# Properties

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

# Related objects

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); Nilpotency class: $-1$ (not nilpotent) 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 $\leq 47$

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

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2$ $396$ $2$ $( 1, 4)( 2, 9)( 3, 7)( 5, 8)( 6,10)(11,12)$ $5, 5, 1, 1$ $9504$ $5$ $( 1, 3, 9, 8, 6)( 2, 5,10, 4, 7)$ $10, 2$ $9504$ $10$ $( 1, 5, 3,10, 9, 4, 8, 7, 6, 2)(11,12)$ $2, 2, 2, 2, 1, 1, 1, 1$ $495$ $2$ $( 1, 9)( 2,12)( 3, 5)( 6,11)$ $3, 3, 3, 1, 1, 1$ $1760$ $3$ $( 1,11, 5)( 3, 9, 6)( 7, 8,10)$ $6, 3, 2, 1$ $15840$ $6$ $( 1, 3,11, 9, 5, 6)( 2,12)( 7,10, 8)$ $4, 4, 1, 1, 1, 1$ $2970$ $4$ $( 1,11, 4,10)( 3,12, 5, 7)$ $8, 2, 1, 1$ $11880$ $8$ $( 1, 3,11,12, 4, 5,10, 7)( 6, 9)$ $4, 4, 2, 2$ $2970$ $4$ $( 1, 6, 9, 4)( 2, 8)( 3, 7)( 5,11,12,10)$ $3, 3, 3, 3$ $2640$ $3$ $( 1, 6, 8)( 2, 4,12)( 3,10,11)( 5, 9, 7)$ $6, 6$ $7920$ $6$ $( 1, 3, 8,11, 6,10)( 2, 5,12, 7, 4, 9)$ $8, 4$ $11880$ $8$ $( 1, 4, 5, 6, 7,12, 3, 8)( 2, 9,11,10)$ $11, 1$ $8640$ $11$ $( 1, 2,12, 6, 9, 4,11, 3, 7, 8,10)$ $11, 1$ $8640$ $11$ $( 1,10, 8, 7, 3,11, 4, 9, 6,12, 2)$

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); Label: 95040.a magma: IdentifyGroup(G);
 Character table:  2 6 6 1 1 2 4 2 1 1 . . 5 3 5 3 3 3 1 3 1 2 1 1 . . . . . . . . 5 1 . . . . 1 . 1 1 . . . . . . 11 1 . . . . . . . . 1 1 . . . . 1a 2a 3a 6a 3b 2b 6b 5a 10a 11a 11b 4a 8a 4b 8b 2P 1a 1a 3a 3a 3b 1a 3b 5a 5a 11b 11a 2a 4a 2a 4b 3P 1a 2a 1a 2a 1a 2b 2b 5a 10a 11a 11b 4a 8a 4b 8b 5P 1a 2a 3a 6a 3b 2b 6b 1a 2b 11a 11b 4a 8a 4b 8b 7P 1a 2a 3a 6a 3b 2b 6b 5a 10a 11b 11a 4a 8a 4b 8b 11P 1a 2a 3a 6a 3b 2b 6b 5a 10a 1a 1a 4a 8a 4b 8b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 11 3 2 . -1 -1 -1 1 -1 . . -1 -1 3 1 X.3 11 3 2 . -1 -1 -1 1 -1 . . 3 1 -1 -1 X.4 16 . -2 . 1 4 1 1 -1 A /A . . . . X.5 16 . -2 . 1 4 1 1 -1 /A A . . . . X.6 45 -3 . . 3 5 -1 . . 1 1 1 -1 1 -1 X.7 54 6 . . . 6 . -1 1 -1 -1 2 . 2 . X.8 55 7 1 1 1 -5 1 . . . . -1 -1 -1 -1 X.9 55 -1 1 -1 1 -5 1 . . . . -1 1 3 -1 X.10 55 -1 1 -1 1 -5 1 . . . . 3 -1 -1 1 X.11 66 2 3 -1 . 6 . 1 1 . . -2 . -2 . X.12 99 3 . . 3 -1 -1 -1 -1 . . -1 1 -1 1 X.13 120 -8 3 1 . . . . . -1 -1 . . . . X.14 144 . . . -3 4 1 -1 -1 1 1 . . . . X.15 176 . -4 . -1 -4 -1 1 1 . . . . . . A = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10 = (-1-Sqrt(-11))/2 = -1-b11 

magma: CharacterTable(G);