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Properties

Label 12T116
Degree $12$
Order $216$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3^2:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(12, 116);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $116$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3^2:S_3$
CHM label:   $[3^{3}]D(4)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7)(3,9)(5,11), (2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$24$:  $(C_6\times C_2):C_2$
$72$:  $C_3^2:D_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: None

Low degree siblings

12T120, 18T103, 18T106, 24T556, 24T560, 27T81, 36T271, 36T272, 36T273, 36T280, 36T281, 36T282, 36T291, 36T293

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 $ $8$ $3$ $( 3, 7,11)( 4,12, 8)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 2, 4)( 6, 8)(10,12)$
$ 6, 3, 1, 1, 1 $ $12$ $6$ $( 2, 4,10,12, 6, 8)( 3, 7,11)$
$ 6, 3, 1, 1, 1 $ $12$ $6$ $( 2, 4, 6, 8,10,12)( 3,11, 7)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 2, 2, 2, 2, 2, 2 $ $18$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 4, 4, 4 $ $54$ $4$ $( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10,11,12)$
$ 6, 6 $ $36$ $6$ $( 1, 2, 5, 6, 9,10)( 3, 4,11,12, 7, 8)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$
$ 6, 6 $ $18$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4,10,12, 6, 8)$
$ 3, 3, 2, 2, 2 $ $12$ $6$ $( 1, 3)( 2, 6,10)( 4,12, 8)( 5, 7)( 9,11)$
$ 6, 3, 3 $ $6$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 6,10)( 4, 8,12)$
$ 6, 3, 3 $ $6$ $6$ $( 1, 3, 9,11, 5, 7)( 2,10, 6)( 4,12, 8)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $216=2^{3} \cdot 3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  216.158
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 3D1 3D-1 3E 4A 6A1 6A-1 6B 6C1 6C-1 6D 6E
Size 1 6 9 18 2 4 4 4 4 8 54 6 6 12 12 12 18 36
2 P 1A 1A 1A 1A 3A 3B 3C 3D-1 3D1 3E 2B 3A 3A 3D-1 3C 3D1 3A 3B
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 4A 2A 2A 2A 2A 2A 2B 2C
Type
216.158.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
216.158.1b R 1 −1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 −1
216.158.1c R 1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1
216.158.1d R 1 1 1 −1 1 1 1 1 1 1 −1 1 1 1 1 1 1 −1
216.158.2a R 2 2 2 0 −1 2 2 −1 −1 −1 0 −1 −1 2 −1 −1 −1 0
216.158.2b R 2 0 −2 0 2 2 2 2 2 2 0 0 0 0 0 0 −2 0
216.158.2c R 2 −2 2 0 −1 2 2 −1 −1 −1 0 1 1 −2 1 1 −1 0
216.158.2d1 C 2 0 −2 0 −1 2 2 −1 −1 −1 0 −1−2ζ3 1+2ζ3 0 −1−2ζ3 1+2ζ3 1 0
216.158.2d2 C 2 0 −2 0 −1 2 2 −1 −1 −1 0 1+2ζ3 −1−2ζ3 0 1+2ζ3 −1−2ζ3 1 0
216.158.4a R 4 0 0 2 4 1 −2 −2 −2 1 0 0 0 0 0 0 0 −1
216.158.4b R 4 2 0 0 4 −2 1 1 1 −2 0 2 2 −1 −1 −1 0 0
216.158.4c R 4 −2 0 0 4 −2 1 1 1 −2 0 −2 −2 1 1 1 0 0
216.158.4d R 4 0 0 −2 4 1 −2 −2 −2 1 0 0 0 0 0 0 0 1
216.158.4e1 C 4 2 0 0 −2 −2 1 −2−3ζ3 1+3ζ3 1 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 0
216.158.4e2 C 4 2 0 0 −2 −2 1 1+3ζ3 −2−3ζ3 1 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 0
216.158.4f1 C 4 −2 0 0 −2 −2 1 −2−3ζ3 1+3ζ3 1 0 −2ζ3−1 −2ζ3 1 ζ3−1 ζ3 0 0
216.158.4f2 C 4 −2 0 0 −2 −2 1 1+3ζ3 −2−3ζ3 1 0 −2ζ3 −2ζ3−1 1 ζ3 ζ3−1 0 0
216.158.8a R 8 0 0 0 −4 2 −4 2 2 −1 0 0 0 0 0 0 0 0

magma: CharacterTable(G);