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Properties

Label 20T17
Degree $20$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^4:C_5$

Related objects

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Group action invariants

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

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 2

Low degree siblings

10T8 x 3, 16T178, 20T17 x 5, 20T23, 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 TypeSizeOrderRepresentative
$ 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5,16)( 6,15)( 7,17)( 8,18)( 9,10)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 3,14)( 4,13)( 5, 6)( 7,18)( 8,17)( 9,10)(15,16)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,14)( 4,13)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)(11,12)$
$ 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