# Properties

 Label 28T39 Degree $28$ Order $224$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^2\times F_8$

## Group action invariants

 Degree $n$: $28$ Transitive number $t$: $39$ Group: $C_2^2\times F_8$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (1,15)(2,16)(3,4)(5,20)(6,19)(7,22)(8,21)(9,23)(10,24)(11,12)(13,14)(17,18)(25,26)(27,28), (1,24,18,11,5,27,21,16,9,3,26,19,13,7)(2,23,17,12,6,28,22,15,10,4,25,20,14,8)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$7$:  $C_7$
$14$:  $C_{14}$ x 3
$28$:  28T2
$56$:  $C_2^3:C_7$
$112$:  14T9 x 3

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: None

Degree 7: $C_7$

Degree 14: $C_{14}$, 14T9 x 2

## Low degree siblings

28T38 x 6, 28T39 x 2, 32T2228

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

## Group invariants

 Order: $224=2^{5} \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [224, 195]
 Character table: not available.