# Properties

 Label 28T9 Degree $28$ Order $56$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^2\times D_7$

## Group action invariants

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

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: $D_{7}$

Degree 14: $D_{14}$ x 3

## Low degree siblings

28T9 x 3

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

## Group invariants

 Order: $56=2^{3} \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [56, 12]
 Character table:  2 3 3 3 3 3 2 3 2 2 2 2 2 2 2 2 2 2 2 3 3 7 1 . 1 . . 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 2d 14a 2e 14b 7a 14c 14d 14e 7b 14f 14g 14h 14i 7c 2f 2g 2P 1a 1a 1a 1a 1a 7a 1a 7a 7b 7b 7c 7c 7c 7c 7b 7b 7a 7a 1a 1a 3P 1a 2a 2b 2c 2d 14d 2e 14e 7c 14i 14g 14h 7a 14c 14a 14b 14f 7b 2f 2g 5P 1a 2a 2b 2c 2d 14g 2e 14h 7b 14f 14a 14b 7c 14i 14d 14e 14c 7a 2f 2g 7P 1a 2a 2b 2c 2d 2f 2e 2g 1a 2b 2f 2g 1a 2b 2f 2g 2b 1a 2f 2g 11P 1a 2a 2b 2c 2d 14d 2e 14e 7c 14i 14g 14h 7a 14c 14a 14b 14f 7b 2f 2g 13P 1a 2a 2b 2c 2d 14a 2e 14b 7a 14c 14d 14e 7b 14f 14g 14h 14i 7c 2f 2g X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 X.3 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 X.4 1 -1 1 -1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.5 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 X.6 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 X.7 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 X.8 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 X.9 2 . -2 . . A . -A -B B C -C -C C B -B A -A -2 2 X.10 2 . -2 . . B . -B -C C A -A -A A C -C B -B -2 2 X.11 2 . -2 . . C . -C -A A B -B -B B A -A C -C -2 2 X.12 2 . -2 . . -C . C -A A -B B -B B -A A C -C 2 -2 X.13 2 . -2 . . -B . B -C C -A A -A A -C C B -B 2 -2 X.14 2 . -2 . . -A . A -B B -C C -C C -B B A -A 2 -2 X.15 2 . 2 . . A . A -B -B C C -C -C B B -A -A -2 -2 X.16 2 . 2 . . B . B -C -C A A -A -A C C -B -B -2 -2 X.17 2 . 2 . . C . C -A -A B B -B -B A A -C -C -2 -2 X.18 2 . 2 . . -C . -C -A -A -B -B -B -B -A -A -C -C 2 2 X.19 2 . 2 . . -B . -B -C -C -A -A -A -A -C -C -B -B 2 2 X.20 2 . 2 . . -A . -A -B -B -C -C -C -C -B -B -A -A 2 2 A = -E(7)-E(7)^6 B = -E(7)^2-E(7)^5 C = -E(7)^3-E(7)^4