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Properties

Label 20T6
Degree $20$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4\times D_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 6);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4\times D_5$
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:  (1,15,2,16)(3,14,4,13)(5,12,6,11)(7,9,8,10)(17,20,18,19), (1,3,5,8,10,11,14,16,18,19,2,4,6,7,9,12,13,15,17,20)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$10$:  $D_{5}$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $D_{5}$

Degree 10: $D_{10}$

Low degree siblings

20T6, 40T9

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

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:   not nilpotent
Label:  40.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 5A1 5A2 10A1 10A3 20A1 20A-1 20A3 20A-3
Size 1 1 5 5 1 1 5 5 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2A 2A 2A 2A 5A2 5A1 5A1 5A2 10A1 10A3 10A1 10A3
5 P 1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 1A 1A 2A 2A 4A-1 4A-1 4A1 4A1
Type
40.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
40.5.1b R 1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
40.5.1c R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 1 1
40.5.1d R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1
40.5.1e1 C 1 −1 −1 1 −i i −i i 1 1 −1 −1 −i i i −i
40.5.1e2 C 1 −1 −1 1 i −i i −i 1 1 −1 −1 i −i −i i
40.5.1f1 C 1 −1 1 −1 −i i i −i 1 1 −1 −1 −i i i −i
40.5.1f2 C 1 −1 1 −1 i −i −i i 1 1 −1 −1 i −i −i i
40.5.2a1 R 2 2 0 0 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
40.5.2a2 R 2 2 0 0 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
40.5.2b1 R 2 2 0 0 −2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
40.5.2b2 R 2 2 0 0 −2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
40.5.2c1 C 2 −2 0 0 −2ζ205 2ζ205 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 −ζ20−4−ζ204 ζ20−2+ζ202 ζ203+ζ207 −ζ203−ζ207 ζ203−ζ205+ζ207 −ζ203+ζ205−ζ207
40.5.2c2 C 2 −2 0 0 2ζ205 −2ζ205 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 −ζ20−4−ζ204 ζ20−2+ζ202 −ζ203−ζ207 ζ203+ζ207 −ζ203+ζ205−ζ207 ζ203−ζ205+ζ207
40.5.2c3 C 2 −2 0 0 −2ζ205 2ζ205 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 ζ20−2+ζ202 −ζ20−4−ζ204 −ζ203+ζ205−ζ207 ζ203−ζ205+ζ207 −ζ203−ζ207 ζ203+ζ207
40.5.2c4 C 2 −2 0 0 2ζ205 −2ζ205 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 ζ20−2+ζ202 −ζ20−4−ζ204 ζ203−ζ205+ζ207 −ζ203+ζ205−ζ207 ζ203+ζ207 −ζ203−ζ207

magma: CharacterTable(G);