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Properties

Label 8T49
Order \(20160\)
n \(8\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $A_8$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $49$
Group :  $A_8$
CHM label :  $A8$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,4,5,6,7,8), (1,2,3)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Low degree siblings

15T72 x 2, 28T433, 35T36

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2 $ $105$ $2$ $(1,6)(2,7)(3,5)(4,8)$
$ 3, 3, 1, 1 $ $1120$ $3$ $(2,5,8)(3,4,7)$
$ 6, 2 $ $3360$ $6$ $(1,6)(2,4,5,7,8,3)$
$ 2, 2, 1, 1, 1, 1 $ $210$ $2$ $( 2, 7)( 3, 5)$
$ 4, 2, 1, 1 $ $2520$ $4$ $( 1, 6)( 2, 3, 7, 5)$
$ 3, 1, 1, 1, 1, 1 $ $112$ $3$ $(1,4,6)$
$ 3, 2, 2, 1 $ $1680$ $6$ $(1,6,4)(2,7)(3,5)$
$ 7, 1 $ $2880$ $7$ $(1,3,8,2,4,7,5)$
$ 7, 1 $ $2880$ $7$ $(1,5,7,4,2,8,3)$
$ 5, 1, 1, 1 $ $1344$ $5$ $(1,5,4,6,7)$
$ 5, 3 $ $1344$ $15$ $(1,4,7,5,6)(2,3,8)$
$ 5, 3 $ $1344$ $15$ $(1,4,7,5,6)(2,8,3)$
$ 4, 4 $ $1260$ $4$ $(1,6,8,5)(2,4,3,7)$

Group invariants

Order:  $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:    
      2  6  .  .  5  3  6  1  1  2  2  .  4   .   .
      3  2  .  .  1  .  1  2  1  2  1  1  .   1   1
      5  1  .  .  .  .  .  .  .  1  .  1  .   1   1
      7  1  1  1  .  .  .  .  .  .  .  .  .   .   .

        1a 7a 7b 2a 4a 2b 3a 6a 3b 6b 5a 4b 15a 15b
     2P 1a 7a 7b 1a 2a 1a 3a 3a 3b 3b 5a 2b 15a 15b
     3P 1a 7b 7a 2a 4a 2b 1a 2b 1a 2a 5a 4b  5a  5a
     5P 1a 7b 7a 2a 4a 2b 3a 6a 3b 6b 1a 4b  3b  3b
     7P 1a 1a 1a 2a 4a 2b 3a 6a 3b 6b 5a 4b 15b 15a
    11P 1a 7a 7b 2a 4a 2b 3a 6a 3b 6b 5a 4b 15b 15a
    13P 1a 7b 7a 2a 4a 2b 3a 6a 3b 6b 5a 4b 15b 15a

X.1      1  1  1  1  1  1  1  1  1  1  1  1   1   1
X.2      7  .  .  3  1 -1  1 -1  4  .  2 -1  -1  -1
X.3     14  .  .  2  .  6  2  . -1 -1 -1  2  -1  -1
X.4     20 -1 -1  4  .  4 -1  1  5  1  .  .   .   .
X.5     21  .  .  1 -1 -3  .  .  6 -2  1  1   1   1
X.6     21  .  .  1 -1 -3  .  . -3  1  1  1   B  /B
X.7     21  .  .  1 -1 -3  .  . -3  1  1  1  /B   B
X.8     28  .  .  4  . -4  1 -1  1  1 -2  .   1   1
X.9     35  .  . -5 -1  3  2  .  5  1  . -1   .   .
X.10    45  A /A -3  1 -3  .  .  .  .  .  1   .   .
X.11    45 /A  A -3  1 -3  .  .  .  .  .  1   .   .
X.12    56  .  .  .  .  8 -1 -1 -4  .  1  .   1   1
X.13    64  1  1  .  .  . -2  .  4  . -1  .  -1  -1
X.14    70  .  .  2  . -2  1  1 -5 -1  . -2   .   .

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
B = -E(15)-E(15)^2-E(15)^4-E(15)^8
  = (-1-Sqrt(-15))/2 = -1-b15