Properties

Label 28T36
Order \(196\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_7^2$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $36$
Group :  $D_7^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,7)(4,8)(5,25)(6,26)(9,22)(10,21)(11,27)(12,28)(13,18)(14,17)(15,24)(16,23)(19,20), (1,12,25,8,22,3,18,27,14,24,10,20,6,16)(2,11,26,7,21,4,17,28,13,23,9,19,5,15)
$|\Aut(F/K)|$:  $14$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: None

Degree 14: 14T13

Low degree siblings

14T13 x 3, 28T36 x 2

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

Group invariants

Order:  $196=2^{2} \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [196, 9]
Character table: Data not available.