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Properties

Label 12T295
Degree $12$
Order $95040$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $M_{12}$

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Show commands: Magma

magma: G := TransitiveGroup(12, 295);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $295$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $M_{12}$
CHM label:   $M(12)$
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:  (1,9,5,12,11,8,2,4)(6,10), (1,11,2,3,4)(5,8,12,6,7)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Low degree siblings

12T295

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{6}$ $396$ $2$ $6$ $( 1,12)( 2, 6)( 3, 8)( 4, 9)( 5, 7)(10,11)$
2B $2^{4},1^{4}$ $495$ $2$ $4$ $( 1, 2)( 3,12)( 5, 9)( 6, 7)$
3A $3^{3},1^{3}$ $1760$ $3$ $6$ $( 1,10, 4)( 3, 5,11)( 7, 9, 8)$
3B $3^{4}$ $2640$ $3$ $8$ $( 1, 2,11)( 3,10, 8)( 4, 6,12)( 5, 7, 9)$
4A $4^{2},2^{2}$ $2970$ $4$ $8$ $( 1,12, 2, 3)( 4, 8)( 5, 6, 9, 7)(10,11)$
4B $4^{2},1^{4}$ $2970$ $4$ $6$ $( 3,12, 8, 7)( 4,11, 9, 6)$
5A $5^{2},1^{2}$ $9504$ $5$ $8$ $( 1, 9, 7, 3,10)( 2, 6,12,11, 4)$
6A $6^{2}$ $7920$ $6$ $10$ $( 1,10, 6,12,11, 2)( 3, 5, 4, 8, 7, 9)$
6B $6,3,2,1$ $15840$ $6$ $8$ $( 1, 6)( 2, 8,11, 4, 9, 7)( 3, 5,10)$
8A $8,4$ $11880$ $8$ $10$ $( 1, 9,12, 7, 2, 5, 3, 6)( 4,11, 8,10)$
8B $8,2,1^{2}$ $11880$ $8$ $8$ $( 1,10, 3, 8, 7,11, 6, 5)( 2,12)$
10A $10,2$ $9504$ $10$ $10$ $( 1, 6, 3,11, 2, 7,12, 8, 9, 4)( 5,10)$
11A1 $11,1$ $8640$ $11$ $10$ $( 1, 8, 2, 4, 6, 5, 3,12,11, 9,10)$
11A-1 $11,1$ $8640$ $11$ $10$ $( 1, 3, 8,12, 2,11, 4, 9, 6,10, 5)$

Malle's constant $a(G)$:     $1/4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $95040=2^{6} \cdot 3^{3} \cdot 5 \cdot 11$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  95040.a
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 3B 4A 4B 5A 6A 6B 8A 8B 10A 11A1 11A-1
Size 1 396 495 1760 2640 2970 2970 9504 7920 15840 11880 11880 9504 8640 8640
2 P 1A 1A 1A 3A 3B 2B 2B 5A 3B 3A 4A 4B 5A 11A-1 11A1
3 P 1A 2A 2B 1A 1A 4A 4B 5A 2A 2B 8A 8B 10A 11A1 11A-1
5 P 1A 2A 2B 3A 3B 4A 4B 1A 6A 6B 8A 8B 2A 11A1 11A-1
11 P 1A 2A 2B 3A 3B 4A 4B 5A 6A 6B 8A 8B 10A 1A 1A
Type

magma: CharacterTable(G);