# Properties

 Label 28T11 Degree $28$ Order $56$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $F_8$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$7$:  $C_7$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 4: None

Degree 7: $C_7$

Degree 14: 14T6

## Low degree siblings

8T25, 14T6

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

## Group invariants

 Order: $56=2^{3} \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [56, 11]
 Character table:  2 3 3 . . . . . . 7 1 . 1 1 1 1 1 1 1a 2a 7a 7b 7c 7d 7e 7f 2P 1a 1a 7b 7d 7f 7a 7c 7e 3P 1a 2a 7c 7f 7b 7e 7a 7d 5P 1a 2a 7e 7c 7a 7f 7d 7b 7P 1a 2a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 X.2 1 1 A B C /C /B /A X.3 1 1 B /C /A A C /B X.4 1 1 C /A B /B A /C X.5 1 1 /C A /B B /A C X.6 1 1 /B C A /A /C B X.7 1 1 /A /B /C C B A X.8 7 -1 . . . . . . A = E(7)^6 B = E(7)^5 C = E(7)^4