 −1 1 1 1 −1 −1 −1
28.3.1d R 1 1 −1 −1 1 1 1 1 1 1
28.3.2a1 R 2 2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73
28.3.2a2 R 2 2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72
28.3.2a3 R 2 2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7
28.3.2b1 R 2 −2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73
28.3.2b2 R 2 −2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72
28.3.2b3 R 2 −2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7

magma: CharacterTable(G);