# Properties

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

# Related objects

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); Nilpotency class: $-1$ (not nilpotent) 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

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

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); Label: 80.50 magma: IdentifyGroup(G);
 Character table:  2 4 4 4 4 4 4 4 4 4 2 4 4 4 2 4 4 2 2 4 4 5 1 . . . 1 . . . . 1 . . . 1 . . 1 1 1 1 1a 4a 4b 2a 2b 4c 4d 2c 2d 10a 4e 4f 2e 10b 4g 4h 5a 10c 2f 2g 2P 1a 2a 2a 1a 1a 2a 2a 1a 1a 5a 2a 2a 1a 5a 2a 2a 5a 5a 1a 1a 3P 1a 4b 4a 2a 2b 4d 4c 2c 2d 10a 4f 4e 2e 10b 4h 4g 5a 10c 2f 2g 5P 1a 4a 4b 2a 2b 4c 4d 2c 2d 2f 4e 4f 2e 2g 4g 4h 1a 2b 2f 2g 7P 1a 4b 4a 2a 2b 4d 4c 2c 2d 10a 4f 4e 2e 10b 4h 4g 5a 10c 2f 2g X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 X.3 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 X.4 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 X.5 1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 X.6 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 1 X.7 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 X.8 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 X.9 1 A -A -1 -1 -A A 1 -1 1 A -A 1 -1 -A A 1 -1 1 -1 X.10 1 -A A -1 -1 A -A 1 -1 1 -A A 1 -1 A -A 1 -1 1 -1 X.11 1 A -A -1 -1 -A A 1 1 -1 -A A -1 1 A -A 1 -1 -1 1 X.12 1 -A A -1 -1 A -A 1 1 -1 A -A -1 1 -A A 1 -1 -1 1 X.13 1 A -A -1 1 A -A -1 -1 1 A -A -1 1 A -A 1 1 1 1 X.14 1 -A A -1 1 -A A -1 -1 1 -A A -1 1 -A A 1 1 1 1 X.15 1 A -A -1 1 A -A -1 1 -1 -A A 1 -1 -A A 1 1 -1 -1 X.16 1 -A A -1 1 -A A -1 1 -1 A -A 1 -1 A -A 1 1 -1 -1 X.17 4 . . . -4 . . . . -1 . . . 1 . . -1 1 4 -4 X.18 4 . . . -4 . . . . 1 . . . -1 . . -1 1 -4 4 X.19 4 . . . 4 . . . . -1 . . . -1 . . -1 -1 4 4 X.20 4 . . . 4 . . . . 1 . . . 1 . . -1 -1 -4 -4 A = -E(4) = -Sqrt(-1) = -i 

magma: CharacterTable(G);