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Properties

Label 12T40
Degree $12$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_3^2:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 40);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times C_3^2:C_4$
CHM label:   $F_{36}(6)[x]2$
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:  (1,5,9)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11), (2,10)(3,11)(4,8)(5,9), (1,7)(2,8,10,4)(3,9,11,5)(6,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $C_3^2:C_4$

Low degree siblings

12T40, 12T41 x 2, 18T27 x 2, 24T76 x 2, 36T35, 36T36

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, 2, 1, 1, 1, 1 $ $9$ $2$ $( 4, 8)( 5, 9)( 6,10)( 7,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 3, 7,11)$
$ 4, 4, 2, 2 $ $9$ $4$ $( 1, 2)( 3,12)( 4, 7, 8,11)( 5, 6, 9,10)$
$ 4, 4, 2, 2 $ $9$ $4$ $( 1, 2)( 3,12)( 4,11, 8, 7)( 5,10, 9, 6)$
$ 4, 4, 2, 2 $ $9$ $4$ $( 1, 3, 5, 7)( 2, 4, 6,12)( 8,10)( 9,11)$
$ 4, 4, 2, 2 $ $9$ $4$ $( 1, 3, 5,11)( 2, 4,10,12)( 6, 8)( 7, 9)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)$
$ 6, 2, 2, 2 $ $4$ $6$ $( 1, 4, 9,12, 5, 8)( 2, 3)( 6, 7)(10,11)$
$ 6, 6 $ $4$ $6$ $( 1, 4, 9,12, 5, 8)( 2, 7,10, 3, 6,11)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 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:  $72=2^{3} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  72.45
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 4A1 4A-1 4B1 4B-1 6A 6B
Size 1 1 9 9 4 4 9 9 9 9 4 4
2 P 1A 1A 1A 1A 3A 3B 2B 2B 2B 2B 3A 3B
3 P 1A 2A 2B 2C 1A 1A 4B-1 4A-1 4A1 4B1 2A 2A
Type
72.45.1a R 1 1 1 1 1 1 1 1 1 1 1 1
72.45.1b R 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1
72.45.1c R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1
72.45.1d R 1 1 1 1 1 1 −1 −1 −1 −1 1 1
72.45.1e1 C 1 −1 −1 1 1 1 −i i −i i −1 −1
72.45.1e2 C 1 −1 −1 1 1 1 i −i i −i −1 −1
72.45.1f1 C 1 1 −1 −1 1 1 −i i i −i 1 1
72.45.1f2 C 1 1 −1 −1 1 1 i −i −i i 1 1
72.45.4a R 4 4 0 0 −2 1 0 0 0 0 −2 1
72.45.4b R 4 4 0 0 1 −2 0 0 0 0 1 −2
72.45.4c R 4 −4 0 0 −2 1 0 0 0 0 2 −1
72.45.4d R 4 −4 0 0 1 −2 0 0 0 0 −1 2

magma: CharacterTable(G);