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Properties

Label 28T49
Degree $28$
Order $392$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_7^2:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(28, 49);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $49$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times C_7^2:C_4$
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:  (1,11,17,28)(2,12,18,27)(3,25,8,21)(4,26,7,22)(5,16,14,24)(6,15,13,23)(9,19,10,20), (1,21,13,6,25,17,9,2,22,14,5,26,18,10)(3,27,24,19,15,11,7,4,28,23,20,16,12,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$196$:  14T12

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 7: None

Degree 14: 14T12

Low degree siblings

28T48 x 4, 28T49 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 $ $49$ $2$ $( 5,25)( 6,26)( 7,28)( 8,27)( 9,22)(10,21)(11,23)(12,24)(13,18)(14,17)(15,20) (16,19)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $7$ $( 3, 7,12,15,20,24,28)( 4, 8,11,16,19,23,27)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $7$ $( 3,12,20,28, 7,15,24)( 4,11,19,27, 8,16,23)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $7$ $( 3,15,28,12,24, 7,20)( 4,16,27,11,23, 8,19)$
$ 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 $ $49$ $2$ $( 1, 2)( 3, 4)( 5,26)( 6,25)( 7,27)( 8,28)( 9,21)(10,22)(11,24)(12,23)(13,17) (14,18)(15,19)(16,20)$
$ 14, 2, 2, 2, 2, 2, 2, 2 $ $4$ $14$ $( 1, 2)( 3, 8,12,16,20,23,28, 4, 7,11,15,19,24,27)( 5, 6)( 9,10)(13,14)(17,18) (21,22)(25,26)$
$ 14, 2, 2, 2, 2, 2, 2, 2 $ $4$ $14$ $( 1, 2)( 3,11,20,27, 7,16,24, 4,12,19,28, 8,15,23)( 5, 6)( 9,10)(13,14)(17,18) (21,22)(25,26)$
$ 14, 2, 2, 2, 2, 2, 2, 2 $ $4$ $14$ $( 1, 2)( 3,16,28,11,24, 8,20, 4,15,27,12,23, 7,19)( 5, 6)( 9,10)(13,14)(17,18) (21,22)(25,26)$
$ 4, 4, 4, 4, 4, 4, 4 $ $49$ $4$ $( 1, 3, 2, 4)( 5, 7,26,27)( 6, 8,25,28)( 9,12,21,23)(10,11,22,24)(13,15,17,19) (14,16,18,20)$
$ 4, 4, 4, 4, 4, 4, 4 $ $49$ $4$ $( 1, 3, 2, 4)( 5,28,26, 8)( 6,27,25, 7)( 9,24,21,11)(10,23,22,12)(13,20,17,16) (14,19,18,15)$
$ 4, 4, 4, 4, 4, 4, 4 $ $49$ $4$ $( 1, 4, 2, 3)( 5, 8,26,28)( 6, 7,25,27)( 9,11,21,24)(10,12,22,23)(13,16,17,20) (14,15,18,19)$
$ 4, 4, 4, 4, 4, 4, 4 $ $49$ $4$ $( 1, 4, 2, 3)( 5,27,26, 7)( 6,28,25, 8)( 9,23,21,12)(10,24,22,11)(13,19,17,15) (14,20,18,16)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 5, 9,13,18,22,25)( 2, 6,10,14,17,21,26)( 3, 7,12,15,20,24,28) ( 4, 8,11,16,19,23,27)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 5, 9,13,18,22,25)( 2, 6,10,14,17,21,26)( 3,12,20,28, 7,15,24) ( 4,11,19,27, 8,16,23)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 5, 9,13,18,22,25)( 2, 6,10,14,17,21,26)( 3,15,28,12,24, 7,20) ( 4,16,27,11,23, 8,19)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 5, 9,13,18,22,25)( 2, 6,10,14,17,21,26)( 3,20, 7,24,12,28,15) ( 4,19, 8,23,11,27,16)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 5, 9,13,18,22,25)( 2, 6,10,14,17,21,26)( 3,24,15, 7,28,20,12) ( 4,23,16, 8,27,19,11)$
$ 14, 14 $ $4$ $14$ $( 1, 6, 9,14,18,21,25, 2, 5,10,13,17,22,26)( 3, 8,12,16,20,23,28, 4, 7,11,15, 19,24,27)$
$ 14, 14 $ $4$ $14$ $( 1, 6, 9,14,18,21,25, 2, 5,10,13,17,22,26)( 3,11,20,27, 7,16,24, 4,12,19,28, 8,15,23)$
$ 14, 14 $ $4$ $14$ $( 1, 6, 9,14,18,21,25, 2, 5,10,13,17,22,26)( 3,16,28,11,24, 8,20, 4,15,27,12, 23, 7,19)$
$ 14, 14 $ $4$ $14$ $( 1, 6, 9,14,18,21,25, 2, 5,10,13,17,22,26)( 3,19, 7,23,12,27,15, 4,20, 8,24, 11,28,16)$
$ 14, 14 $ $4$ $14$ $( 1, 6, 9,14,18,21,25, 2, 5,10,13,17,22,26)( 3,23,15, 8,28,19,12, 4,24,16, 7, 27,20,11)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 9,18,25, 5,13,22)( 2,10,17,26, 6,14,21)( 3,12,20,28, 7,15,24) ( 4,11,19,27, 8,16,23)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 9,18,25, 5,13,22)( 2,10,17,26, 6,14,21)( 3,15,28,12,24, 7,20) ( 4,16,27,11,23, 8,19)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 9,18,25, 5,13,22)( 2,10,17,26, 6,14,21)( 3,20, 7,24,12,28,15) ( 4,19, 8,23,11,27,16)$
$ 14, 14 $ $4$ $14$ $( 1,10,18,26, 5,14,22, 2, 9,17,25, 6,13,21)( 3,11,20,27, 7,16,24, 4,12,19,28, 8,15,23)$
$ 14, 14 $ $4$ $14$ $( 1,10,18,26, 5,14,22, 2, 9,17,25, 6,13,21)( 3,16,28,11,24, 8,20, 4,15,27,12, 23, 7,19)$
$ 14, 14 $ $4$ $14$ $( 1,10,18,26, 5,14,22, 2, 9,17,25, 6,13,21)( 3,19, 7,23,12,27,15, 4,20, 8,24, 11,28,16)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1,13,25, 9,22, 5,18)( 2,14,26,10,21, 6,17)( 3,15,28,12,24, 7,20) ( 4,16,27,11,23, 8,19)$
$ 14, 14 $ $4$ $14$ $( 1,14,25,10,22, 6,18, 2,13,26, 9,21, 5,17)( 3,16,28,11,24, 8,20, 4,15,27,12, 23, 7,19)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $392=2^{3} \cdot 7^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  392.40
magma: IdentifyGroup(G);
 
Character table:    32 x 32 character table

magma: CharacterTable(G);