Properties

Label 8T25
Order \(56\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_2^3:C_7$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $25$
Group :  $C_2^3:C_7$
CHM label :  $E(8):7=F_{56}(8)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3)(2,8)(4,6)(5,7), (1,2,6,3,4,5,7), (1,8)(2,3)(4,5)(6,7), (1,5)(2,6)(3,7)(4,8)
$|\Aut(F/K)|$:  $1$

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

Low degree siblings

14T6, 28T11

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 $ $1$ $1$ $()$
$ 7, 1 $ $8$ $7$ $(2,3,6,5,7,8,4)$
$ 7, 1 $ $8$ $7$ $(2,4,8,7,5,6,3)$
$ 7, 1 $ $8$ $7$ $(2,5,4,6,8,3,7)$
$ 7, 1 $ $8$ $7$ $(2,6,7,4,3,5,8)$
$ 7, 1 $ $8$ $7$ $(2,7,3,8,6,4,5)$
$ 7, 1 $ $8$ $7$ $(2,8,5,3,4,7,6)$
$ 2, 2, 2, 2 $ $7$ $2$ $(1,2)(3,8)(4,7)(5,6)$

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 7a 7b 7c 7d 7e 7f 2a
    2P 1a 7d 7f 7b 7e 7a 7c 1a
    3P 1a 7c 7e 7d 7b 7f 7a 2a
    5P 1a 7f 7d 7a 7c 7b 7e 2a
    7P 1a 1a 1a 1a 1a 1a 1a 2a

X.1     1  1  1  1  1  1  1  1
X.2     1  A /A  C  B /C /B  1
X.3     1  B /B /A /C  A  C  1
X.4     1  C /C  B /A /B  A  1
X.5     1 /C  C /B  A  B /A  1
X.6     1 /B  B  A  C /A /C  1
X.7     1 /A  A /C /B  C  B  1
X.8     7  .  .  .  .  .  . -1

A = E(7)^6
B = E(7)^5
C = E(7)^4