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Properties

Label 28T9
Degree $28$
Order $56$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times D_{14}$

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Show commands: Magma

magma: G := TransitiveGroup(28, 9);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times D_{14}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (3,28)(4,27)(5,25)(6,26)(7,23)(8,24)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28), (1,28)(2,27)(3,25)(4,26)(5,23)(6,24)(7,22)(8,21)(9,20)(10,19)(11,17)(12,18)(13,16)(14,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$
$14$:  $D_{7}$
$28$:  $D_{14}$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: $D_{7}$

Degree 14: $D_{14}$ x 3

Low degree siblings

28T9 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $56=2^{3} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  56.12
magma: IdentifyGroup(G);
 
Character table:    
      2  3  3  3  3  3   2  3   2  2   2   2   2  2   2   2   2   2  2  3  3
      7  1  .  1  .  .   1  .   1  1   1   1   1  1   1   1   1   1  1  1  1

        1a 2a 2b 2c 2d 14a 2e 14b 7a 14c 14d 14e 7b 14f 14g 14h 14i 7c 2f 2g
     2P 1a 1a 1a 1a 1a  7a 1a  7a 7b  7b  7c  7c 7c  7c  7b  7b  7a 7a 1a 1a
     3P 1a 2a 2b 2c 2d 14d 2e 14e 7c 14i 14g 14h 7a 14c 14a 14b 14f 7b 2f 2g
     5P 1a 2a 2b 2c 2d 14g 2e 14h 7b 14f 14a 14b 7c 14i 14d 14e 14c 7a 2f 2g
     7P 1a 2a 2b 2c 2d  2f 2e  2g 1a  2b  2f  2g 1a  2b  2f  2g  2b 1a 2f 2g
    11P 1a 2a 2b 2c 2d 14d 2e 14e 7c 14i 14g 14h 7a 14c 14a 14b 14f 7b 2f 2g
    13P 1a 2a 2b 2c 2d 14a 2e 14b 7a 14c 14d 14e 7b 14f 14g 14h 14i 7c 2f 2g

X.1      1  1  1  1  1   1  1   1  1   1   1   1  1   1   1   1   1  1  1  1
X.2      1 -1 -1  1 -1   1  1  -1  1  -1   1  -1  1  -1   1  -1  -1  1  1 -1
X.3      1 -1 -1  1  1  -1 -1   1  1  -1  -1   1  1  -1  -1   1  -1  1 -1  1
X.4      1 -1  1 -1 -1   1 -1   1  1   1   1   1  1   1   1   1   1  1  1  1
X.5      1 -1  1 -1  1  -1  1  -1  1   1  -1  -1  1   1  -1  -1   1  1 -1 -1
X.6      1  1 -1 -1 -1  -1  1   1  1  -1  -1   1  1  -1  -1   1  -1  1 -1  1
X.7      1  1 -1 -1  1   1 -1  -1  1  -1   1  -1  1  -1   1  -1  -1  1  1 -1
X.8      1  1  1  1 -1  -1 -1  -1  1   1  -1  -1  1   1  -1  -1   1  1 -1 -1
X.9      2  . -2  .  .   A  .  -A -B   B   C  -C -C   C   B  -B   A -A -2  2
X.10     2  . -2  .  .   B  .  -B -C   C   A  -A -A   A   C  -C   B -B -2  2
X.11     2  . -2  .  .   C  .  -C -A   A   B  -B -B   B   A  -A   C -C -2  2
X.12     2  . -2  .  .  -C  .   C -A   A  -B   B -B   B  -A   A   C -C  2 -2
X.13     2  . -2  .  .  -B  .   B -C   C  -A   A -A   A  -C   C   B -B  2 -2
X.14     2  . -2  .  .  -A  .   A -B   B  -C   C -C   C  -B   B   A -A  2 -2
X.15     2  .  2  .  .   A  .   A -B  -B   C   C -C  -C   B   B  -A -A -2 -2
X.16     2  .  2  .  .   B  .   B -C  -C   A   A -A  -A   C   C  -B -B -2 -2
X.17     2  .  2  .  .   C  .   C -A  -A   B   B -B  -B   A   A  -C -C -2 -2
X.18     2  .  2  .  .  -C  .  -C -A  -A  -B  -B -B  -B  -A  -A  -C -C  2  2
X.19     2  .  2  .  .  -B  .  -B -C  -C  -A  -A -A  -A  -C  -C  -B -B  2  2
X.20     2  .  2  .  .  -A  .  -A -B  -B  -C  -C -C  -C  -B  -B  -A -A  2  2

A = -E(7)-E(7)^6
B = -E(7)^2-E(7)^5
C = -E(7)^3-E(7)^4

magma: CharacterTable(G);