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Properties

Label 12T87
Degree $12$
Order $192$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^4:A_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 87);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $87$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4:A_4$
CHM label:   $[2^{5}]6$
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)(2,3), (1,3,5,7,9,11)(2,4,6,8,10,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $A_4$, $C_6\times C_2$
$24$:  $A_4\times C_2$ x 3
$48$:  $C_2^2 \times A_4$
$96$:  $C_2^4:C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$

Low degree siblings

12T87, 12T88 x 2, 16T416 x 2, 24T441 x 2, 24T442 x 2, 24T443 x 4, 24T444 x 2, 24T445 x 2, 24T446 x 2, 24T447 x 2, 24T448 x 4, 24T449, 24T450, 24T451 x 4, 24T452 x 4, 32T2183 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 $ $6$ $2$ $( 8, 9)(10,11)$
$ 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 $ $3$ $2$ $( 4, 5)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 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 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 6, 6 $ $16$ $6$ $( 1, 2, 4, 6, 8,10)( 3, 5, 7, 9,11,12)$
$ 6, 6 $ $16$ $6$ $( 1, 2, 4, 6, 8,11)( 3, 5, 7, 9,10,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 4, 8)( 2, 6,10)( 3, 7,11)( 5, 9,12)$
$ 6, 6 $ $16$ $6$ $( 1, 4, 8,12, 5, 9)( 2, 6,10, 3, 7,11)$
$ 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 6)( 2, 8)( 3, 9)( 4,10)( 5,11)( 7,12)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 6)( 2, 8, 3, 9)( 4,10, 5,11)( 7,12)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 6,12, 7)( 2, 8)( 3, 9)( 4,10, 5,11)$
$ 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 6)( 2, 8)( 3, 9)( 4,11)( 5,10)( 7,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 8, 4)( 2,10, 6)( 3,11, 7)( 5,12, 9)$
$ 6, 6 $ $16$ $6$ $( 1, 8, 5,12, 9, 4)( 2,10, 7, 3,11, 6)$
$ 6, 6 $ $16$ $6$ $( 1,10, 8, 6, 4, 2)( 3,12,11, 9, 7, 5)$
$ 6, 6 $ $16$ $6$ $( 1,10, 9, 6, 4, 2)( 3,12,11, 8, 7, 5)$
$ 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:  $192=2^{6} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  192.1000
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A1 3A-1 4A 4B 6A1 6A-1 6B1 6B-1 6C1 6C-1
Size 1 1 3 3 4 4 6 6 6 6 16 16 12 12 16 16 16 16 16 16
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 3A-1 3A1 2C 2C 3A1 3A-1 3A1 3A-1 3A1 3A-1
3 P 1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 1A 1A 4A 4B 2E 2A 2D 2D 2A 2E
Type
192.1000.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
192.1000.1b R 1 −1 −1 1 −1 1 −1 1 1 −1 1 1 1 −1 −1 −1 −1 −1 1 1
192.1000.1c R 1 −1 −1 1 1 −1 −1 1 1 −1 1 1 −1 1 −1 −1 1 1 −1 −1
192.1000.1d R 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1
192.1000.1e1 C 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
192.1000.1e2 C 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
192.1000.1f1 C 1 −1 −1 1 −1 1 −1 1 1 −1 ζ3−1 ζ3 1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3
192.1000.1f2 C 1 −1 −1 1 −1 1 −1 1 1 −1 ζ3 ζ3−1 1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1
192.1000.1g1 C 1 −1 −1 1 1 −1 −1 1 1 −1 ζ3−1 ζ3 −1 1 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3
192.1000.1g2 C 1 −1 −1 1 1 −1 −1 1 1 −1 ζ3 ζ3−1 −1 1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1
192.1000.1h1 C 1 1 1 1 −1 −1 1 1 1 1 ζ3−1 ζ3 −1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
192.1000.1h2 C 1 1 1 1 −1 −1 1 1 1 1 ζ3 ζ3−1 −1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1
192.1000.3a R 3 3 3 3 3 3 −1 −1 −1 −1 0 0 −1 −1 0 0 0 0 0 0
192.1000.3b R 3 −3 −3 3 −3 3 1 −1 −1 1 0 0 −1 1 0 0 0 0 0 0
192.1000.3c R 3 −3 −3 3 3 −3 1 −1 −1 1 0 0 1 −1 0 0 0 0 0 0
192.1000.3d R 3 3 3 3 −3 −3 −1 −1 −1 −1 0 0 1 1 0 0 0 0 0 0
192.1000.6a R 6 6 −2 −2 0 0 −2 −2 2 2 0 0 0 0 0 0 0 0 0 0
192.1000.6b R 6 6 −2 −2 0 0 2 2 −2 −2 0 0 0 0 0 0 0 0 0 0
192.1000.6c R 6 −6 2 −2 0 0 −2 2 −2 2 0 0 0 0 0 0 0 0 0 0
192.1000.6d R 6 −6 2 −2 0 0 2 −2 2 −2 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);