Properties

Label 28T49
Order \(392\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_7:D_7.C_2$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $49$
Group :  $C_2\times C_7:D_7.C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $2$

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

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

Group invariants

Order:  $392=2^{3} \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [392, 40]
Character table: Data not available.