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Properties

Label 35T36
Order \(20160\)
n \(35\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $A_8$

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Group action invariants

Degree $n$ :  $35$
Transitive number $t$ :  $36$
Group :  $A_8$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,4,8,3,6,12)(5,10,15,19,25,29,22)(7,14,18,24,28,34,9)(11,16,20,27,23,13,17)(21,26,32,35,30,31,33), (1,3,7)(2,6,5)(9,10,12)(11,14,13)(16,18,21)(17,22,23)(19,26,24)(25,31,28)(27,33,29)(30,35,32)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Degree 7: None

Low degree siblings

8T49, 15T72 x 2, 28T433

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, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 7, 7, 7, 7, 7 $ $2880$ $7$ $( 1,15, 7,13, 6,29,35)( 2,21,10,22, 4,33, 9)( 3,16,26,12,34,31,11) ( 5,14,23,28,18, 8,30)(17,20,25,27,19,24,32)$
$ 7, 7, 7, 7, 7 $ $2880$ $7$ $( 1,35,29, 6,13, 7,15)( 2, 9,33, 4,22,10,21)( 3,11,31,34,12,26,16) ( 5,30, 8,18,28,23,14)(17,32,24,19,27,25,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $105$ $2$ $( 1,34)( 3,30)( 4,11)( 6,24)( 7,32)( 8,35)(12,23)(13,21)(14,16)(18,20)(27,31) (28,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $1120$ $3$ $( 1,31, 7)( 2, 9,26)( 3,35,33)( 4,13,24)( 6,11,21)( 8,28,30)(10,22,19) (12,20,16)(14,23,18)(15,29,25)(27,32,34)$
$ 6, 6, 6, 6, 3, 3, 3, 1, 1 $ $3360$ $6$ $( 1,32,31,34, 7,27)( 2,26, 9)( 3,28,35,30,33, 8)( 4, 6,13,11,24,21)(10,19,22) (12,14,20,23,16,18)(15,25,29)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1 $ $1260$ $4$ $( 1,24,21,31)( 2,26, 9,17)( 3,20)( 4, 7,11,32)( 6,13,27,34)( 8,23,14,28) (12,16,33,35)(15,25)(18,30)(19,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $210$ $2$ $( 3,22)( 6,34)( 7,17)( 8,18)( 9,21)(10,16)(11,24)(12,15)(13,32)(14,30)(19,33) (20,25)(26,27)(29,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $112$ $3$ $( 1,28,31)( 2, 4, 5)( 3,25, 7)( 8,27,33)( 9,11,10)(12,13,14)(15,32,30) (16,21,24)(17,22,20)(18,26,19)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $1680$ $6$ $( 1,31,28)( 2, 5, 4)( 3,17,25,22, 7,20)( 6,34)( 8,19,27,18,33,26) ( 9,16,11,21,10,24)(12,30,13,15,14,32)(29,35)$
$ 5, 5, 5, 5, 5, 5, 5 $ $1344$ $5$ $( 1, 6,28, 5,25)( 2, 4,31, 7, 3)( 8,19,16,21,33)( 9,30,15,14,10) (11,12,35,13,32)(17,23,22,34,20)(18,29,26,27,24)$
$ 15, 15, 5 $ $1344$ $15$ $( 1,35,26, 5,11,18, 6,13,27,25,12,29,28,32,24)( 2,15,19, 7, 9,33, 4,14,16, 3, 30, 8,31,10,21)(17,23,22,34,20)$
$ 15, 15, 5 $ $1344$ $15$ $( 1,29,13, 5,24,12, 6,26,32,25,18,35,28,27,11)( 2, 8,14, 7,21,30, 4,19,10, 3, 33,15,31,16, 9)(17,23,22,34,20)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $2520$ $4$ $( 1,31)( 2, 4)( 3,21,22, 9)( 5,28)( 6,16,34,10)( 7,29,17,35)( 8,14,18,30) (11,25,24,20)(12,19,15,33)(13,27,32,26)$

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  .  2   .   .  6  1  1  4  5  3  .  .  2
      3  2  1  2   1   1  1  2  1  .  1  .  .  .  1
      5  1  1  1   1   1  .  .  .  .  .  .  .  .  .
      7  1  .  .   .   .  .  .  .  .  .  .  1  1  .

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

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

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