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Properties

Label 28T5
Degree $28$
Order $56$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7\times D_4$

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Show commands: Magma

magma: G := TransitiveGroup(28, 5);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7\times D_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $14$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,19,12,16,7,26,3,22,13,17,10,27,6,23,2,20,11,15,8,25,4,21,14,18,9,28,5,24), (1,6,9,13,4,7,11)(2,5,10,14,3,8,12)(15,20,23,27,17,22,26,16,19,24,28,18,21,25)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$7$:  $C_7$
$8$:  $D_{4}$
$14$:  $C_{14}$ x 3
$28$:  28T2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: $C_7$

Degree 14: $C_{14}$

Low degree siblings

28T5

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

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:  $2$
Label:  56.9
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);