Properties

Label 28T50
Order \(392\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_{14}\times D_7):C_2$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $50$
Group :  $(C_{14}\times D_7):C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,14,28,25,11,10,24,22,8,5,20,17,4,2,15,13,27,26,12,9,23,21,7,6,19,18,3), (1,16,17,27,6,11,22,23,9,8,25,19,13,4)(2,15,18,28,5,12,21,24,10,7,26,20,14,3)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$
14:  $D_{7}$ x 2
28:  $D_{14}$ x 2
56:  28T6, $D_{28}$
196:  14T13

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: None

Degree 14: 14T13

Low degree siblings

28T50 x 5

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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, 2, 1, 1 $ $98$ $2$ $( 3, 4)( 5,26)( 6,25)( 7,27)( 8,28)( 9,22)(10,21)(11,24)(12,23)(13,17)(14,18) (15,19)(16,20)$
$ 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$
$ 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 $ $4$ $14$ $( 1, 2)( 3,16,28,11,24, 8,20, 4,15,27,12,23, 7,19)( 5, 6)( 9,10)(13,14)(17,18) (21,22)(25,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $14$ $2$ $( 1, 3)( 2, 4)( 5,27)( 6,28)( 7,25)( 8,26)( 9,24)(10,23)(11,21)(12,22)(13,20) (14,19)(15,17)(16,18)$
$ 4, 4, 4, 4, 4, 4, 4 $ $14$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,23,22,24) (25,28,26,27)$
$ 28 $ $14$ $28$ $( 1, 3, 5, 8, 9,12,14,16,17,20,21,23,25,28, 2, 4, 6, 7,10,11,13,15,18,19,22, 24,26,27)$
$ 14, 14 $ $14$ $14$ $( 1, 3, 6,28, 9,24,13,20,17,15,22,12,25, 7)( 2, 4, 5,27,10,23,14,19,18,16,21, 11,26, 8)$
$ 14, 14 $ $14$ $14$ $( 1, 3, 9,24,17,15,25, 7, 6,28,13,20,22,12)( 2, 4,10,23,18,16,26, 8, 5,27,14, 19,21,11)$
$ 28 $ $14$ $28$ $( 1, 3,10,11,17,20,26,27, 6, 7,14,16,22,24, 2, 4, 9,12,18,19,25,28, 5, 8,13, 15,21,23)$
$ 14, 14 $ $14$ $14$ $( 1, 3,13,20,25, 7, 9,24,22,12, 6,28,17,15)( 2, 4,14,19,26, 8,10,23,21,11, 5, 27,18,16)$
$ 28 $ $14$ $28$ $( 1, 3,14,16,25,28,10,11,22,24, 5, 8,17,20, 2, 4,13,15,26,27, 9,12,21,23, 6, 7,18,19)$
$ 14, 14 $ $14$ $14$ $( 1, 3,17,15, 6,28,22,12, 9,24,25, 7,13,20)( 2, 4,18,16, 5,27,21,11,10,23,26, 8,14,19)$
$ 28 $ $14$ $28$ $( 1, 3,18,19, 6, 7,21,23, 9,12,26,27,13,15, 2, 4,17,20, 5, 8,22,24,10,11,25, 28,14,16)$
$ 28 $ $14$ $28$ $( 1, 3,21,23,13,15, 5, 8,25,28,18,19, 9,12, 2, 4,22,24,14,16, 6, 7,26,27,17, 20,10,11)$
$ 14, 14 $ $14$ $14$ $( 1, 3,22,12,13,20, 6,28,25, 7,17,15, 9,24)( 2, 4,21,11,14,19, 5,27,26, 8,18, 16,10,23)$
$ 14, 14 $ $14$ $14$ $( 1, 3,25, 7,22,12,17,15,13,20, 9,24, 6,28)( 2, 4,26, 8,21,11,18,16,14,19,10, 23, 5,27)$
$ 28 $ $14$ $28$ $( 1, 3,26,27,22,24,18,19,13,15,10,11, 6, 7, 2, 4,25,28,21,23,17,20,14,16, 9, 12, 5, 8)$
$ 14, 14 $ $2$ $14$ $( 1, 5, 9,14,17,21,25, 2, 6,10,13,18,22,26)( 3, 8,12,16,20,23,28, 4, 7,11,15, 19,24,27)$
$ 14, 14 $ $4$ $14$ $( 1, 5, 9,14,17,21,25, 2, 6,10,13,18,22,26)( 3,11,20,27, 7,16,24, 4,12,19,28, 8,15,23)$
