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Properties

Label 12T135
Degree $12$
Order $384$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\wr S_3$

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Show commands: Magma

magma: G := TransitiveGroup(12, 135);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $135$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\wr S_3$
CHM label:   $[2^{6}]D_{6}=2wrD_{6}(6)$
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), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,11)(2,8)(3,9)(4,6)(5,7)(10,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$24$:  $S_4$ x 3, $(C_6\times C_2):C_2$
$48$:  $S_4\times C_2$ x 3
$96$:  $V_4^2:S_3$, 12T49 x 3
$192$:  12T100

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$

Low degree siblings

12T135, 12T148 x 6, 16T765 x 2, 24T1003 x 6, 24T1004 x 3, 24T1005 x 3, 24T1008 x 3, 24T1010 x 3, 24T1123 x 6, 24T1173 x 6, 24T1174 x 6, 24T1175 x 2, 24T1176 x 12, 24T1177 x 12, 24T1178 x 2, 24T1179 x 6, 24T1180 x 3, 24T1181 x 2, 24T1182 x 2, 24T1183, 24T1250 x 12, 24T1251 x 12, 24T1252 x 3, 24T1253 x 3, 32T9368

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 8, 9)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 6, 7)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $(6,7)(8,9)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 2, 3)( 6, 7)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $(2,3)(6,7)(8,9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 2, 3)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $(2,3)(4,5)(6,7)(8,9)$
$ 2, 2, 2, 2, 2, 1, 1 $ $6$ $2$ $( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1, 2)( 3,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 2)( 3,12)( 4,10, 5,11)( 6, 8)( 7, 9)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 2)( 3,12)( 4,10)( 5,11)( 6, 8, 7, 9)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 2)( 3,12)( 4,10, 5,11)( 6, 8, 7, 9)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 2,12, 3)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 2,12, 3)( 4,10, 5,11)( 6, 8)( 7, 9)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 2,12, 3)( 4,10)( 5,11)( 6, 8, 7, 9)$
$ 4, 4, 4 $ $24$ $4$ $( 1, 2,12, 3)( 4,10, 5,11)( 6, 8, 7, 9)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 4, 8)( 2, 6,10)( 3, 7,11)( 5, 9,12)$
$ 6, 3, 3 $ $32$ $6$ $( 1, 4, 8)( 2, 6,10, 3, 7,11)( 5, 9,12)$
$ 6, 3, 3 $ $32$ $6$ $( 1, 4, 8,12, 5, 9)( 2, 6,10)( 3, 7,11)$
$ 6, 6 $ $32$ $6$ $( 1, 4, 8,12, 5, 9)( 2, 6,10, 3, 7,11)$
$ 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:  384.20100
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 2P 3A 4A 4B 4C 4D 4E 4F 4G 6A 6B1 6B-1
Size 1 1 2 3 3 3 3 3 3 6 6 6 6 6 6 6 24 32 24 24 24 24 24 24 24 32 32 32
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 3A 2C 2G 2D 2H 2F 2A 2E 3A 3A 3A
3 P 1A 2A 2B 2C 2F 2D 2H 2E 2G 2I 2N 2K 2O 2M 2J 2L 2P 1A 4A 4F 4C 4G 4E 4B 4D 2B 2B 2A
Type
384.20100.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
384.20100.1b R 1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 −1 1 1 1 −1 1 1 −1 −1 1 −1 −1
384.20100.1c R 1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 1 −1 −1 1 −1 −1 1 1 1 −1 −1
384.20100.1d R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 1 −1 −1 −1 −1 −1 −1 −1 1 1 1
384.20100.2a R 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 −1 0 0 0 0 0 0 0 −1 −1 −1
384.20100.2b R 2 −2 0 −2 2 −2 −2 2 2 −2 0 0 0 0 0 2 0 2 0 0 0 0 0 0 0 −2 0 0
384.20100.2c R 2 2 −2 2 2 2 2 2 2 2 −2 −2 −2 −2 −2 2 0 −1 0 0 0 0 0 0 0 −1 1 1
384.20100.2d1 C 2 −2 0 −2 2 −2 −2 2 2 −2 0 0 0 0 0 2 0 −1 0 0 0 0 0 0 0 1 −1−2ζ3 1+2ζ3
384.20100.2d2 C 2 −2 0 −2 2 −2 −2 2 2 −2 0 0 0 0 0 2 0 −1 0 0 0 0 0 0 0 1 1+2ζ3 −1−2ζ3
384.20100.3a R 3 3 3 −1 −1 3 −1 −1 3 −1 3 −1 −1 −1 −1 −1 1 0 −1 1 −1 1 −1 −1 1 0 0 0
384.20100.3b R 3 3 3 −1 3 −1 3 −1 −1 −1 −1 −1 −1 3 −1 −1 1 0 −1 1 1 −1 1 −1 −1 0 0 0
384.20100.3c R 3 3 3 3 −1 −1 −1 3 −1 −1 −1 3 −1 −1 −1 −1 1 0 1 1 −1 −1 −1 1 −1 0 0 0
384.20100.3d R 3 3 3 −1 −1 3 −1 −1 3 −1 3 −1 −1 −1 −1 −1 −1 0 1 −1 1 −1 1 1 −1 0 0 0
384.20100.3e R 3 3 3 −1 3 −1 3 −1 −1 −1 −1 −1 −1 3 −1 −1 −1 0 1 −1 −1 1 −1 1 1 0 0 0
384.20100.3f R 3 3 3 3 −1 −1 −1 3 −1 −1 −1 3 −1 −1 −1 −1 −1 0 −1 −1 1 1 1 −1 1 0 0 0
384.20100.3g R 3 3 −3 −1 −1 3 −1 −1 3 −1 −3 1 1 1 1 −1 −1 0 −1 1 1 1 −1 1 −1 0 0 0
384.20100.3h R 3 3 −3 −1 −1 3 −1 −1 3 −1 −3 1 1 1 1 −1 1 0 1 −1 −1 −1 1 −1 1 0 0 0
384.20100.3i R 3 3 −3 −1 3 −1 3 −1 −1 −1 1 1 1 −3 1 −1 −1 0 −1 1 −1 −1 1 1 1 0 0 0
384.20100.3j R 3 3 −3 −1 3 −1 3 −1 −1 −1 1 1 1 −3 1 −1 1 0 1 −1 1 1 −1 −1 −1 0 0 0
384.20100.3k R 3 3 −3 3 −1 −1 −1 3 −1 −1 1 −3 1 1 1 −1 −1 0 1 1 1 −1 −1 −1 1 0 0 0
384.20100.3l R 3 3 −3 3 −1 −1 −1 3 −1 −1 1 −3 1 1 1 −1 1 0 −1 −1 −1 1 1 1 −1 0 0 0
384.20100.6a R 6 6 6 −2 −2 −2 −2 −2 −2 2 −2 −2 2 −2 2 2 0 0 0 0 0 0 0 0 0 0 0 0
384.20100.6b R 6 −6 0 −6 −2 2 2 6 −2 2 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0
384.20100.6c R 6 −6 0 2 −2 −6 2 −2 6 2 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0
384.20100.6d R 6 −6 0 2 6 2 −6 −2 −2 2 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0
384.20100.6e R 6 6 −6 −2 −2 −2 −2 −2 −2 2 2 2 −2 2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
384.20100.6f R 6 −6 0 2 −2 2 2 −2 −2 −2 0 0 −4 0 4 2 0 0 0 0 0 0 0 0 0 0 0 0
384.20100.6g R 6 −6 0 2 −2 2 2 −2 −2 −2 0 0 4 0 −4 2 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);