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Properties

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

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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 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, 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