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Properties

Label 20T88
Order \(320\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^4:C_5:C_4$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $88$
Group :  $C_2^4:C_5:C_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,2,4)(5,20,16,10,6,19,15,9)(7,18,14,11,8,17,13,12), (1,8,16,10,2,7,15,9)(3,5,14,11,4,6,13,12)(17,19,18,20)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
20:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $F_5$

Degree 10: $F_5$

Low degree siblings

10T24, 10T25, 16T711, 20T77, 20T78, 20T79, 20T80, 20T83, 32T9312, 40T206, 40T207, 40T296, 40T297, 40T298, 40T299, 40T300, 40T301, 40T302, 40T303

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 5,15)( 6,16)( 7,13)( 8,14)( 9,19,10,20)(11,18,12,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7,13)( 8,14)( 9,19)(10,20)(11,18)(12,17)$
$ 4, 4, 4, 4, 2, 2 $ $20$ $4$ $( 1, 2)( 3, 4)( 5,15, 6,16)( 7,13, 8,14)( 9,19,10,20)(11,18,12,17)$
$ 4, 4, 4, 4, 4 $ $40$ $4$ $( 1, 3, 2, 4)( 5, 9,15,19)( 6,10,16,20)( 7,12,13,17)( 8,11,14,18)$
$ 8, 8, 4 $ $40$ $8$ $( 1, 3, 2, 4)( 5,19,15,10, 6,20,16, 9)( 7,17,13,11, 8,18,14,12)$
$ 8, 8, 4 $ $40$ $8$ $( 1, 4, 2, 3)( 5, 9,15,19, 6,10,16,20)( 7,12,13,17, 8,11,14,18)$
$ 4, 4, 4, 4, 4 $ $40$ $4$ $( 1, 4, 2, 3)( 5,19,16, 9)( 6,20,15,10)( 7,17,14,12)( 8,18,13,11)$
$ 5, 5, 5, 5 $ $64$ $5$ $( 1, 5,17,11,15)( 2, 6,18,12,16)( 3, 7,20, 9,13)( 4, 8,19,10,14)$

Group invariants

Order:  $320=2^{6} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [320, 1635]
Character table:    
      2  6  5  6  3  4  4  3  3  3  3  .
      5  1  .  .  .  .  .  .  .  .  .  1

        1a 2a 2b 4a 2c 4b 4c 8a 8b 4d 5a
     2P 1a 1a 1a 2a 1a 2b 2c 4b 4b 2c 5a
     3P 1a 2a 2b 4a 2c 4b 4d 8b 8a 4c 5a
     5P 1a 2a 2b 4a 2c 4b 4c 8a 8b 4d 1a
     7P 1a 2a 2b 4a 2c 4b 4d 8b 8a 4c 5a

X.1      1  1  1  1  1  1  1  1  1  1  1
X.2      1  1  1  1  1  1 -1 -1 -1 -1  1
X.3      1  1  1 -1 -1 -1  A -A  A -A  1
X.4      1  1  1 -1 -1 -1 -A  A -A  A  1
X.5      4  4  4  .  .  .  .  .  .  . -1
X.6      5  1 -3 -1  1  1 -1  1  1 -1  .
X.7      5  1 -3 -1  1  1  1 -1 -1  1  .
X.8      5  1 -3  1 -1 -1  A  A -A -A  .
X.9      5  1 -3  1 -1 -1 -A -A  A  A  .
X.10    10 -2  2  . -2  2  .  .  .  .  .
X.11    10 -2  2  .  2 -2  .  .  .  .  .

A = -E(4)
  = -Sqrt(-1) = -i