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Properties

Label 21T27
Order \(1008\)
n \(21\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $27$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,19)(2,18,20,3,17,21)(4,7,13)(5,9,14,6,8,15)(11,12), (1,18,8)(2,16,9)(3,17,7)(4,6,5)(10,21,14)(11,19,15)(12,20,13)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
168:  $\GL(3,2)$
336:  14T17

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: $\GL(3,2)$

Low degree siblings

21T27, 24T2671, 42T169 x 2, 42T170 x 2, 42T171 x 2, 42T175 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)$
$ 6, 6, 3, 3, 3 $ $42$ $6$ $( 1, 2, 3)( 4,11, 6,10, 5,12)( 7,20, 9,19, 8,21)(13,14,15)(16,17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $63$ $2$ $( 1, 3)( 4,12)( 5,11)( 6,10)( 7,21)( 8,20)( 9,19)(13,15)(16,18)$
$ 12, 6, 3 $ $84$ $12$ $( 1, 3, 2)( 4,12, 5,10, 6,11)( 7,18,20,13, 9,17,19,15, 8,16,21,14)$
$ 4, 4, 4, 2, 2, 2, 1, 1, 1 $ $42$ $4$ $( 4,10)( 5,11)( 6,12)( 7,16,19,13)( 8,17,20,14)( 9,18,21,15)$
$ 4, 4, 4, 2, 2, 2, 2, 1 $ $126$ $4$ $( 1, 2)( 4,11)( 5,10)( 6,12)( 7,17,19,14)( 8,16,20,13)( 9,18,21,15)$
$ 3, 3, 3, 3, 3, 3, 3 $ $112$ $3$ $( 1, 2, 3)( 4,17,21)( 5,18,19)( 6,16,20)( 7,11,15)( 8,12,13)( 9,10,14)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $56$ $3$ $( 4,16,19)( 5,17,20)( 6,18,21)( 7,10,13)( 8,11,14)( 9,12,15)$
$ 6, 6, 3, 3, 2, 1 $ $168$ $6$ $( 1, 3)( 4,18,19, 6,16,21)( 5,17,20)( 7,12,13, 9,10,15)( 8,11,14)$
$ 7, 7, 7 $ $24$ $7$ $( 1,16,19, 7,10,13, 4)( 2,17,20, 8,11,14, 5)( 3,18,21, 9,12,15, 6)$
$ 21 $ $48$ $21$ $( 1,18,20, 7,12,14, 4, 3,17,19, 9,11,13, 6, 2,16,21, 8,10,15, 5)$
$ 14, 7 $ $72$ $14$ $( 1,16,19, 7,10,13, 4)( 2,18,20, 9,11,15, 5, 3,17,21, 8,12,14, 6)$
$ 21 $ $48$ $21$ $( 1,17,21,13, 5,12, 7, 2,18,19,14, 6,10, 8, 3,16,20,15, 4,11, 9)$
$ 7, 7, 7 $ $24$ $7$ $( 1,16,19,13, 4,10, 7)( 2,17,20,14, 5,11, 8)( 3,18,21,15, 6,12, 9)$
$ 14, 7 $ $72$ $14$ $( 1,18,19,15, 4,12, 7, 3,16,21,13, 6,10, 9)( 2,17,20,14, 5,11, 8)$

Group invariants

Order:  $1008=2^{4} \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [1008, 883]
Character table:    
      2  4  3  4  3  4  4   2  3  3  .  1  1  1   .   1   .  1   1
      3  2  2  1  1  1  .   1  1  .  2  2  1  1   1   .   1  1   .
      7  1  1  1  .  .  .   .  .  .  .  .  .  1   1   1   1  1   1

        1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 7a 21a 14a 21b 7b 14b
     2P 1a 3a 1a 3a 1a 1a  6a 2b 2b 3b 3c 3c 7a 21a  7a 21b 7b  7b
     3P 1a 1a 2a 2b 2b 2c  4a 4a 4b 1a 1a 2a 7b  7b 14b  7a 7a 14a
     5P 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 7b 21b 14b 21a 7a 14a
     7P 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 1a  3a  2a  3a 1a  2a
    11P 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 7a 21a 14a 21b 7b 14b
    13P 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 7b 21b 14b 21a 7a 14a
    17P 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 7b 21b 14b 21a 7a 14a
    19P 1a 3a 2a 6a 2b 2c 12a 4a 4b 3b 3c 6b 7b 21b 14b 21a 7a 14a

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

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
B = 2*E(7)^3+2*E(7)^5+2*E(7)^6
  = -1-Sqrt(-7) = -1-i7