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Properties

Label 20T12
Degree $20$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times D_4$

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Show commands: Magma

magma: G := TransitiveGroup(20, 12);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5\times D_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $10$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,13,8,20,4,15,10,12,5,18,2,14,7,19,3,16,9,11,6,17), (1,19,9,18,7,15,5,13,4,11)(2,20,10,17,8,16,6,14,3,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$5$:  $C_5$
$8$:  $D_{4}$
$10$:  $C_{10}$ x 3
$20$:  20T3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $C_5$

Degree 10: $C_{10}$

Low degree siblings

20T12, 40T6

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, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 10, 5, 5 $ $2$ $10$ $( 1, 3, 5, 8, 9, 2, 4, 6, 7,10)(11,13,15,18,19)(12,14,16,17,20)$
$ 10, 10 $ $1$ $10$ $( 1, 3, 5, 8, 9, 2, 4, 6, 7,10)(11,14,15,17,19,12,13,16,18,20)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1, 4, 5, 7, 9)( 2, 3, 6, 8,10)(11,13,15,18,19)(12,14,16,17,20)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1, 5, 9, 4, 7)( 2, 6,10, 3, 8)(11,15,19,13,18)(12,16,20,14,17)$
$ 10, 5, 5 $ $2$ $10$ $( 1, 5, 9, 4, 7)( 2, 6,10, 3, 8)(11,16,19,14,18,12,15,20,13,17)$
$ 10, 10 $ $1$ $10$ $( 1, 6, 9, 3, 7, 2, 5,10, 4, 8)(11,16,19,14,18,12,15,20,13,17)$
$ 10, 5, 5 $ $2$ $10$ $( 1, 7, 4, 9, 5)( 2, 8, 3,10, 6)(11,17,13,20,15,12,18,14,19,16)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1, 7, 4, 9, 5)( 2, 8, 3,10, 6)(11,18,13,19,15)(12,17,14,20,16)$
$ 10, 10 $ $1$ $10$ $( 1, 8, 4,10, 5, 2, 7, 3, 9, 6)(11,17,13,20,15,12,18,14,19,16)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1, 9, 7, 5, 4)( 2,10, 8, 6, 3)(11,19,18,15,13)(12,20,17,16,14)$
$ 10, 5, 5 $ $2$ $10$ $( 1, 9, 7, 5, 4)( 2,10, 8, 6, 3)(11,20,18,16,13,12,19,17,15,14)$
$ 10, 10 $ $1$ $10$ $( 1,10, 7, 6, 4, 2, 9, 8, 5, 3)(11,20,18,16,13,12,19,17,15,14)$
$ 20 $ $2$ $20$ $( 1,11, 3,14, 5,15, 8,17, 9,19, 2,12, 4,13, 6,16, 7,18,10,20)$
$ 10, 10 $ $2$ $10$ $( 1,11, 4,13, 5,15, 7,18, 9,19)( 2,12, 3,14, 6,16, 8,17,10,20)$
$ 10, 10 $ $2$ $10$ $( 1,13, 7,19, 4,15, 9,11, 5,18)( 2,14, 8,20, 3,16,10,12, 6,17)$
$ 20 $ $2$ $20$ $( 1,13, 8,20, 4,15,10,12, 5,18, 2,14, 7,19, 3,16, 9,11, 6,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,15)( 2,16)( 3,17)( 4,18)( 5,19)( 6,20)( 7,11)( 8,12)( 9,13)(10,14)$
$ 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,15, 2,16)( 3,17, 4,18)( 5,19, 6,20)( 7,11, 8,12)( 9,13,10,14)$
$ 20 $ $2$ $20$ $( 1,17, 6,11, 9,16, 3,19, 7,14, 2,18, 5,12,10,15, 4,20, 8,13)$
$ 10, 10 $ $2$ $10$ $( 1,17, 5,12, 9,16, 4,20, 7,14)( 2,18, 6,11,10,15, 3,19, 8,13)$
$ 10, 10 $ $2$ $10$ $( 1,19, 9,18, 7,15, 5,13, 4,11)( 2,20,10,17, 8,16, 6,14, 3,12)$
$ 20 $ $2$ $20$ $( 1,19,10,17, 7,15, 6,14, 4,11, 2,20, 9,18, 8,16, 5,13, 3,12)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $40=2^{3} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  40.10
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);