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Properties

Label 20T16
Degree $20$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2\times F_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 16);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $16$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2\times F_5$
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:  (1,16,14,20)(2,15,13,19)(3,10,11,5)(4,9,12,6)(7,17)(8,18), (1,9,14,6)(2,10,13,5)(3,15,11,19)(4,16,12,20)(7,8)(17,18), (1,16,10,3,17,11,5,20,14,7)(2,15,9,4,18,12,6,19,13,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $C_4\times C_2$ x 6, $C_2^3$
$16$:  $C_4\times C_2^2$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $F_5$

Degree 10: $F_{5}\times C_2$ x 3

Low degree siblings

20T16 x 3, 40T44 x 3, 40T56

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

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:  80.50
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1 5A 10A 10B 10C
Size 1 1 1 1 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2D 2D 2D 2D 2D 2D 2D 2D 5A 5A 5A 5A
5 P 1A 2A 2B 2C 2D 2E 2F 2G 4A-1 4A1 4D-1 4C1 4B-1 4D1 4C-1 4B1 1A 2A 2B 2C
Type
80.50.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.50.1b R 1 −1 −1 1 1 −1 −1 1 −1 −1 1 1 −1 −1 1 1 1 −1 −1 1
80.50.1c R 1 −1 −1 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 1
80.50.1d R 1 −1 1 −1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 1 −1 1 −1
80.50.1e R 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 −1 1 −1
80.50.1f R 1 1 −1 −1 1 −1 1 −1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1
80.50.1g R 1 1 −1 −1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1
80.50.1h R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
80.50.1i1 C 1 −1 −1 1 −1 1 1 −1 −i i i −i −i i −i i 1 −1 −1 1
80.50.1i2 C 1 −1 −1 1 −1 1 1 −1 i −i −i i i −i i −i 1 −1 −1 1
80.50.1j1 C 1 −1 1 −1 −1 −1 1 1 −i i i −i i −i i −i 1 −1 1 −1
80.50.1j2 C 1 −1 1 −1 −1 −1 1 1 i −i −i i −i i −i i 1 −1 1 −1
80.50.1k1 C 1 1 −1 −1 −1 1 −1 1 −i i −i i i −i −i i 1 1 −1 −1
80.50.1k2 C 1 1 −1 −1 −1 1 −1 1 i −i i −i −i i i −i 1 1 −1 −1
80.50.1l1 C 1 1 1 1 −1 −1 −1 −1 −i i −i i −i i i −i 1 1 1 1
80.50.1l2 C 1 1 1 1 −1 −1 −1 −1 i −i i −i i −i −i i 1 1 1 1
80.50.4a R 4 4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1 −1
80.50.4b R 4 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 1 −1
80.50.4c R 4 −4 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 −1 1
80.50.4d R 4 4 −4 −4 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 1

magma: CharacterTable(G);