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Properties

Label 10T31
Degree $10$
Order $720$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $M_{10}$

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Show commands: Magma

magma: G := TransitiveGroup(10, 31);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $M_{10}$
CHM label:  $M(10)=L(10)'2$
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), (1,4,2,8)(3,7,6,5), (1,2,10)(3,4,5)(6,7,8), (1,3,2,6)(4,5,8,7)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: None

Low degree siblings

12T181, 20T148, 20T150 x 2, 30T162, 36T1253, 40T591, 45T109

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $720=2^{4} \cdot 3^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Label:  720.765
magma: IdentifyGroup(G);
 
Character table:    
     2  4  2  4  3  .  3  3  .
     3  2  .  .  .  2  .  .  .
     5  1  .  .  .  .  .  .  1

       1a 4a 2a 4b 3a 8a 8b 5a
    2P 1a 2a 1a 2a 3a 4b 4b 5a
    3P 1a 4a 2a 4b 1a 8a 8b 5a
    5P 1a 4a 2a 4b 3a 8b 8a 1a
    7P 1a 4a 2a 4b 3a 8b 8a 5a

X.1     1  1  1  1  1  1  1  1
X.2     1 -1  1  1  1 -1 -1  1
X.3     9 -1  1  1  .  1  1 -1
X.4     9  1  1  1  . -1 -1 -1
X.5    10  .  2 -2  1  .  .  .
X.6    10  . -2  .  1  A -A  .
X.7    10  . -2  .  1 -A  A  .
X.8    16  .  .  . -2  .  .  1

A = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2

magma: CharacterTable(G);