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Properties

Label 20T21
Degree $20$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_4\times D_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 21);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $21$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4\times D_5$
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:  (1,13,7,20,3,15,9,11,6,17)(2,14,8,19,4,16,10,12,5,18), (1,14,2,13)(3,12,4,11)(5,20,6,19)(7,18,8,17)(9,16,10,15), (1,11,4,14,6,15,8,18,9,20,2,12,3,13,5,16,7,17,10,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$10$:  $D_{5}$
$16$:  $D_4\times C_2$
$20$:  $D_{10}$ x 3
$40$:  20T8

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 10: $D_{10}$

Low degree siblings

20T21 x 3, 40T22 x 2, 40T39 x 2, 40T40 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, 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, 1, 1 $ $10$ $2$ $( 3, 9)( 4,10)( 5, 8)( 6, 7)(11,19)(12,20)(13,18)(14,17)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 3, 9)( 4,10)( 5, 8)( 6, 7)(11,20)(12,19)(13,17)(14,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,10)( 4, 9)( 5, 7)( 6, 8)(11,19)(12,20)(13,18)(14,17)(15,16)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1, 3, 6, 7, 9)( 2, 4, 5, 8,10)(11,13,15,17,20)(12,14,16,18,19)$
$ 10, 5, 5 $ $4$ $10$ $( 1, 3, 6, 7, 9)( 2, 4, 5, 8,10)(11,14,15,18,20,12,13,16,17,19)$
$ 10, 10 $ $2$ $10$ $( 1, 4, 6, 8, 9, 2, 3, 5, 7,10)(11,14,15,18,20,12,13,16,17,19)$
$ 10, 5, 5 $ $4$ $10$ $( 1, 5, 9, 4, 7, 2, 6,10, 3, 8)(11,15,20,13,17)(12,16,19,14,18)$
$ 10, 10 $ $2$ $10$ $( 1, 5, 9, 4, 7, 2, 6,10, 3, 8)(11,16,20,14,17,12,15,19,13,18)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1, 6, 9, 3, 7)( 2, 5,10, 4, 8)(11,15,20,13,17)(12,16,19,14,18)$
$ 10, 10 $ $4$ $10$ $( 1,11, 3,13, 6,15, 7,17, 9,20)( 2,12, 4,14, 5,16, 8,18,10,19)$
$ 20 $ $4$ $20$ $( 1,11, 4,14, 6,15, 8,18, 9,20, 2,12, 3,13, 5,16, 7,17,10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1,11)( 2,12)( 3,20)( 4,19)( 5,18)( 6,17)( 7,15)( 8,16)( 9,13)(10,14)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1,11, 2,12)( 3,20, 4,19)( 5,18, 6,17)( 7,15, 8,16)( 9,13,10,14)$
$ 20 $ $4$ $20$ $( 1,13, 8,19, 3,15,10,12, 6,17, 2,14, 7,20, 4,16, 9,11, 5,18)$
$ 10, 10 $ $4$ $10$ $( 1,13, 7,20, 3,15, 9,11, 6,17)( 2,14, 8,19, 4,16,10,12, 5,18)$
$ 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 $ $2$ $4$ $( 1,15, 2,16)( 3,17, 4,18)( 5,19, 6,20)( 7,11, 8,12)( 9,13,10,14)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $80=2^{4} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  80.39
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A1 5A2 10A1 10A3 10B1 10B3 10C1 10C3 20A1 20A3
Size 1 1 2 2 5 5 10 10 2 10 2 2 2 2 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 5A2 5A1 5A1 5A2 5A1 5A2 5A1 5A2 10A1 10A3
5 P 1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 1A 1A 2A 2A 2B 2B 2C 2C 4A 4A
Type
80.39.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.39.1b R 1 1 −1 −1 −1 −1 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1
80.39.1c R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 1
80.39.1d R 1 1 −1 1 −1 −1 −1 1 −1 1 1 1 1 1 −1 −1 1 1 −1 −1
80.39.1e R 1 1 −1 1 1 1 1 −1 −1 −1 1 1 1 1 −1 −1 1 1 −1 −1
80.39.1f R 1 1 1 −1 −1 −1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
80.39.1g R 1 1 1 −1 1 1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
80.39.1h R 1 1 1 1 −1 −1 −1 −1 1 −1 1 1 1 1 1 1 1 1 1 1
80.39.2a R 2 −2 0 0 −2 2 0 0 0 0 2 2 −2 −2 0 0 0 0 0 0
80.39.2b R 2 −2 0 0 2 −2 0 0 0 0 2 2 −2 −2 0 0 0 0 0 0
80.39.2c1 R 2 2 2 2 0 0 0 0 2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
80.39.2c2 R 2 2 2 2 0 0 0 0 2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
80.39.2d1 R 2 2 −2 −2 0 0 0 0 2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5
80.39.2d2 R 2 2 −2 −2 0 0 0 0 2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52
80.39.2e1 R 2 2 −2 2 0 0 0 0 −2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.39.2e2 R 2 2 −2 2 0 0 0 0 −2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
80.39.2f1 R 2 2 2 −2 0 0 0 0 −2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.39.2f2 R 2 2 2 −2 0 0 0 0 −2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
80.39.4a1 R 4 −4 0 0 0 0 0 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 0 0 0 0 0
80.39.4a2 R 4 −4 0 0 0 0 0 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 0 0 0 0 0

magma: CharacterTable(G);