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Properties

Label 20T65
Order \(240\)
n \(20\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_2\times S_5$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $65$
Group :  $C_2\times S_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,3,15,8,17)(2,11,4,16,7,18)(5,19,14,6,20,13), (1,18,2,17)(3,10,15,19)(4,9,16,20)(5,8,11,13)(6,7,12,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
120:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: None

Degree 10: $S_5$

Low degree siblings

10T22 x 2, 12T123 x 2, 20T62 x 2, 20T65, 20T70, 24T570, 24T577, 30T58 x 2, 30T60 x 2, 40T173 x 2, 40T180, 40T181, 40T187 x 2

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

Group invariants

Order:  $240=2^{4} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [240, 189]
Character table:    
      2  4  3  2  4  2  3  4  2  4  2   1  3  3  1
      3  1  1  1  .  1  1  1  1  .  1   .  .  .  .
      5  1  .  .  .  .  .  1  .  .  .   1  .  .  1

        1a 2a 3a 2b 6a 2c 2d 6b 2e 6c 10a 4a 4b 5a
     2P 1a 1a 3a 1a 3a 1a 1a 3a 1a 3a  5a 2e 2e 5a
     3P 1a 2a 1a 2b 2c 2c 2d 2d 2e 2a 10a 4a 4b 5a
     5P 1a 2a 3a 2b 6a 2c 2d 6b 2e 6c  2d 4a 4b 1a
     7P 1a 2a 3a 2b 6a 2c 2d 6b 2e 6c 10a 4a 4b 5a

X.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
X.3      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
X.5      4 -2  1  .  1 -2  4  1  .  1  -1  .  . -1
X.6      4  2  1  . -1  2  4  1  . -1  -1  .  . -1
X.7      4 -2  1  . -1  2 -4 -1  .  1   1  .  . -1
X.8      4  2  1  .  1 -2 -4 -1  . -1   1  .  . -1
X.9      5  1 -1  1  1  1  5 -1  1  1   . -1 -1  .
X.10     5 -1 -1  1 -1 -1  5 -1  1 -1   .  1  1  .
X.11     5  1 -1 -1 -1 -1 -5  1  1  1   .  1 -1  .
X.12     5 -1 -1 -1  1  1 -5  1  1 -1   . -1  1  .
X.13     6  .  . -2  .  .  6  . -2  .   1  .  .  1
X.14     6  .  .  2  .  . -6  . -2  .  -1  .  .  1