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Properties

Label 12T10
Degree $12$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3 \times C_2^2$

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Show commands: Magma

magma: G := TransitiveGroup(12, 10);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $10$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3 \times C_2^2$
CHM label:   $S(3)[x]E(4)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 3

Low degree siblings

12T10 x 3, 24T11

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 6)( 3,11)( 5, 9)( 8,12)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 6, 6 $ $2$ $6$ $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$
$ 6, 6 $ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 3)( 2, 8)( 4, 6)( 5,11)( 7, 9)(10,12)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 6, 6 $ $2$ $6$ $( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $24=2^{3} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  24.14
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 3A 6A 6B 6C
Size 1 1 1 1 3 3 3 3 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 3A 3A 3A
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 2A 2B 2C
Type
24.14.1a R 1 1 1 1 1 1 1 1 1 1 1 1
24.14.1b R 1 −1 −1 1 −1 1 −1 1 1 −1 −1 1
24.14.1c R 1 −1 −1 1 1 −1 1 −1 1 −1 −1 1
24.14.1d R 1 −1 1 −1 −1 1 1 −1 1 −1 1 −1
24.14.1e R 1 −1 1 −1 1 −1 −1 1 1 −1 1 −1
24.14.1f R 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1
24.14.1g R 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1
24.14.1h R 1 1 1 1 −1 −1 −1 −1 1 1 1 1
24.14.2a R 2 2 2 2 0 0 0 0 −1 −1 −1 −1
24.14.2b R 2 −2 −2 2 0 0 0 0 −1 1 1 −1
24.14.2c R 2 −2 2 −2 0 0 0 0 −1 1 −1 1
24.14.2d R 2 2 −2 −2 0 0 0 0 −1 −1 1 1

magma: CharacterTable(G);