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Properties

Label 20T201
Order \(1440\)
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 $t$ :  $201$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,16,12,5,4,17,9)(2,14,15,11,6,3,18,10)(7,8)(19,20), (3,17,19)(4,18,20)(5,8,10,11,14,16)(6,7,9,12,13,15)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: None

Degree 10: $(A_6 : C_2) : C_2$

Low degree siblings

10T35, 12T220, 20T204, 20T208, 24T2960, 30T264, 36T2341, 40T1198, 40T1199, 40T1201, 45T187

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 $ $45$ $2$ $( 1,10)( 2, 9)( 5,16)( 6,15)(11,17)(12,18)(13,20)(14,19)$
$ 4, 4, 4, 4, 2, 2 $ $180$ $4$ $( 1, 6,10,15)( 2, 5, 9,16)( 3, 4)( 7, 8)(11,20,17,13)(12,19,18,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $36$ $2$ $( 1,20)( 2,19)( 3, 6)( 4, 5)( 7,17)( 8,18)( 9,11)(10,12)(13,16)(14,15)$
$ 5, 5, 5, 5 $ $144$ $5$ $( 1,14, 3,17,11)( 2,13, 4,18,12)( 5, 8,10,19,16)( 6, 7, 9,20,15)$
$ 10, 10 $ $144$ $10$ $( 1, 7,14, 9, 3,20,17,15,11, 6)( 2, 8,13,10, 4,19,18,16,12, 5)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $30$ $2$ $( 1,19)( 2,20)( 5, 8)( 6, 7)(11,14)(12,13)$
$ 3, 3, 3, 3, 3, 3, 1, 1 $ $80$ $3$ $( 1,11, 5)( 2,12, 6)( 7,20,13)( 8,19,14)( 9,15,18)(10,16,17)$
$ 6, 6, 3, 3, 1, 1 $ $240$ $6$ $( 1, 8,11,19, 5,14)( 2, 7,12,20, 6,13)( 9,18,15)(10,17,16)$
$ 4, 4, 4, 4, 2, 2 $ $90$ $4$ $( 1,17,10,11)( 2,18, 9,12)( 3, 8)( 4, 7)( 5,14,16,19)( 6,13,15,20)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $90$ $4$ $( 1,11,10,17)( 2,12, 9,18)( 5,14,16,19)( 6,13,15,20)$
$ 8, 8, 2, 2 $ $180$ $8$ $( 1,20,11, 6,10,13,17,15)( 2,19,12, 5, 9,14,18,16)( 3, 4)( 7, 8)$
$ 8, 8, 2, 2 $ $180$ $8$ $( 1,13,17, 6,10,20,11,15)( 2,14,18, 5, 9,19,12,16)( 3, 7)( 4, 8)$

Group invariants

Order:  $1440=2^{5} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [1440, 5841]
Character table:    
      2  5  1  4  1  1  3  5  4  4  3  3   1  3
      3  2  2  1  1  .  .  .  .  .  .  .   .  .
      5  1  .  .  .  1  1  .  .  .  .  .   1  .

        1a 3a 2a 6a 5a 2b 2c 4a 4b 8a 8b 10a 4c
     2P 1a 3a 1a 3a 5a 1a 1a 2c 2c 4b 4b  5a 2c
     3P 1a 1a 2a 2a 5a 2b 2c 4a 4b 8a 8b 10a 4c
     5P 1a 3a 2a 6a 1a 2b 2c 4a 4b 8a 8b  2b 4c
     7P 1a 3a 2a 6a 5a 2b 2c 4a 4b 8a 8b 10a 4c

X.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
X.3      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
X.5      9  . -3  . -1 -1  1  1  1  1  1  -1 -1
X.6      9  . -3  . -1  1  1  1  1 -1 -1   1  1
X.7      9  .  3  . -1 -1  1 -1  1 -1  1  -1  1
X.8      9  .  3  . -1  1  1 -1  1  1 -1   1 -1
X.9     10  1  2 -1  .  .  2  2 -2  .  .   .  .
X.10    10  1 -2  1  .  .  2 -2 -2  .  .   .  .
X.11    16 -2  .  .  1 -4  .  .  .  .  .   1  .
X.12    16 -2  .  .  1  4  .  .  .  .  .  -1  .
X.13    20  2  .  .  .  . -4  .  .  .  .   .  .