# Properties

 Label 28T4 Order $$28$$ n $$28$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_{14}$

## Group action invariants

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

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: $D_{7}$

Degree 14: $D_{7}$, $D_{14}$ x 2

## Low degree siblings

14T3 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

## Group invariants

 Order: $28=2^{2} \cdot 7$ Cyclic: No Abelian: No Solvable: Yes GAP id: [28, 3]
 Character table: 2 2 2 2 1 1 1 1 1 1 2 7 1 . . 1 1 1 1 1 1 1 1a 2a 2b 14a 7a 14b 7b 14c 7c 2c 2P 1a 1a 1a 7a 7b 7c 7c 7b 7a 1a 3P 1a 2a 2b 14b 7c 14c 7a 14a 7b 2c 5P 1a 2a 2b 14c 7b 14a 7c 14b 7a 2c 7P 1a 2a 2b 2c 1a 2c 1a 2c 1a 2c 11P 1a 2a 2b 14b 7c 14c 7a 14a 7b 2c 13P 1a 2a 2b 14a 7a 14b 7b 14c 7c 2c X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 1 1 1 X.3 1 -1 1 -1 1 -1 1 -1 1 -1 X.4 1 1 -1 -1 1 -1 1 -1 1 -1 X.5 2 . . A -B C -C B -A -2 X.6 2 . . B -C A -A C -B -2 X.7 2 . . C -A B -B A -C -2 X.8 2 . . -C -A -B -B -A -C 2 X.9 2 . . -B -C -A -A -C -B 2 X.10 2 . . -A -B -C -C -B -A 2 A = -E(7)-E(7)^6 B = -E(7)^2-E(7)^5 C = -E(7)^3-E(7)^4