$ 14, 14 $ $4$ $14$ $( 1, 5, 9,14,17,21,25, 2, 6,10,13,18,22,26)( 3,16,28,11,24, 8,20, 4,15,27,12, 23, 7,19)$
$ 14, 14 $ $4$ $14$ $( 1, 5, 9,14,17,21,25, 2, 6,10,13,18,22,26)( 3,19, 7,23,12,27,15, 4,20, 8,24, 11,28,16)$
$ 14, 14 $ $4$ $14$ $( 1, 5, 9,14,17,21,25, 2, 6,10,13,18,22,26)( 3,23,15, 8,28,19,12, 4,24,16, 7, 27,20,11)$
$ 14, 14 $ $2$ $14$ $( 1, 5, 9,14,17,21,25, 2, 6,10,13,18,22,26)( 3,27,24,19,15,11, 7, 4,28,23,20, 16,12, 8)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 6, 9,13,17,22,25)( 2, 5,10,14,18,21,26)( 3, 7,12,15,20,24,28) ( 4, 8,11,16,19,23,27)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 6, 9,13,17,22,25)( 2, 5,10,14,18,21,26)( 3,12,20,28, 7,15,24) ( 4,11,19,27, 8,16,23)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 6, 9,13,17,22,25)( 2, 5,10,14,18,21,26)( 3,15,28,12,24, 7,20) ( 4,16,27,11,23, 8,19)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 6, 9,13,17,22,25)( 2, 5,10,14,18,21,26)( 3,20, 7,24,12,28,15) ( 4,19, 8,23,11,27,16)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 6, 9,13,17,22,25)( 2, 5,10,14,18,21,26)( 3,24,15, 7,28,20,12) ( 4,23,16, 8,27,19,11)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 6, 9,13,17,22,25)( 2, 5,10,14,18,21,26)( 3,28,24,20,15,12, 7) ( 4,27,23,19,16,11, 8)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 9,17,25, 6,13,22)( 2,10,18,26, 5,14,21)( 3,12,20,28, 7,15,24) ( 4,11,19,27, 8,16,23)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 9,17,25, 6,13,22)( 2,10,18,26, 5,14,21)( 3,15,28,12,24, 7,20) ( 4,16,27,11,23, 8,19)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 9,17,25, 6,13,22)( 2,10,18,26, 5,14,21)( 3,20, 7,24,12,28,15) ( 4,19, 8,23,11,27,16)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 9,17,25, 6,13,22)( 2,10,18,26, 5,14,21)( 3,24,15, 7,28,20,12) ( 4,23,16, 8,27,19,11)$
$ 14, 14 $ $2$ $14$ $( 1,10,17,26, 6,14,22, 2, 9,18,25, 5,13,21)( 3,11,20,27, 7,16,24, 4,12,19,28, 8,15,23)$
$ 14, 14 $ $4$ $14$ $( 1,10,17,26, 6,14,22, 2, 9,18,25, 5,13,21)( 3,16,28,11,24, 8,20, 4,15,27,12, 23, 7,19)$
$ 14, 14 $ $4$ $14$ $( 1,10,17,26, 6,14,22, 2, 9,18,25, 5,13,21)( 3,19, 7,23,12,27,15, 4,20, 8,24, 11,28,16)$
$ 14, 14 $ $2$ $14$ $( 1,10,17,26, 6,14,22, 2, 9,18,25, 5,13,21)( 3,23,15, 8,28,19,12, 4,24,16, 7, 27,20,11)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1,13,25, 9,22, 6,17)( 2,14,26,10,21, 5,18)( 3,15,28,12,24, 7,20) ( 4,16,27,11,23, 8,19)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1,13,25, 9,22, 6,17)( 2,14,26,10,21, 5,18)( 3,20, 7,24,12,28,15) ( 4,19, 8,23,11,27,16)$
$ 14, 14 $ $2$ $14$ $( 1,14,25,10,22, 5,17, 2,13,26, 9,21, 6,18)( 3,16,28,11,24, 8,20, 4,15,27,12, 23, 7,19)$
$ 14, 14 $ $2$ $14$ $( 1,14,25,10,22, 5,17, 2,13,26, 9,21, 6,18)( 3,19, 7,23,12,27,15, 4,20, 8,24, 11,28,16)$

Group invariants

Order:  $392=2^{3} \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [392, 21]
Character table: Data not available.