↑

Properties

Label 12T103
Degree $12$
Order $192$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^3:S_4$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(12, 103);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $103$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^3:S_4$
CHM label:   $1/2[E(4)^{3}]S(3)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,11)(2,4)(3,9)(5,7)(6,12)(8,10), (3,12)(6,9), (1,7)(3,9)(4,10)(6,12), (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
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$ x 3
$48$:  $S_4\times C_2$ x 3
$96$:  $V_4^2:S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$, $S_4\times C_2$ x 2

Low degree siblings

12T100 x 3, 12T101 x 6, 12T103 x 5, 12T106, 16T429, 24T432 x 3, 24T485 x 3, 24T486 x 6, 24T487 x 6, 24T488 x 3, 24T489 x 3, 24T490 x 3, 24T491 x 2, 24T492 x 6, 24T493 x 6, 24T508 x 3, 24T509 x 6, 24T510, 24T511, 32T2212 x 2

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $192=2^{6} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  192.1538
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A 4A 4B 4C 4D 4E 4F 6A
Size 1 1 3 3 3 3 3 3 6 6 12 12 32 12 12 12 12 12 12 32
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 3A 2B 2D 2D 2B 2C 2C 3A
3 P 1A 2A 2B 2F 2E 2C 2G 2D 2H 2I 2J 2K 1A 4E 4F 4C 4A 4B 4D 2A
Type
192.1538.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
192.1538.1b R 1 −1 1 1 1 −1 −1 −1 −1 1 −1 1 1 −1 −1 −1 1 1 1 −1
192.1538.1c R 1 −1 1 1 1 −1 −1 −1 −1 1 1 −1 1 1 1 1 −1 −1 −1 −1
192.1538.1d R 1 1 1 1 1 1 1 1 1 1 −1 −1 1 −1 −1 −1 −1 −1 −1 1
192.1538.2a R 2 2 2 2 2 2 2 2 2 2 0 0 −1 0 0 0 0 0 0 −1
192.1538.2b R 2 −2 2 2 2 −2 −2 −2 −2 2 0 0 −1 0 0 0 0 0 0 1
192.1538.3a R 3 3 −1 −1 3 −1 3 −1 −1 −1 1 1 0 −1 −1 1 −1 −1 1 0
192.1538.3b R 3 3 −1 3 −1 −1 −1 3 −1 −1 1 1 0 −1 1 −1 1 −1 −1 0
192.1538.3c R 3 3 3 −1 −1 3 −1 −1 −1 −1 1 1 0 1 −1 −1 −1 1 −1 0
192.1538.3d R 3 3 −1 −1 3 −1 3 −1 −1 −1 −1 −1 0 1 1 −1 1 1 −1 0
192.1538.3e R 3 3 −1 3 −1 −1 −1 3 −1 −1 −1 −1 0 1 −1 1 −1 1 1 0
192.1538.3f R 3 3 3 −1 −1 3 −1 −1 −1 −1 −1 −1 0 −1 1 1 1 −1 1 0
192.1538.3g R 3 −3 −1 −1 3 1 −3 1 1 −1 −1 1 0 1 1 −1 −1 −1 1 0
192.1538.3h R 3 −3 −1 −1 3 1 −3 1 1 −1 1 −1 0 −1 −1 1 1 1 −1 0
192.1538.3i R 3 −3 −1 3 −1 1 1 −3 1 −1 −1 1 0 1 −1 1 1 −1 −1 0
192.1538.3j R 3 −3 −1 3 −1 1 1 −3 1 −1 1 −1 0 −1 1 −1 −1 1 1 0
192.1538.3k R 3 −3 3 −1 −1 −3 1 1 1 −1 −1 1 0 −1 1 1 −1 1 −1 0
192.1538.3l R 3 −3 3 −1 −1 −3 1 1 1 −1 1 −1 0 1 −1 −1 1 −1 1 0
192.1538.6a R 6 6 −2 −2 −2 −2 −2 −2 2 2 0 0 0 0 0 0 0 0 0 0
192.1538.6b R 6 −6 −2 −2 −2 2 2 2 −2 2 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);