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Properties

Label 12T49
Degree $12$
Order $96$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\GL(2,\mathbb{Z}/4)$

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Show commands: Magma

magma: G := TransitiveGroup(12, 49);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $49$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\GL(2,\mathbb{Z}/4)$
CHM label:   $[2]2S_{4}(6)_{2}$
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:  (1,2,9,7)(3,11,5,12)(4,8,10,6), (1,9)(2,7)(3,5)(4,10)(6,8)(11,12), (1,2)(3,5)(4,6)(7,9)(8,10), (1,3,6,12)(2,4,7,10)(5,8,11,9)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$24$:  $S_4$, $(C_6\times C_2):C_2$
$48$:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4\times C_2$

Low degree siblings

12T49, 12T50, 12T52, 16T186, 16T193, 24T153, 24T154, 24T155 x 2, 24T156, 24T157, 24T158, 24T159, 24T165, 24T166, 32T392

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $96=2^{5} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  96.195
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 3A 4A 4B 4C 6A 6B1 6B-1
Size 1 1 2 3 3 6 12 8 12 12 12 8 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 3A 2A 2C 2D 3A 3A 3A
3 P 1A 2A 2B 2C 2D 2E 2F 1A 4A 4B 4C 2A 2B 2B
Type
96.195.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
96.195.1b R 1 1 −1 1 1 −1 −1 1 1 −1 1 1 −1 −1
96.195.1c R 1 1 −1 1 1 −1 1 1 −1 1 −1 1 −1 −1
96.195.1d R 1 1 1 1 1 1 −1 1 −1 −1 −1 1 1 1
96.195.2a R 2 2 2 2 2 2 0 −1 0 0 0 −1 −1 −1
96.195.2b R 2 −2 0 2 −2 0 0 2 0 0 0 −2 0 0
96.195.2c R 2 2 −2 2 2 −2 0 −1 0 0 0 −1 1 1
96.195.2d1 C 2 −2 0 2 −2 0 0 −1 0 0 0 1 −1−2ζ3 1+2ζ3
96.195.2d2 C 2 −2 0 2 −2 0 0 −1 0 0 0 1 1+2ζ3 −1−2ζ3
96.195.3a R 3 3 3 −1 −1 −1 1 0 1 −1 −1 0 0 0
96.195.3b R 3 3 3 −1 −1 −1 −1 0 −1 1 1 0 0 0
96.195.3c R 3 3 −3 −1 −1 1 −1 0 1 1 −1 0 0 0
96.195.3d R 3 3 −3 −1 −1 1 1 0 −1 −1 1 0 0 0
96.195.6a R 6 −6 0 −2 2 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);