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

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  1  4   1  1  4  1  3  2  4  3  3  3   2  .
      3  2  .  1   .  2  1  1  2  1  1  .  1  1   1  .
      5  1  1  1   1  .  .  .  .  .  .  .  .  .   .  .
      7  1  .  .   .  .  .  .  .  .  .  .  .  .   .  1

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

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