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Properties

Label 20T123
Degree $20$
Order $480$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_4\times S_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 123);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $123$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4\times S_5$
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:  (1,15,18,3)(2,16,17,4)(5,8,13,11)(6,7,14,12)(9,19,10,20), (1,4,10,8,6,11,14,20,17,15,2,3,9,7,5,12,13,19,18,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$120$:  $S_5$
$240$:  $S_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $S_5$

Degree 10: $S_5\times C_2$

Low degree siblings

20T123, 24T1347 x 2, 24T1348 x 2, 40T408 x 2, 40T409 x 2, 40T430

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 7,19)( 8,20)( 9,17)(10,18)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 5,10,18)( 6, 9,17)( 7,15,19)( 8,16,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $15$ $2$ $( 3, 8)( 4, 7)( 5,10)( 6, 9)(13,17)(14,18)(15,19)(16,20)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $30$ $4$ $( 3, 8,16,20)( 4, 7,15,19)( 5,10,14,18)( 6, 9,13,17)$
$ 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 $ $10$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,20)( 8,19)( 9,18)(10,17)(11,12)(13,14)(15,16)$
$ 6, 6, 2, 2, 2, 2 $ $20$ $6$ $( 1, 2)( 3, 4)( 5, 9,18, 6,10,17)( 7,16,19, 8,15,20)(11,12)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3, 7)( 4, 8)( 5, 9)( 6,10)(11,12)(13,18)(14,17)(15,20)(16,19)$
$ 4, 4, 4, 4, 2, 2 $ $30$ $4$ $( 1, 2)( 3, 7,16,19)( 4, 8,15,20)( 5, 9,14,17)( 6,10,13,18)(11,12)$
$ 4, 4, 4, 4, 4 $ $15$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,20,10,19)(11,13,12,14)(15,17,16,18)$
$ 12, 4, 4 $ $20$ $12$ $( 1, 3, 2, 4)( 5, 7, 9,16,18,19, 6, 8,10,15,17,20)(11,13,12,14)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 3, 2, 4)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)(11,13,12,14)$
$ 4, 4, 4, 4, 4 $ $30$ $4$ $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,20,10,19)(11,13,16,18)(12,14,15,17)$
$ 20 $ $24$ $20$ $( 1, 3, 5, 7, 9,12,14,15,17,20, 2, 4, 6, 8,10,11,13,16,18,19)$
$ 12, 4, 4 $ $20$ $12$ $( 1, 3, 5,11,13,16, 2, 4, 6,12,14,15)( 7,17, 8,18)( 9,20,10,19)$
$ 4, 4, 4, 4, 4 $ $15$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,19,10,20)(11,14,12,13)(15,18,16,17)$
$ 12, 4, 4 $ $20$ $12$ $( 1, 4, 2, 3)( 5, 8, 9,15,18,20, 6, 7,10,16,17,19)(11,14,12,13)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 4, 2, 3)( 5,16, 6,15)( 7,18, 8,17)( 9,19,10,20)(11,14,12,13)$
$ 4, 4, 4, 4, 4 $ $30$ $4$ $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,19,10,20)(11,14,16,17)(12,13,15,18)$
$ 20 $ $24$ $20$ $( 1, 4, 5, 8, 9,11,14,16,17,19, 2, 3, 6, 7,10,12,13,15,18,20)$
$ 12, 4, 4 $ $20$ $12$ $( 1, 4, 5,12,13,15, 2, 3, 6,11,14,16)( 7,18, 8,17)( 9,19,10,20)$
$ 6, 6, 2, 2, 2, 2 $ $20$ $6$ $( 1, 5)( 2, 6)( 3, 7,20, 4, 8,19)( 9,14,17,10,13,18)(11,16)(12,15)$
$ 10, 10 $ $24$ $10$ $( 1, 5, 9,14,17, 2, 6,10,13,18)( 3, 7,12,15,20, 4, 8,11,16,19)$
$ 3, 3, 3, 3, 2, 2, 2, 2 $ $20$ $6$ $( 1, 6)( 2, 5)( 3, 8,20)( 4, 7,19)( 9,13,17)(10,14,18)(11,15)(12,16)$
$ 5, 5, 5, 5 $ $24$ $5$ $( 1, 6, 9,13,17)( 2, 5,10,14,18)( 3, 8,12,16,20)( 4, 7,11,15,19)$
