Properties

 Label 20T5 Degree $20$ Order $20$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $F_5$

Related objects

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$

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

5T3, 10T4

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

Group invariants

 Order: $20=2^{2} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [20, 3]
 Character table:  2 2 2 2 2 . 5 1 . . . 1 1a 2a 4a 4b 5a 2P 1a 1a 2a 2a 5a 3P 1a 2a 4b 4a 5a 5P 1a 2a 4a 4b 1a X.1 1 1 1 1 1 X.2 1 1 -1 -1 1 X.3 1 -1 A -A 1 X.4 1 -1 -A A 1 X.5 4 . . . -1 A = -E(4) = -Sqrt(-1) = -i