# Properties

 Label 20T50 Order $$200$$ n $$20$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_5\wr C_2$

# Related objects

## Group action invariants

 Degree $n$ : $20$ Transitive number $t$ : $50$ Group : $D_5\wr C_2$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,7,14,20)(2,8,13,19)(3,18)(4,17)(5,12,9,16)(6,11,10,15), (1,13,6,18,10,2,14,5,17,9)(3,7)(4,8)(11,19)(12,20)(15,16) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: None

Degree 10: $D_5^2 : C_2$

## Low degree siblings

10T19, 10T21 x 2, 20T48 x 2, 20T50, 20T55, 20T57 x 2, 25T21, 40T167 x 2, 40T170

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $25$ $2$ $( 5,18)( 6,17)( 7,20)( 8,19)( 9,13)(10,14)(11,15)(12,16)$ $5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $5$ $( 3, 8,12,16,19)( 4, 7,11,15,20)$ $5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $5$ $( 3,12,19, 8,16)( 4,11,20, 7,15)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $10$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,19)( 8,20)( 9,10)(11,16)(12,15)(13,14)(17,18)$ $10, 2, 2, 2, 2, 2$ $20$ $10$ $( 1, 2)( 3, 7,12,15,19, 4, 8,11,16,20)( 5,17)( 6,18)( 9,14)(10,13)$ $10, 2, 2, 2, 2, 2$ $20$ $10$ $( 1, 2)( 3,11,19, 7,16, 4,12,20, 8,15)( 5,17)( 6,18)( 9,14)(10,13)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $10$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)$ $10, 10$ $20$ $10$ $( 1, 3, 6, 8,10,12,14,16,17,19)( 2, 4, 5, 7, 9,11,13,15,18,20)$ $10, 10$ $20$ $10$ $( 1, 3,10,12,17,19, 6, 8,14,16)( 2, 4, 9,11,18,20, 5, 7,13,15)$ $4, 4, 4, 4, 2, 2$ $50$ $4$ $( 1, 4)( 2, 3)( 5, 8,18,19)( 6, 7,17,20)( 9,12,13,16)(10,11,14,15)$ $5, 5, 5, 5$ $4$ $5$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3, 8,12,16,19)( 4, 7,11,15,20)$ $5, 5, 5, 5$ $8$ $5$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,12,19, 8,16)( 4,11,20, 7,15)$ $5, 5, 5, 5$ $4$ $5$ $( 1,10,17, 6,14)( 2, 9,18, 5,13)( 3,12,19, 8,16)( 4,11,20, 7,15)$

## Group invariants

 Order: $200=2^{3} \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [200, 43]
 Character table:  2 3 3 1 1 2 1 1 2 1 1 2 1 . 1 5 2 . 2 2 1 1 1 1 1 1 . 2 2 2 1a 2a 5a 5b 2b 10a 10b 2c 10c 10d 4a 5c 5d 5e 2P 1a 1a 5b 5a 1a 5b 5a 1a 5c 5e 2a 5e 5d 5c 3P 1a 2a 5b 5a 2b 10b 10a 2c 10d 10c 4a 5e 5d 5c 5P 1a 2a 1a 1a 2b 2b 2b 2c 2c 2c 4a 1a 1a 1a 7P 1a 2a 5b 5a 2b 10b 10a 2c 10d 10c 4a 5e 5d 5c X.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 X.3 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 X.5 2 -2 2 2 . . . . . . . 2 2 2 X.6 4 . A *A -2 C *C . . . . B -1 *B X.7 4 . *A A -2 *C C . . . . *B -1 B X.8 4 . A *A 2 -C -*C . . . . B -1 *B X.9 4 . *A A 2 -*C -C . . . . *B -1 B X.10 4 . B *B . . . -2 C *C . *A -1 A X.11 4 . *B B . . . -2 *C C . A -1 *A X.12 4 . B *B . . . 2 -C -*C . *A -1 A X.13 4 . *B B . . . 2 -*C -C . A -1 *A X.14 8 . -2 -2 . . . . . . . -2 3 -2 A = -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4 = (3-Sqrt(5))/2 = 1-b5 B = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 C = -E(5)^2-E(5)^3 = (1+Sqrt(5))/2 = 1+b5