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Properties

Label 12T183
Degree $12$
Order $720$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_6$

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Show commands: Magma

magma: G := TransitiveGroup(12, 183);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $183$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_6$
CHM label:   $S_{6}(12)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3,5)(2,4,12), (2,4,6,8,10)(3,5,7,9,11), (1,12)(2,3)(4,5)(6,7)(8,11)(9,10)
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: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $S_6$

Low degree siblings

6T16 x 2, 10T32, 12T183, 15T28 x 2, 20T145, 20T149 x 2, 30T164 x 2, 30T166 x 2, 30T176 x 2, 36T1252, 40T589, 40T592 x 2, 45T96

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $40$ $3$ $( 6, 8,10)( 7, 9,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $45$ $2$ $( 4, 6)( 5, 7)( 8,10)( 9,11)$
$ 5, 5, 1, 1 $ $144$ $5$ $( 2, 4, 6, 8,10)( 3, 5, 7, 9,11)$
$ 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 6, 2, 2, 2 $ $120$ $6$ $( 1, 2)( 3,12)( 4, 5)( 6, 9,10, 7, 8,11)$
$ 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3,12)( 4, 7)( 5, 6)( 8,11)( 9,10)$
$ 4, 4, 2, 2 $ $90$ $4$ $( 1, 2, 5, 6)( 3, 4, 7,12)( 8, 9)(10,11)$
$ 6, 6 $ $120$ $6$ $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$
$ 4, 4, 2, 2 $ $90$ $4$ $( 1, 3, 5, 7)( 2, 4, 6,12)( 8,10)( 9,11)$
$ 3, 3, 3, 3 $ $40$ $3$ $( 1, 3, 5)( 2, 4,12)( 6, 8,10)( 7, 9,11)$

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);
 
Nilpotency class:   not nilpotent
Label:  720.763
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 4A 4B 5A 6A 6B
Size 1 15 15 45 40 40 90 90 144 120 120
2 P 1A 1A 1A 1A 3A 3B 2C 2C 5A 3A 3B
3 P 1A 2A 2B 2C 1A 1A 4A 4B 5A 2A 2B
5 P 1A 2A 2B 2C 3A 3B 4A 4B 1A 6A 6B
Type
720.763.1a R 1 1 1 1 1 1 1 1 1 1 1
720.763.1b R 1 −1 −1 1 1 1 1 −1 1 −1 −1
720.763.5a R 5 −1 3 1 −1 2 −1 1 0 −1 0
720.763.5b R 5 3 −1 1 2 −1 −1 1 0 0 −1
720.763.5c R 5 −3 1 1 2 −1 −1 −1 0 0 1
720.763.5d R 5 1 −3 1 −1 2 −1 −1 0 1 0
720.763.9a R 9 3 3 1 0 0 1 −1 −1 0 0
720.763.9b R 9 −3 −3 1 0 0 1 1 −1 0 0
720.763.10a R 10 −2 2 −2 1 1 0 0 0 1 −1
720.763.10b R 10 2 −2 −2 1 1 0 0 0 −1 1
720.763.16a R 16 0 0 0 −2 −2 0 0 1 0 0

magma: CharacterTable(G);