Properties

Label 21T38
Order \(5040\)
n \(21\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $38$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,12,16,19,21,6)(2,8,13,17,20,5,11)(3,9,14,18,4,10,15), (2,3)(4,5,6)(7,8)(9,10,11)(13,17,15,16,14,18)(19,21,20)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 7: None

Low degree siblings

7T7, 14T46, 30T565, 35T31, 42T411, 42T412, 42T413, 42T418

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 2, 5)( 7,10)(12,17)(13,19)(15,21)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $70$ $3$ $( 1, 8, 3)( 2, 7,12)( 4, 9,16)( 5,10,17)( 6,11,18)$
$ 6, 3, 3, 3, 2, 2, 1, 1 $ $420$ $6$ $( 1, 3, 8)( 2,17, 7, 5,12,10)( 4,16, 9)( 6,18,11)(13,19)(15,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $105$ $2$ $( 2, 5)( 4, 6)( 7,10)( 9,11)(12,17)(13,21)(15,19)(16,18)$
$ 6, 6, 3, 2, 2, 1, 1 $ $210$ $6$ $( 1, 8, 3)( 2,10,12, 5, 7,17)( 4,11,16, 6, 9,18)(13,21)(15,19)$
$ 4, 4, 4, 4, 2, 1, 1, 1 $ $210$ $4$ $( 2, 6, 5, 4)( 7,11,10, 9)(12,18,17,16)(13,15,21,19)(14,20)$
$ 12, 4, 3, 2 $ $420$ $12$ $( 1, 8, 3)( 2, 9,17, 6, 7,16, 5,11,12, 4,10,18)(13,19,21,15)(14,20)$
$ 4, 4, 4, 4, 2, 2, 1 $ $630$ $4$ $( 1, 3)( 2, 6, 5, 4)( 7,18,10,16)( 9,12,11,17)(13,15,21,19)(14,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $105$ $2$ $( 1, 8)( 2,17)( 4,18)( 5,12)( 6,16)( 7,10)( 9,11)(13,21)(15,19)$
$ 3, 3, 3, 3, 3, 3, 3 $ $280$ $3$ $( 1, 7, 9)( 2,13, 4)( 3,14,20)( 5,15,16)( 6,12,19)( 8,10,11)(17,21,18)$
$ 6, 6, 6, 3 $ $840$ $6$ $( 1,11, 7, 8, 9,10)( 2,18,13,17, 4,21)( 3,20,14)( 5, 6,15,12,16,19)$
$ 5, 5, 5, 5, 1 $ $504$ $5$ $( 1,16, 6, 9,18)( 2,13,15, 7,12)( 3, 4,20,11, 8)( 5,19,21,10,17)$
$ 10, 5, 5, 1 $ $504$ $10$ $( 1, 9,16,18, 6)( 2,10,13,17,15, 5, 7,19,12,21)( 3,11, 4, 8,20)$
$ 7, 7, 7 $ $720$ $7$ $( 1,16,14, 6, 9,17,15)( 2, 4,19,21,11, 8,12)( 3,13, 5,20,10,18, 7)$

Group invariants

Order:  $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:   
      2  4  4  1  1  4  3  3  3  4  1   2  3  .   1  2
      3  2  1  2  1  1  1  2  1  1  .   1  .  .   .  1
      5  1  .  .  .  .  .  .  .  1  1   .  .  .   1  .
      7  1  .  .  .  .  .  .  .  .  .   .  .  1   .  .

        1a 2a 3a 6a 2b 4a 3b 6b 2c 5a 12a 4b 7a 10a 6c
     2P 1a 1a 3a 3a 1a 2b 3b 3b 1a 5a  6b 2b 7a  5a 3b
     3P 1a 2a 1a 2a 2b 4a 1a 2b 2c 5a  4a 4b 7a 10a 2c
     5P 1a 2a 3a 6a 2b 4a 3b 6b 2c 1a 12a 4b 7a  2c 6c
     7P 1a 2a 3a 6a 2b 4a 3b 6b 2c 5a 12a 4b 1a 10a 6c
    11P 1a 2a 3a 6a 2b 4a 3b 6b 2c 5a 12a 4b 7a 10a 6c

X.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
X.3      6  .  .  .  2  2  3 -1  4  1  -1  . -1  -1  1
X.4      6  .  .  .  2 -2  3 -1 -4  1   1  . -1   1 -1
X.5     14  .  2  .  2 -2 -1 -1  4 -1   1  .  .  -1  1
X.6     14 -2 -1  1  2  .  2  2 -6 -1   .  .  .  -1  .
X.7     14  .  2  .  2  2 -1 -1 -4 -1  -1  .  .   1 -1
X.8     14  2 -1 -1  2  .  2  2  6 -1   .  .  .   1  .
X.9     15  3  .  . -1 -1  3 -1 -5  .  -1 -1  1   .  1
X.10    15 -3  .  . -1  1  3 -1  5  .   1 -1  1   . -1
X.11    20  .  2  . -4  .  2  2  .  .   .  . -1   .  .
X.12    21 -3  .  .  1 -1 -3  1  1  1  -1 -1  .   1  1
X.13    21  3  .  .  1  1 -3  1 -1  1   1 -1  .  -1 -1
X.14    35 -1 -1 -1 -1  1 -1 -1 -5  .   1  1  .   .  1
X.15    35  1 -1  1 -1 -1 -1 -1  5  .  -1  1  .   . -1