# Properties

 Label 20T2 Order $$20$$ n $$20$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5:C_4$

# Related objects

## Group action invariants

 Degree $n$ : $20$ Transitive number $t$ : $2$ Group : $C_5:C_4$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,19,2,20)(3,17,4,18)(5,15,6,16)(7,14,8,13)(9,12,10,11), (1,12,2,11)(3,10,4,9)(5,7,6,8)(13,19,14,20)(15,17,16,18) $|\Aut(F/K)|$: $20$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
10:  $D_{5}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $D_{5}$

Degree 10: $D_5$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

## Group invariants

 Order: $20=2^{2} \cdot 5$ Cyclic: No Abelian: No Solvable: Yes GAP id: [20, 1]
 Character table:  2 2 2 2 2 1 1 1 1 5 1 1 . . 1 1 1 1 1a 2a 4a 4b 5a 10a 5b 10b 2P 1a 1a 2a 2a 5b 5b 5a 5a 3P 1a 2a 4b 4a 5b 10b 5a 10a 5P 1a 2a 4a 4b 1a 2a 1a 2a 7P 1a 2a 4b 4a 5b 10b 5a 10a X.1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 1 1 X.3 1 -1 A -A 1 -1 1 -1 X.4 1 -1 -A A 1 -1 1 -1 X.5 2 -2 . . B -B *B -*B X.6 2 -2 . . *B -*B B -B X.7 2 2 . . B B *B *B X.8 2 2 . . *B *B B B A = -E(4) = -Sqrt(-1) = -i B = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5