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Properties

Label 20T226
Order \(1920\)
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 $t$ :  $226$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20)(2,19)(3,14)(4,13)(5,6)(7,18)(8,17)(9,12)(10,11)(15,16), (1,14,5,19,18)(2,13,6,20,17)(3,16,10,7,12)(4,15,9,8,11)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $S_5$

Degree 10: $S_5$, $(C_2^4:A_5) : C_2$, $(C_2^4:A_5) : C_2$

Low degree siblings

10T37, 10T38, 16T1328, 20T218, 20T219, 20T222, 20T223, 30T329, 30T332, 30T333, 30T341, 32T97736, 40T1581, 40T1582, 40T1583, 40T1584, 40T1587, 40T1588, 40T1595, 40T1596, 40T1658, 40T1659, 40T1676, 40T1677, 40T1678

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 7,17)( 8,18)( 9,19)(10,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 1,11)( 2,12)( 3,13)( 4,14)( 7,17)( 8,18)( 9,19)(10,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1, 6)( 2, 5)( 3,14)( 4,13)( 7, 8)( 9,20)(10,19)(11,16)(12,15)(17,18)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $60$ $4$ $( 1, 6,11,16)( 2, 5,12,15)( 3,14)( 4,13)( 7, 8)( 9,10)(17,18)(19,20)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $20$ $4$ $( 1, 6,11,16)( 2, 5,12,15)( 3,14)( 4,13)( 7,18)( 8,17)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 6)( 2, 5)( 3, 4)( 7, 8)( 9,10)(11,16)(12,15)(13,14)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $60$ $2$ $( 1, 5)( 2, 6)( 7,20)( 8,19)( 9,18)(10,17)(11,15)(12,16)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $60$ $4$ $( 1, 5,11,15)( 2, 6,12,16)( 7,10,17,20)( 8, 9,18,19)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $120$ $4$ $( 1, 5)( 2, 6)( 3,13)( 4,14)( 7,10,17,20)( 8, 9,18,19)(11,15)(12,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $80$ $3$ $( 1,15,19)( 2,16,20)( 5, 9,11)( 6,10,12)$
$ 6, 6, 2, 2, 1, 1, 1, 1 $ $160$ $6$ $( 1,15, 9,11, 5,19)( 2,16,10,12, 6,20)( 7,17)( 8,18)$
$ 3, 3, 3, 3, 2, 2, 2, 2 $ $80$ $6$ $( 1,15,19)( 2,16,20)( 3,13)( 4,14)( 5, 9,11)( 6,10,12)( 7,17)( 8,18)$
$ 6, 6, 2, 2, 2, 2 $ $160$ $6$ $( 1, 6, 9, 2, 5,10)( 3, 8)( 4, 7)(11,16,19,12,15,20)(13,18)(14,17)$
$ 6, 6, 4, 4 $ $160$ $12$ $( 1, 6,19,12,15,10)( 2, 5,20,11,16, 9)( 3,18,13, 8)( 4,17,14, 7)$
$ 8, 8, 2, 2 $ $240$ $8$ $( 1,16,19,17,11, 6, 9, 7)( 2,15,20,18,12, 5,10, 8)( 3,14)( 4,13)$
$ 4, 4, 4, 4, 2, 2 $ $240$ $4$ $( 1,16, 9, 7)( 2,15,10, 8)( 3, 4)( 5,20,18,12)( 6,19,17,11)(13,14)$
$ 5, 5, 5, 5 $ $384$ $5$ $( 1, 5, 9,18, 4)( 2, 6,10,17, 3)( 7,13,12,16,20)( 8,14,11,15,19)$

Group invariants

Order:  $1920=2^{7} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:    
      2  7  6  7  5  5  5  5  5  5  4  3  2  3  2   2  3  3  .
      3  1  1  1  .  .  1  1  .  .  .  1  1  1  1   1  .  .  .
      5  1  .  .  .  .  .  .  .  .  .  .  .  .  .   .  .  .  1

        1a 2a 2b 2c 4a 4b 2d 2e 4c 4d 3a 6a 6b 6c 12a 8a 4e 5a
     2P 1a 1a 1a 1a 2a 2a 1a 1a 2b 2a 3a 3a 3a 3a  6b 4c 2e 5a
     3P 1a 2a 2b 2c 4a 4b 2d 2e 4c 4d 1a 2b 2a 2d  4b 8a 4e 5a
     5P 1a 2a 2b 2c 4a 4b 2d 2e 4c 4d 3a 6a 6b 6c 12a 8a 4e 1a
     7P 1a 2a 2b 2c 4a 4b 2d 2e 4c 4d 3a 6a 6b 6c 12a 8a 4e 5a
    11P 1a 2a 2b 2c 4a 4b 2d 2e 4c 4d 3a 6a 6b 6c 12a 8a 4e 5a

X.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
X.3      4  4  4 -2 -2 -2 -2  .  .  .  1  1  1  1   1  .  . -1
X.4      4  4  4  2  2  2  2  .  .  .  1  1  1 -1  -1  .  . -1
X.5      5  5  5 -1 -1 -1 -1  1  1  1 -1 -1 -1 -1  -1  1  1  .
X.6      5  5  5  1  1  1  1  1  1  1 -1 -1 -1  1   1 -1 -1  .
X.7      5  1 -3 -1  1 -3  3  1  1 -1  2  . -2  .   . -1  1  .
X.8      5  1 -3  1 -1  3 -3  1  1 -1  2  . -2  .   .  1 -1  .
X.9      6  6  6  .  .  .  . -2 -2 -2  .  .  .  .   .  .  .  1
X.10    10 -2  2  .  2 -2 -4  2 -2  .  1 -1  1 -1   1  .  .  .
X.11    10 -2  2  2  . -4 -2 -2  2  .  1 -1  1  1  -1  .  .  .
X.12    10 -2  2  . -2  2  4  2 -2  .  1 -1  1  1  -1  .  .  .
X.13    10 -2  2 -2  .  4  2 -2  2  .  1 -1  1 -1   1  .  .  .
X.14    10  2 -6  .  .  .  .  2  2 -2 -2  .  2  .   .  .  .  .
X.15    15  3 -9  1 -1  3 -3 -1 -1  1  .  .  .  .   . -1  1  .
X.16    15  3 -9 -1  1 -3  3 -1 -1  1  .  .  .  .   .  1 -1  .
X.17    20 -4  4 -2  2  2 -2  .  .  . -1  1 -1  1  -1  .  .  .
X.18    20 -4  4  2 -2 -2  2  .  .  . -1  1 -1 -1   1  .  .  .