↑

Properties

Label 20T145
Order \(720\)
n \(20\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

Learn more about

Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $145$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,16,17)(2,5,15,18)(7,9,20,11)(8,10,19,12), (1,18)(2,17)(3,7,19,14)(4,8,20,13)(5,9,15,11)(6,10,16,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: None

Degree 10: $S_{6}$

Low degree siblings

6T16 x 2, 10T32, 12T183 x 2, 15T28 x 2, 20T149 x 2, 30T164 x 2, 30T166 x 2, 30T176 x 2, 36T1252, 40T589, 40T592 x 2, 45T96

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, 2, 1, 1, 1, 1 $ $45$ $2$ $( 5,12)( 6,11)( 7, 9)( 8,10)(13,15)(14,16)(17,20)(18,19)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $90$ $4$ $( 5,18,12,19)( 6,17,11,20)( 7,16, 9,14)( 8,15,10,13)$
$ 3, 3, 3, 3, 3, 3, 1, 1 $ $40$ $3$ $( 3, 5,12)( 4, 6,11)( 7,17,14)( 8,18,13)( 9,16,20)(10,15,19)$
$ 3, 3, 3, 3, 3, 3, 1, 1 $ $40$ $3$ $( 3, 8,10)( 4, 7, 9)( 5,18,15)( 6,17,16)(11,14,20)(12,13,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,15)( 8,16)( 9,13)(10,14)(11,12)(17,19)(18,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3, 4)( 5,17)( 6,18)( 7,10)( 8, 9)(11,19)(12,20)(13,14)(15,16)$
$ 6, 6, 6, 2 $ $120$ $6$ $( 1, 2)( 3, 6,10,16, 8,17)( 4, 5, 9,15, 7,18)(11,13,20,12,14,19)$
$ 6, 6, 6, 2 $ $120$ $6$ $( 1, 2)( 3, 6,12, 4, 5,11)( 7,19,14,15,17,10)( 8,20,13,16,18, 9)$
$ 4, 4, 4, 4, 2, 2 $ $90$ $4$ $( 1, 3)( 2, 4)( 5,17,12,20)( 6,18,11,19)( 7,13, 9,15)( 8,14,10,16)$
$ 5, 5, 5, 5 $ $144$ $5$ $( 1, 4, 6, 7,17)( 2, 3, 5, 8,18)( 9,14,20,11,16)(10,13,19,12,15)$

Group invariants

Order:  $720=2^{4} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [720, 763]
Character table:    
      2  4  4  3  1  1  4  4  1  1  3  .
      3  2  .  .  2  2  1  1  1  1  .  .
      5  1  .  .  .  .  .  .  .  .  .  1

        1a 2a 4a 3a 3b 2b 2c 6a 6b 4b 5a
     2P 1a 1a 2a 3a 3b 1a 1a 3b 3a 2a 5a
     3P 1a 2a 4a 1a 1a 2b 2c 2c 2b 4b 5a
     5P 1a 2a 4a 3a 3b 2b 2c 6a 6b 4b 1a

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      5  1 -1  2 -1 -3  1  1  . -1  .
X.4      5  1 -1  2 -1  3 -1 -1  .  1  .
X.5      5  1 -1 -1  2 -1  3  . -1  1  .
X.6      5  1 -1 -1  2  1 -3  .  1 -1  .
X.7      9  1  1  .  . -3 -3  .  .  1 -1
X.8      9  1  1  .  .  3  3  .  . -1 -1
X.9     10 -2  .  1  1 -2  2 -1  1  .  .
X.10    10 -2  .  1  1  2 -2  1 -1  .  .
X.11    16  .  . -2 -2  .  .  .  .  .  1