$ 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,11, 2,12)( 3,13, 4,14)( 5,16, 6,15)( 7,18, 8,17)( 9,19,10,20)$
$ 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,12, 2,11)( 3,14, 4,13)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $480=2^{5} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  480.943
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 3A 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D 4E 4F1 4F-1 5A 6A 6B 6C 10A 12A1 12A-1 12B1 12B-1 20A1 20A-1
Size 1 1 10 10 15 15 20 1 1 10 10 15 15 30 30 30 30 24 20 20 20 24 20 20 20 20 24 24
2 P 1A 1A 1A 1A 1A 1A 3A 2A 2A 2A 2A 2A 2A 2E 2E 2D 2D 5A 3A 3A 3A 5A 6A 6A 6A 6A 10A 10A
3 P 1A 2A 2B 2C 2D 2E 1A 4A-1 4A1 4B-1 4B1 4C-1 4C1 4D 4E 4F-1 4F1 5A 2B 2C 2A 10A 4A1 4A-1 4B1 4B-1 20A-1 20A1
5 P 1A 2A 2B 2C 2D 2E 3A 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D 4E 4F1 4F-1 1A 6B 6C 6A 2A 12A1 12A-1 12B1 12B-1 4A1 4A-1
Type
480.943.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
480.943.1b R 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 1 −1 −1 1 1 −1 −1
480.943.1c R 1 1 −1 −1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 −1 −1 1 1 1 −1 −1 1 1
480.943.1d R 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
480.943.1e1 C 1 −1 −1 1 −1 1 1 −i i i −i i −i 1 −1 i −i 1 −1 −1 1 −1 i −i −i i −i i
480.943.1e2 C 1 −1 −1 1 −1 1 1 i −i −i i −i i 1 −1 −i i 1 −1 −1 1 −1 −i i i −i i −i
480.943.1f1 C 1 −1 1 −1 −1 1 1 −i i −i i i −i −1 1 −i i 1 −1 1 −1 −1 i −i i −i −i i
480.943.1f2 C 1 −1 1 −1 −1 1 1 i −i i −i −i i −1 1 i −i 1 −1 1 −1 −1 −i i −i i i −i
480.943.4a R 4 4 2 2 0 0 1 4 4 2 2 0 0 0 0 0 0 −1 1 −1 −1 −1 1 1 −1 −1 −1 −1
480.943.4b R 4 4 −2 −2 0 0 1 4 4 −2 −2 0 0 0 0 0 0 −1 1 1 1 −1 1 1 1 1 −1 −1
480.943.4c R 4 4 −2 −2 0 0 1 −4 −4 2 2 0 0 0 0 0 0 −1 1 1 1 −1 −1 −1 −1 −1 1 1
480.943.4d R 4 4 2 2 0 0 1 −4 −4 −2 −2 0 0 0 0 0 0 −1 1 −1 −1 −1 −1 −1 1 1 1 1
480.943.4e1 C 4 −4 −2 2 0 0 1 −4i 4i 2i −2i 0 0 0 0 0 0 −1 −1 1 −1 1 i −i i −i i −i
480.943.4e2 C 4 −4 −2 2 0 0 1 4i −4i −2i 2i 0 0 0 0 0 0 −1 −1 1 −1 1 −i i −i i −i i
480.943.4f1 C 4 −4 2 −2 0 0 1 −4i 4i −2i 2i 0 0 0 0 0 0 −1 −1 −1 1 1 i −i −i i i −i
480.943.4f2 C 4 −4 2 −2 0 0 1 4i −4i 2i −2i 0 0 0 0 0 0 −1 −1 −1 1 1 −i i i −i −i i
480.943.5a R 5 5 −1 −1 1 1 −1 5 5 −1 −1 1 1 1 1 1 1 0 −1 −1 −1 0 −1 −1 −1 −1 0 0
480.943.5b R 5 5 1 1 1 1 −1 5 5 1 1 1 1 −1 −1 −1 −1 0 −1 1 1 0 −1 −1 1 1 0 0
480.943.5c R 5 5 −1 −1 1 1 −1 −5 −5 1 1 −1 −1 1 1 −1 −1 0 −1 −1 −1 0 1 1 1 1 0 0
480.943.5d R 5 5 1 1 1 1 −1 −5 −5 −1 −1 −1 −1 −1 −1 1 1 0 −1 1 1 0 1 1 −1 −1 0 0
480.943.5e1 C 5 −5 −1 1 −1 1 −1 −5i 5i i −i i −i −1 1 −i i 0 1 −1 1 0 −i i −i i 0 0
480.943.5e2 C 5 −5 −1 1 −1 1 −1 5i −5i −i i −i i −1 1 i −i 0 1 −1 1 0 i −i i −i 0 0
480.943.5f1 C 5 −5 1 −1 −1 1 −1 −5i 5i −i i i −i 1 −1 i −i 0 1 1 −1 0 −i i i −i 0 0
480.943.5f2 C 5 −5 1 −1 −1 1 −1 5i −5i i −i −i i 1 −1 −i i 0 1 1 −1 0 i −i −i i 0 0
480.943.6a R 6 6 0 0 −2 −2 0 6 6 0 0 −2 −2 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1
480.943.6b R 6 6 0 0 −2 −2 0 −6 −6 0 0 2 2 0 0 0 0 1 0 0 0 1 0 0 0 0 −1 −1
480.943.6c1 C 6 −6 0 0 2 −2 0 −6i 6i 0 0 −2i 2i 0 0 0 0 1 0 0 0 −1 0 0 0 0 −i i
480.943.6c2 C 6 −6 0 0 2 −2 0 6i −6i 0 0 2i −2i 0 0 0 0 1 0 0 0 −1 0 0 0 0 i −i

magma: CharacterTable(G);