↑

Properties

Label 12T149
Degree $12$
Order $384$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2.\GL(2,\mathbb{Z}/4)$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(12, 149);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $149$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2.\GL(2,\mathbb{Z}/4)$
CHM label:   $[2^{4}]S_{4}(6c)_{4}$
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:  (1,12)(6,7), (1,12)(2,3)(4,5), (2,8)(3,9)(4,10)(5,11)(6,7), (1,3)(2,12)(4,10)(5,11)(6,8)(7,9), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$24$:  $S_4$, $(C_6\times C_2):C_2$
$48$:  $S_4\times C_2$
$96$:  12T49
$192$:  12T112

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$

Low degree siblings

12T147 x 2, 12T149, 24T759, 24T838, 24T852, 24T1245, 24T1246 x 2, 24T1247, 24T1248 x 4, 24T1249 x 4, 24T1254, 24T1255

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, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 8, 1, 1, 1, 1 $ $24$ $8$ $( 2, 4, 8,11, 3, 5, 9,10)$
$ 8, 2, 1, 1 $ $24$ $8$ $( 2, 4, 8,10, 3, 5, 9,11)( 6, 7)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 8, 3, 9)( 4,10, 5,11)$
$ 2, 2, 2, 2, 2, 1, 1 $ $24$ $2$ $( 2, 8)( 3, 9)( 4,10)( 5,11)( 6, 7)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 8, 3, 9)( 4,11, 5,10)$
$ 8, 2, 1, 1 $ $24$ $8$ $( 2,10, 8, 5, 3,11, 9, 4)( 6, 7)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 2)( 3,12)( 4,10, 5,11)( 6, 8)( 7, 9)$
$ 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1, 2)( 3,12)( 4,10)( 5,11)( 6, 9)( 7, 8)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 2, 4)( 3, 5,12)( 6, 8,11)( 7, 9,10)$
$ 6, 3, 3 $ $32$ $6$ $( 1, 2, 4)( 3, 5,12)( 6, 9,11, 7, 8,10)$
$ 6, 6 $ $32$ $6$ $( 1, 2, 4,12, 3, 5)( 6, 8,11, 7, 9,10)$
$ 6, 3, 3 $ $32$ $6$ $( 1, 2, 4,12, 3, 5)( 6, 9,11)( 7, 8,10)$
$ 8, 2, 2 $ $24$ $8$ $( 1, 2, 6, 8,12, 3, 7, 9)( 4, 5)(10,11)$
$ 4, 4, 4 $ $24$ $4$ $( 1, 2,12, 3)( 4,10, 5,11)( 6, 8, 7, 9)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 2,12, 3)( 4,10)( 5,11)( 6, 9, 7, 8)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 6,12, 7)( 2, 3)( 4,10, 5,11)( 8, 9)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 6,12, 7)( 2, 3)( 4,11, 5,10)( 8, 9)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $384=2^{7} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  384.5660
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 3A 4A 4B 4C 4D 4E 4F 4G 6A 6B1 6B-1 8A 8B 8C1 8C-1
Size 1 1 3 3 8 24 24 32 6 6 6 6 24 24 24 32 32 32 24 24 24 24
2 P 1A 1A 1A 1A 1A 1A 1A 3A 2B 2B 2B 2B 2C 2B 2A 3A 3A 3A 4C 4A 4A 4C
3 P 1A 2A 2B 2C 2D 2E 2F 1A 4B 4A 4D 4C 4E 4F 4G 2D 2D 2A 8A 8C-1 8C1 8B
Type
384.5660.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
384.5660.1b R 1 1 1 1 −1 −1 −1 1 1 1 1 1 1 −1 1 1 −1 −1 1 1 −1 −1
384.5660.1c R 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1 1 1
384.5660.1d R 1 1 1 1 1 1 −1 1 1 1 1 1 −1 −1 −1 1 1 1 −1 −1 −1 −1
384.5660.2a R 2 2 2 2 2 2 0 −1 2 2 2 2 0 0 0 −1 −1 −1 0 0 0 0
384.5660.2b R 2 −2 2 −2 0 0 0 2 2 2 −2 −2 0 0 0 −2 0 0 0 0 0 0
384.5660.2c R 2 2 2 2 −2 −2 0 −1 2 2 2 2 0 0 0 −1 1 1 0 0 0 0
384.5660.2d1 C 2 −2 2 −2 0 0 0 −1 2 2 −2 −2 0 0 0 1 −1−2ζ3 1+2ζ3 0 0 0 0
384.5660.2d2 C 2 −2 2 −2 0 0 0 −1 2 2 −2 −2 0 0 0 1 1+2ζ3 −1−2ζ3 0 0 0 0
384.5660.3a R 3 3 3 3 3 −1 1 0 −1 −1 −1 −1 1 1 1 0 0 0 −1 −1 −1 −1
384.5660.3b R 3 3 3 3 3 −1 −1 0 −1 −1 −1 −1 −1 −1 −1 0 0 0 1 1 1 1
384.5660.3c R 3 3 3 3 −3 1 −1 0 −1 −1 −1 −1 1 −1 1 0 0 0 −1 −1 1 1
384.5660.3d R 3 3 3 3 −3 1 1 0 −1 −1 −1 −1 −1 1 −1 0 0 0 1 1 −1 −1
384.5660.6a R 6 −6 6 −6 0 0 0 0 −2 −2 2 2 0 0 0 0 0 0 0 0 0 0
384.5660.6b R 6 6 −2 −2 0 0 −2 0 −2 2 2 −2 0 2 0 0 0 0 0 0 0 0
384.5660.6c R 6 6 −2 −2 0 0 0 0 2 −2 −2 2 2 0 −2 0 0 0 0 0 0 0
384.5660.6d R 6 6 −2 −2 0 0 0 0 2 −2 −2 2 −2 0 2 0 0 0 0 0 0 0
384.5660.6e R 6 6 −2 −2 0 0 2 0 −2 2 2 −2 0 −2 0 0 0 0 0 0 0 0
384.5660.6f R 6 −6 −2 2 0 0 0 0 2 −2 2 −2 0 0 0 0 0 0 −2 2 0 0
384.5660.6g R 6 −6 −2 2 0 0 0 0 2 −2 2 −2 0 0 0 0 0 0 2 −2 0 0
384.5660.6h1 C 6 −6 −2 2 0 0 0 0 −2 2 −2 2 0 0 0 0 0 0 0 0 −2i 2i
384.5660.6h2 C 6 −6 −2 2 0 0 0 0 −2 2 −2 2 0 0 0 0 0 0 0 0 2i −2i

magma: CharacterTable(G);