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Properties

Label 20T149
Order \(720\)
n \(20\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $149$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,3,5)(2,12,4,6)(7,10,16,13)(8,9,15,14)(17,20)(18,19), (1,4,18,16,14,6)(2,3,17,15,13,5)(9,11,19)(10,12,20)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: None

Degree 10: $S_{6}$

Low degree siblings

6T16 x 2, 10T32, 12T183 x 2, 15T28 x 2, 20T145, 20T149, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 7,15)( 8,16)( 9,13)(10,14)(17,19)(18,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $45$ $2$ $( 5,11)( 6,12)( 7, 9)( 8,10)(13,15)(14,16)(17,19)(18,20)$
$ 3, 3, 3, 3, 3, 3, 1, 1 $ $40$ $3$ $( 3, 5,11)( 4, 6,12)( 7,17,13)( 8,18,14)( 9,15,19)(10,16,20)$
$ 6, 6, 3, 3, 1, 1 $ $120$ $6$ $( 3, 5,11)( 4, 6,12)( 7,19,13,15,17, 9)( 8,20,14,16,18,10)$
$ 3, 3, 3, 3, 3, 3, 1, 1 $ $40$ $3$ $( 3, 8,10)( 4, 7, 9)( 5,18,16)( 6,17,15)(11,14,20)(12,13,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3, 4)( 5,17)( 6,18)( 7,10)( 8, 9)(11,19)(12,20)(13,14)(15,16)$
$ 4, 4, 4, 4, 2, 2 $ $90$ $4$ $( 1, 2)( 3, 4)( 5,17,11,19)( 6,18,12,20)( 7,16, 9,14)( 8,15,10,13)$
$ 6, 6, 6, 2 $ $120$ $6$ $( 1, 2)( 3, 6,10,15, 8,17)( 4, 5, 9,16, 7,18)(11,13,20,12,14,19)$
$ 4, 4, 4, 4, 2, 2 $ $90$ $4$ $( 1, 3, 5,11)( 2, 4, 6,12)( 7,17,16,20)( 8,18,15,19)( 9,14)(10,13)$
$ 5, 5, 5, 5 $ $144$ $5$ $( 1, 3, 7,17,16)( 2, 4, 8,18,15)( 5,13,10,11,20)( 6,14, 9,12,19)$

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  4  1  1  1  4  3  1  3  .
      3  2  1  .  2  1  2  1  .  1  .  .
      5  1  .  .  .  .  .  .  .  .  .  1

        1a 2a 2b 3a 6a 3b 2c 4a 6b 4b 5a
     2P 1a 1a 1a 3a 3a 3b 1a 2b 3b 2b 5a
     3P 1a 2a 2b 1a 2a 1a 2c 4a 2c 4b 5a
     5P 1a 2a 2b 3a 6a 3b 2c 4a 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 -3  1  2  . -1  1 -1  1 -1  .
X.4      5  3  1  2  . -1 -1 -1 -1  1  .
X.5      5 -1  1 -1 -1  2  3 -1  .  1  .
X.6      5  1  1 -1  1  2 -3 -1  . -1  .
X.7      9 -3  1  .  .  . -3  1  .  1 -1
X.8      9  3  1  .  .  .  3  1  . -1 -1
X.9     10 -2 -2  1  1  1  2  . -1  .  .
X.10    10  2 -2  1 -1  1 -2  .  1  .  .
X.11    16  .  . -2  . -2  .  .  .  .  1