Properties

 Label 20T23 Degree $20$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^4:C_5$

Related objects

Group action invariants

 Degree $n$: $20$ Transitive number $t$: $23$ Group: $C_2^4:C_5$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (3,14)(4,13)(5,6)(7,8)(9,20)(10,19)(15,16)(17,18), (1,6,9,13,17)(2,5,10,14,18)(3,7,11,16,19)(4,8,12,15,20)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$5$:  $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $C_5$

Degree 10: $C_2^4 : C_5$ x 3

Low degree siblings

10T8 x 3, 16T178, 20T17 x 6, 40T57 x 3

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

Group invariants

 Order: $80=2^{4} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [80, 49]
 Character table:  2 4 4 4 4 . . . . 5 1 . . . 1 1 1 1 1a 2a 2b 2c 5a 5b 5c 5d 2P 1a 1a 1a 1a 5b 5d 5a 5c 3P 1a 2a 2b 2c 5c 5a 5d 5b 5P 1a 2a 2b 2c 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 A B /B /A X.3 1 1 1 1 B /A A /B X.4 1 1 1 1 /B A /A B X.5 1 1 1 1 /A /B B A X.6 5 -3 1 1 . . . . X.7 5 1 -3 1 . . . . X.8 5 1 1 -3 . . . . A = E(5)^4 B = E(5)^3