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Properties

Label 10T35
Degree $10$
Order $1440$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $(A_6 : C_2) : C_2$

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Show commands: Magma

magma: G := TransitiveGroup(10, 35);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $35$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(A_6 : C_2) : C_2$
CHM label:  $L(10).2^{2}=P|L(2,9)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2)(4,7)(5,8)(9,10), (1,2,10)(3,4,5)(6,7,8), (1,7,3,4,2,5,6,8), (3,6)(4,7)(5,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: None

Low degree siblings

12T220, 20T201, 20T204, 20T208, 24T2960, 30T264, 36T2341, 40T1198, 40T1199, 40T1201, 45T187

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1 $ $45$ $2$ $( 1, 5)( 2, 9)( 4, 6)( 8,10)$
$ 4, 4, 1, 1 $ $180$ $4$ $( 1, 4, 5, 6)( 2,10, 9, 8)$
$ 2, 2, 2, 2, 2 $ $36$ $2$ $( 1, 9)( 2, 5)( 3, 7)( 4, 8)( 6,10)$
$ 4, 4, 2 $ $90$ $4$ $( 1,10, 5, 8)( 2, 4, 9, 6)( 3, 7)$
$ 2, 2, 2, 1, 1, 1, 1 $ $30$ $2$ $( 1, 5)( 3, 7)( 8,10)$
$ 3, 3, 3, 1 $ $80$ $3$ $( 1, 3, 8)( 2, 4, 6)( 5, 7,10)$
$ 6, 3, 1 $ $240$ $6$ $( 1, 2, 8, 6, 3, 4)( 5, 7,10)$
$ 4, 4, 1, 1 $ $90$ $4$ $( 1, 5, 7, 3)( 2, 8, 6,10)$
$ 8, 1, 1 $ $180$ $8$ $( 1, 6, 3, 8, 7, 2, 5,10)$
$ 8, 2 $ $180$ $8$ $( 1, 2, 5, 8, 7, 6, 3,10)( 4, 9)$
$ 5, 5 $ $144$ $5$ $( 1, 5, 8, 9, 7)( 2, 6, 4, 3,10)$
$ 10 $ $144$ $10$ $( 1, 4, 9, 2, 5, 3, 7, 6, 8,10)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1440=2^{5} \cdot 3^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Label:  1440.5841
magma: IdentifyGroup(G);
 
Character table:    
      2  5  1  4  1  3  1   1  5  4  3  4  3  3
      3  2  2  1  1  .  .   .  .  .  .  .  .  .
      5  1  .  .  .  1  1   1  .  .  .  .  .  .

        1a 3a 2a 6a 2b 5a 10a 2c 4a 8a 4b 4c 8b
     2P 1a 3a 1a 3a 1a 5a  5a 1a 2c 4a 2c 2c 4a
     3P 1a 1a 2a 2a 2b 5a 10a 2c 4a 8a 4b 4c 8b
     5P 1a 3a 2a 6a 2b 1a  2b 2c 4a 8a 4b 4c 8b
     7P 1a 3a 2a 6a 2b 5a 10a 2c 4a 8a 4b 4c 8b

X.1      1  1  1  1  1  1   1  1  1  1  1  1  1
X.2      1  1 -1 -1 -1  1  -1  1  1 -1 -1  1  1
X.3      1  1 -1 -1  1  1   1  1  1  1 -1 -1 -1
X.4      1  1  1  1 -1  1  -1  1  1 -1  1 -1 -1
X.5      9  . -3  . -1 -1  -1  1  1  1  1 -1  1
X.6      9  . -3  .  1 -1   1  1  1 -1  1  1 -1
X.7      9  .  3  . -1 -1  -1  1  1  1 -1  1 -1
X.8      9  .  3  .  1 -1   1  1  1 -1 -1 -1  1
X.9     10  1  2 -1  .  .   .  2 -2  .  2  .  .
X.10    10  1 -2  1  .  .   .  2 -2  . -2  .  .
X.11    16 -2  .  . -4  1   1  .  .  .  .  .  .
X.12    16 -2  .  .  4  1  -1  .  .  .  .  .  .
X.13    20  2  .  .  .  .   . -4  .  .  .  .  .

magma: CharacterTable(G);