# Properties

 Label 20T9 Degree $20$ Order $40$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2\times F_5$

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## Group action invariants

 Degree $n$: $20$ Transitive number $t$: $9$ Group: $C_2\times F_5$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (1,17,13,9,6,2,18,14,10,5)(3,20,16,11,8,4,19,15,12,7), (1,15,17,3)(2,16,18,4)(5,8,13,11)(6,7,14,12)(9,19,10,20)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$20$:  $F_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $F_5$

Degree 10: $F_5$

## Low degree siblings

10T5 x 2, 20T13, 40T14

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

## Group invariants

 Order: $40=2^{3} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [40, 12]
 Character table:  2 3 3 3 3 3 3 3 3 1 1 5 1 . 1 . . . . . 1 1 1a 2a 2b 2c 4a 4b 4c 4d 10a 5a 2P 1a 1a 1a 1a 2c 2c 2c 2c 5a 5a 3P 1a 2a 2b 2c 4d 4c 4b 4a 10a 5a 5P 1a 2a 2b 2c 4a 4b 4c 4d 2b 1a 7P 1a 2a 2b 2c 4d 4c 4b 4a 10a 5a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 1 -1 -1 1 X.3 1 -1 -1 1 1 -1 -1 1 -1 1 X.4 1 1 1 1 -1 -1 -1 -1 1 1 X.5 1 -1 1 -1 A -A A -A 1 1 X.6 1 -1 1 -1 -A A -A A 1 1 X.7 1 1 -1 -1 A A -A -A -1 1 X.8 1 1 -1 -1 -A -A A A -1 1 X.9 4 . -4 . . . . . 1 -1 X.10 4 . 4 . . . . . -1 -1 A = -E(4) = -Sqrt(-1) = -i