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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

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, 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:

1A 2A 2B 2C 2D 2E 2F 2G 7A1 7A2 7A3 14A1 14A3 14A5 14B1 14B3 14B5 14C1 14C3 14C5
Size 1 1 1 1 7 7 7 7 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 7A1 7A2 7A3 7A2 7A3 7A3 7A2 7A3 7A2 7A1 7A1 7A1
7 P 1A 2A 2B 2C 2D 2E 2F 2G 7A2 7A3 7A1 14C5 14B1 14C3 14B3 14A5 14A1 14A3 14B5 14C1
Type
56.12.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
56.12.1b R 1 −1 −1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1
56.12.1c R 1 −1 −1 1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1
56.12.1d R 1 −1 1 −1 −1 1 −1 1 1 1 1 −1 −1 −1 1 1 1 −1 −1 −1
56.12.1e R 1 −1 1 −1 1 −1 1 −1 1 1 1 −1 −1 −1 1 1 1 −1 −1 −1
56.12.1f R 1 1 −1 −1 −1 1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
56.12.1g R 1 1 −1 −1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
56.12.1h R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1
56.12.2a1 R 2 2 2 2 0 0 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72
56.12.2a2 R 2 2 2 2 0 0 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7
56.12.2a3 R 2 2 2 2 0 0 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73
56.12.2b1 R 2 −2 −2 2 0 0 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72
56.12.2b2 R 2 −2 −2 2 0 0 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7
56.12.2b3 R 2 −2 −2 2 0 0 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73
56.12.2c1 R 2 −2 2 −2 0 0 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72
56.12.2c2 R 2 −2 2 −2 0 0 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7
56.12.2c3 R 2 −2 2 −2 0 0 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73
56.12.2d1 R 2 2 −2 −2 0 0 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72
56.12.2d2 R 2 2 −2 −2 0 0 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7
56.12.2d3 R 2 2 −2 −2 0 0 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73

magma: CharacterTable(G);