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Properties

Label 21T85
Order \(40320\)
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 $t$ :  $85$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,2,16,20,18,19,12)(3,10,8,9,11,14,4,21)(5,13,6,15), (1,12,16,15,18,2,21)(3,5,6,9,7,4,13)(8,20,11,14,19,10,17)
$|\Aut(F/K)|$:  $1$

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

21T85, 42T930 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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, 1, 1, 1, 1, 1, 1, 1 $ $120$ $2$ $( 2,19)( 3,11)( 5, 6)( 7,16)( 9,14)(10,21)(12,18)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $2240$ $3$ $( 1, 8,15)( 2, 5, 9)( 3,21,18)( 4,17,13)( 6,14,19)(10,12,11)$
$ 6, 6, 3, 3, 2, 1 $ $6720$ $6$ $( 1,15, 8)( 2,14, 5,19, 9, 6)( 3,12,21,11,18,10)( 4,13,17)( 7,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $315$ $2$ $( 3,18)( 4, 7)( 5,17)( 8, 9)(11,21)(13,15)(14,16)(19,20)$
$ 4, 4, 4, 4, 2, 2, 1 $ $2520$ $4$ $( 1,10)( 2,12)( 3,15,18,13)( 4,16, 7,14)( 5, 8,17, 9)(11,19,21,20)$
$ 4, 4, 4, 4, 2, 1, 1, 1 $ $2520$ $4$ $( 3, 5,11,19)( 4,14, 9,13)( 7,16, 8,15)(10,12)(17,21,20,18)$
$ 4, 4, 4, 4, 2, 2, 1 $ $1260$ $4$ $( 2, 6)( 3,19,11, 5)( 4, 7, 9, 8)(10,12)(13,15,14,16)(17,21,20,18)$
$ 8, 8, 4, 1 $ $5040$ $8$ $( 1,10, 2,12)( 3,14,11,17,18,16,21, 5)( 4,15, 8,20, 7,13, 9,19)$
$ 7, 7, 7 $ $2880$ $7$ $( 1, 6,11, 3,18, 2,21)( 4,17, 9,13,15,19,12)( 5,10, 7,20, 8,16,14)$
$ 7, 7, 7 $ $2880$ $7$ $( 1,21, 2,18, 3,11, 6)( 4,12,19,15,13, 9,17)( 5,14,16, 8,20, 7,10)$
$ 14, 7 $ $2880$ $14$ $( 1,11,18,21, 6, 3, 2)( 4, 8,15,10,17,16,19, 7, 9,14,12,20,13, 5)$
$ 14, 7 $ $2880$ $14$ $( 1,21, 2,18, 3,11, 6)( 4,10,19,14,13, 8,17, 7,12, 5,15,16, 9,20)$
$ 5, 5, 5, 5, 1 $ $8064$ $5$ $( 1, 4,14,13, 9)( 2, 8, 5,15,10)( 3,21,20, 7,16)( 6,19,12,17,11)$

Group invariants

Order:  $40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:    
      2   7  7  4  4  4  1  1  5  1  1   1   1  .  3
      3   2  .  .  .  1  2  1  .  .  .   .   .  .  .
      5   1  .  .  .  .  .  .  .  .  .   .   .  1  .
      7   1  .  .  .  1  .  .  .  1  1   1   1  .  .

         1a 2a 4a 4b 2b 3a 6a 4c 7a 7b 14a 14b 5a 8a
     2P  1a 1a 2a 2a 1a 3a 3a 2a 7a 7b  7a  7b 5a 4c
     3P  1a 2a 4a 4b 2b 1a 2b 4c 7b 7a 14b 14a 5a 8a
     5P  1a 2a 4a 4b 2b 3a 6a 4c 7b 7a 14b 14a 1a 8a
     7P  1a 2a 4a 4b 2b 3a 6a 4c 1a 1a  2b  2b 5a 8a
    11P  1a 2a 4a 4b 2b 3a 6a 4c 7a 7b 14a 14b 5a 8a
    13P  1a 2a 4a 4b 2b 3a 6a 4c 7b 7a 14b 14a 5a 8a

X.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
X.3      20  4  . -2 -6  2  .  . -1 -1   1   1  .  .
X.4      20  4  .  2  6  2  .  . -1 -1  -1  -1  .  .
X.5      35  3 -1  1 -7 -1 -1  3  .  .   .   .  .  1
X.6      35  3 -1 -1  7 -1  1  3  .  .   .   .  . -1
X.7      45 -3  1 -1  3  .  .  1  A /A   A  /A  .  1
X.8      45 -3  1 -1  3  .  .  1 /A  A  /A   A  .  1
X.9      45 -3  1  1 -3  .  .  1  A /A  -A -/A  . -1
X.10     45 -3  1  1 -3  .  .  1 /A  A -/A  -A  . -1
X.11     64  .  .  .  8  1 -1  .  1  1   1   1 -1  .
X.12     64  .  .  . -8  1  1  .  1  1  -1  -1 -1  .
X.13     70  6  2  .  . -2  . -2  .  .   .   .  .  .
X.14    126 -2 -2  .  .  .  . -2  .  .   .   .  1  .

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7