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Properties

Label 12T153
Degree $12$
Order $384$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^5.D_6$

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Show commands: Magma

magma: G := TransitiveGroup(12, 153);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $153$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^5.D_6$
CHM label:  $[(1/2.2^{2})^{3}]2S_{4}(6)_{2}{S_{4}(6d)}$
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)(2,3), (1,3)(2,12)(4,8,6,10)(5,9,7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,2,12,3)(4,6,5,7)(8,10,9,11)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$24$:  $S_4$, $S_3 \times C_4$
$48$:  $S_4\times C_2$
$96$:  12T53
$192$:  $V_4^2:(S_3\times C_2)$

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

12T153, 12T155 x 2, 16T725 x 2, 24T720, 24T836, 24T860, 24T1136, 24T1142, 24T1145, 24T1146, 24T1265 x 2, 24T1266, 24T1267, 24T1274, 24T1275, 24T1276 x 2, 24T1277 x 2, 24T1278 x 2, 24T1279 x 2, 32T9333, 32T9454

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 4, 6)( 5, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 4, 6)( 5, 7)( 8,11)( 9,10)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 4, 8)( 5, 9)( 6,10)( 7,11)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 4, 8, 5, 9)( 6,10, 7,11)$
$ 4, 2, 2, 1, 1, 1, 1 $ $12$ $4$ $( 2, 3)( 6, 7)( 8,10, 9,11)$
$ 4, 2, 2, 1, 1, 1, 1 $ $12$ $4$ $( 2, 3)( 6, 7)( 8,11, 9,10)$
$ 4, 4, 2, 1, 1 $ $24$ $4$ $( 2, 3)( 4, 8, 6,11)( 5, 9, 7,10)$
$ 4, 4, 2, 1, 1 $ $24$ $4$ $( 2, 3)( 4, 8, 7,10)( 5, 9, 6,11)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6, 7)( 8,11)( 9,10)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 2)( 3,12)( 4, 8, 6,10)( 5, 9, 7,11)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 2)( 3,12)( 4, 8, 7,11)( 5, 9, 6,10)$
$ 4, 4, 4 $ $4$ $4$ $( 1, 2,12, 3)( 4, 6, 5, 7)( 8,10, 9,11)$
$ 4, 4, 4 $ $4$ $4$ $( 1, 2,12, 3)( 4, 6, 5, 7)( 8,11, 9,10)$
$ 4, 2, 2, 2, 2 $ $12$ $4$ $( 1, 2,12, 3)( 4, 8)( 5, 9)( 6,11)( 7,10)$
$ 4, 4, 4 $ $12$ $4$ $( 1, 2,12, 3)( 4, 8, 5, 9)( 6,11, 7,10)$
$ 4, 4, 4 $ $12$ $4$ $( 1, 2,12, 3)( 4,10, 5,11)( 6, 9, 7, 8)$
$ 4, 2, 2, 2, 2 $ $12$ $4$ $( 1, 2,12, 3)( 4,10)( 5,11)( 6, 9)( 7, 8)$
$ 12 $ $32$ $12$ $( 1, 4, 8, 2, 6,11,12, 5, 9, 3, 7,10)$
$ 12 $ $32$ $12$ $( 1, 4, 8, 3, 7,10,12, 5, 9, 2, 6,11)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 4)( 2, 7)( 3, 6)( 5,12)( 8, 9)(10,11)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 4, 8)( 2, 7,11)( 3, 6,10)( 5, 9,12)$
$ 6, 6 $ $32$ $6$ $( 1, 4, 8,12, 5, 9)( 2, 7,11, 3, 6,10)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 4,12, 5)( 2, 7, 3, 6)( 8, 9)(10,11)$
$ 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:  384.5566
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);