# Properties

 Label 10T3 Order $$20$$ n $$10$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_{10}$

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## Group action invariants

 Degree $n$ : $10$ Transitive number $t$ : $3$ Group : $D_{10}$ CHM label : $D_{10}(10)=[D(5)]2$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,3,4,5,6,7,8,9,10), (1,8)(2,7)(3,6)(4,5)(9,10) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
10:  $D_{5}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 5: $D_{5}$

## Low degree siblings

10T3, 20T4

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1$ $5$ $2$ $( 2,10)( 3, 9)( 4, 8)( 5, 7)$ $2, 2, 2, 2, 2$ $5$ $2$ $( 1, 2)( 3,10)( 4, 9)( 5, 8)( 6, 7)$ $10$ $2$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$ $5, 5$ $2$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ $10$ $2$ $10$ $( 1, 4, 7,10, 3, 6, 9, 2, 5, 8)$ $5, 5$ $2$ $5$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$ $2, 2, 2, 2, 2$ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$

## Group invariants

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