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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 invariants

Abstract group:  $M_{12}$
magma: IdentifyGroup(G);
 
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
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $295$
magma: t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $M(12)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(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 $2^{6}$ $396$ $2$ $6$ $( 1, 6)( 2,12)( 3,11)( 4, 8)( 5,10)( 7, 9)$
2B $2^{4},1^{4}$ $495$ $2$ $4$ $( 1,10)( 3, 7)( 6,11)( 8,12)$
3A $3^{3},1^{3}$ $1760$ $3$ $6$ $( 1,10,11)( 3, 4, 5)( 6,12, 8)$
3B $3^{4}$ $2640$ $3$ $8$ $( 1,11, 5)( 2, 3, 9)( 4,10,12)( 6, 8, 7)$
4A $4^{2},2^{2}$ $2970$ $4$ $8$ $( 1,11,10, 6)( 2, 5)( 3, 8, 7,12)( 4, 9)$
4B $4^{2},1^{4}$ $2970$ $4$ $6$ $( 1, 5,11, 8)( 2, 7, 9,10)$
5A $5^{2},1^{2}$ $9504$ $5$ $8$ $( 2, 8, 9,11, 5)( 3,10,12, 4, 7)$
6A $6^{2}$ $7920$ $6$ $10$ $( 1, 4,11,10, 5,12)( 2, 7, 3, 6, 9, 8)$
6B $6,3,2,1$ $15840$ $6$ $8$ $( 1,11,10)( 3, 6, 4,12, 5, 8)( 7, 9)$
8A $8,4$ $11880$ $8$ $10$ $( 1, 8,11, 7,10,12, 6, 3)( 2, 4, 5, 9)$
8B $8,2,1^{2}$ $11880$ $8$ $8$ $( 1, 9, 5,10,11, 2, 8, 7)( 3,12)$
10A $10,2$ $9504$ $10$ $10$ $( 1, 6)( 2, 3, 8,10, 9,12,11, 4, 5, 7)$
11A1 $11,1$ $8640$ $11$ $10$ $( 1, 6,11, 8, 5, 9, 2, 3,10, 4,12)$
11A-1 $11,1$ $8640$ $11$ $10$ $( 1,12, 4,10, 3, 2, 9, 5, 8,11, 6)$

Malle's constant $a(G)$:     $1/4$

magma: ConjugacyClasses(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
95040.a.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
95040.a.11a R 11 −1 3 2 −1 −1 3 1 −1 0 −1 1 −1 0 0
95040.a.11b R 11 −1 3 2 −1 3 −1 1 −1 0 1 −1 −1 0 0
95040.a.16a1 C 16 4 0 −2 1 0 0 1 1 0 0 0 −1 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
95040.a.16a2 C 16 4 0 −2 1 0 0 1 1 0 0 0 −1 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
95040.a.45a R 45 5 −3 0 3 1 1 0 −1 0 −1 −1 0 1 1
95040.a.54a R 54 6 6 0 0 2 2 −1 0 0 0 0 1 −1 −1
95040.a.55a R 55 −5 −1 1 1 −1 3 0 1 −1 1 −1 0 0 0
95040.a.55b R 55 −5 −1 1 1 3 −1 0 1 −1 −1 1 0 0 0
95040.a.55c R 55 −5 7 1 1 −1 −1 0 1 1 −1 −1 0 0 0
95040.a.66a R 66 6 2 3 0 −2 −2 1 0 −1 0 0 1 0 0
95040.a.99a R 99 −1 3 0 3 −1 −1 −1 −1 0 1 1 −1 0 0
95040.a.120a R 120 0 −8 3 0 0 0 0 0 1 0 0 0 −1 −1
95040.a.144a R 144 4 0 0 −3 0 0 −1 1 0 0 0 −1 1 1
95040.a.176a R 176 −4 0 −4 −1 0 0 1 −1 0 0 0 1 0 0

magma: CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{12} + 44 x^{11} + 754 x^{10} + 6060 x^{9} + 18870 x^{8} - 28356 x^{7} - 272184 x^{6} - 57864 x^{5} + 1574445 x^{4} - 92960 x^{3} + \left(-1968300 t - 1214416\right) x^{2} + \left(1968300 t + 1216456\right) x + \left(-492075 t - 304119\right)$ Copy content Toggle raw display