Properties

Label 35T36
Degree $35$
Order $20160$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_8$

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Show commands: Magma

magma: G := TransitiveGroup(35, 36);
 

Group action invariants

Degree $n$:  $35$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $36$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_8$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  20160.a
magma: IdentifyGroup(G);
 
Character table:   
      2  6  5  3  6  1  1  4  .  .  2  .   .   .  2
      3  2  1  .  1  2  1  .  .  .  2  1   1   1  1
      5  1  .  .  .  .  .  .  .  .  1  1   1   1  .
      7  1  .  .  .  .  .  .  1  1  .  .   .   .  .

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

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

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

magma: CharacterTable(G);