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Properties

Label 20T120
Order \(480\)
n \(20\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_4:S_5$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $120$
Group :  $C_4:S_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,10,15,17,12,14,19,6,8,2,4,9,16,18,11,13,20,5,7), (1,10,13,5)(2,9,14,6)(3,16,12,20)(4,15,11,19)(17,18)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$
120:  $S_5$
240:  $S_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $S_5$

Degree 10: $S_5\times C_2$

Low degree siblings

20T120, 24T1352 x 2, 40T411 x 2, 40T412 x 2, 40T431

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 5,10,18)( 6, 9,17)( 7,15,19)( 8,16,20)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $20$ $2$ $( 3, 4)( 7,20)( 8,19)( 9,17)(10,18)(11,12)(15,16)$
$ 4, 4, 4, 4, 2, 1, 1 $ $60$ $4$ $( 3, 7,16,19)( 4, 8,15,20)( 5,10,14,18)( 6, 9,13,17)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $15$ $2$ $( 3, 8)( 4, 7)( 5,10)( 6, 9)(13,17)(14,18)(15,19)(16,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 6, 6, 2, 2, 2, 2 $ $20$ $6$ $( 1, 2)( 3, 4)( 5, 9,18, 6,10,17)( 7,16,19, 8,15,20)(11,12)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3, 7)( 4, 8)( 5, 9)( 6,10)(11,12)(13,18)(14,17)(15,20)(16,19)$
$ 6, 6, 2, 2, 2, 2 $ $40$ $6$ $( 1, 3)( 2, 4)( 5, 7,10,15,18,19)( 6, 8, 9,16,17,20)(11,14)(12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,18)( 8,17)( 9,20)(10,19)(11,14)(12,13)$
$ 4, 4, 4, 4, 4 $ $30$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,20,10,19)(11,13,12,14)(15,17,16,18)$
$ 20 $ $24$ $20$ $( 1, 3, 5, 7, 9,12,14,15,17,20, 2, 4, 6, 8,10,11,13,16,18,19)$
$ 12, 4, 4 $ $40$ $12$ $( 1, 3, 5,11,13,16, 2, 4, 6,12,14,15)( 7,17, 8,18)( 9,20,10,19)$
$ 20 $ $24$ $20$ $( 1, 3, 5,19,17,12,14,15, 9, 8, 2, 4, 6,20,18,11,13,16,10, 7)$
$ 4, 4, 4, 4, 2, 2 $ $60$ $4$ $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,20)(10,19)(11,14,15,18)(12,13,16,17)$
$ 10, 10 $ $24$ $10$ $( 1, 5, 9,14,17, 2, 6,10,13,18)( 3, 7,12,15,20, 4, 8,11,16,19)$
$ 6, 3, 3, 2, 2, 2, 2 $ $40$ $6$ $( 1, 5)( 2, 6)( 3, 8,20)( 4, 7,19)( 9,14,17,10,13,18)(11,15)(12,16)$
$ 5, 5, 5, 5 $ $24$ $5$ $( 1, 6, 9,13,17)( 2, 5,10,14,18)( 3, 8,12,16,20)( 4, 7,11,15,19)$
$ 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,11, 2,12)( 3,13, 4,14)( 5,16, 6,15)( 7,18, 8,17)( 9,19,10,20)$

Group invariants

Order:  $480=2^{5} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [480, 944]
Character table:    
      2  5  3  3  3  5   5  3  5  2  3  4   2   2   2  3   2  2  2  4
      3  1  1  1  .  .   1  1  .  1  1  .   .   1   .  .   .  1  .  1
      5  1  .  .  .  .   1  .  .  .  .  .   1   .   1  .   1  .  1  1

        1a 3a 2a 4a 2b  2c 6a 2d 6b 2e 4b 20a 12a 20b 4c 10a 6c 5a 4d
     2P 1a 3a 1a 2b 1a  1a 3a 1a 3a 1a 2c 10a  6a 10a 2b  5a 3a 5a 2c
     3P 1a 1a 2a 4a 2b  2c 2c 2d 2e 2e 4b 20a  4d 20b 4c 10a 2a 5a 4d
     5P 1a 3a 2a 4a 2b  2c 6a 2d 6b 2e 4b  4d 12a  4d 4c  2c 6c 1a 4d
     7P 1a 3a 2a 4a 2b  2c 6a 2d 6b 2e 4b 20a 12a 20b 4c 10a 6c 5a 4d
    11P 1a 3a 2a 4a 2b  2c 6a 2d 6b 2e 4b 20b 12a 20a 4c 10a 6c 5a 4d
    13P 1a 3a 2a 4a 2b  2c 6a 2d 6b 2e 4b 20b 12a 20a 4c 10a 6c 5a 4d
    17P 1a 3a 2a 4a 2b  2c 6a 2d 6b 2e 4b 20b 12a 20a 4c 10a 6c 5a 4d
    19P 1a 3a 2a 4a 2b  2c 6a 2d 6b 2e 4b 20b 12a 20a 4c 10a 6c 5a 4d

X.1      1  1  1  1  1   1  1  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   1   1   1 -1   1 -1  1  1
X.3      1  1 -1 -1  1   1  1  1  1  1 -1  -1  -1  -1  1   1 -1  1 -1
X.4      1  1  1  1  1   1  1  1 -1 -1 -1  -1  -1  -1 -1   1  1  1 -1
X.5      2  2  .  .  2  -2 -2 -2  .  .  .   .   .   .  .  -2  .  2  .
X.6      4  1 -2  .  .   4  1  . -1  2  .   1  -1   1  .  -1  1 -1 -4
X.7      4  1 -2  .  .   4  1  .  1 -2  .  -1   1  -1  .  -1  1 -1  4
X.8      4  1  2  .  .   4  1  . -1  2  .  -1   1  -1  .  -1 -1 -1  4
X.9      4  1  2  .  .   4  1  .  1 -2  .   1  -1   1  .  -1 -1 -1 -4
X.10     5 -1 -1  1  1   5 -1  1 -1 -1  1   .  -1   .  1   . -1  .  5
X.11     5 -1 -1  1  1   5 -1  1  1  1 -1   .   1   . -1   . -1  . -5
X.12     5 -1  1 -1  1   5 -1  1 -1 -1 -1   .   1   .  1   .  1  . -5
X.13     5 -1  1 -1  1   5 -1  1  1  1  1   .  -1   . -1   .  1  .  5
X.14     6  .  .  . -2   6  . -2  .  . -2   1   .   1  .   1  .  1  6
X.15     6  .  .  . -2   6  . -2  .  .  2  -1   .  -1  .   1  .  1 -6
X.16     6  .  .  . -2  -6  .  2  .  .  .   A   .  -A  .  -1  .  1  .
X.17     6  .  .  . -2  -6  .  2  .  .  .  -A   .   A  .  -1  .  1  .
X.18     8  2  .  .  .  -8 -2  .  .  .  .   .   .   .  .   2  . -2  .
X.19    10 -2  .  .  2 -10  2 -2  .  .  .   .   .   .  .   .  .  .  .

A = -E(20)-E(20)^9+E(20)^13+E(20)^17
  = -Sqrt(-5) = -i5