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Properties

Label 21T22
Degree $21$
Order $504$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_3\times \PSL(2,7)$

Related objects

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Group action invariants

Degree $n$:  $21$
Transitive number $t$:  $22$
Group:  $C_3\times \PSL(2,7)$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $3$
Generators:  (1,2,3)(4,9,6,8,5,7)(10,20,12,19,11,21)(13,14,15)(16,17,18), (1,8,17,15,4,21,12)(2,9,18,13,5,19,10)(3,7,16,14,6,20,11)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$168$:  $\GL(3,2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

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

Low degree siblings

21T22, 24T1355 x 2, 24T1356, 42T96 x 2, 42T103 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 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
$ 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$
$ 6, 6, 3, 3, 3 $ $21$ $6$ $( 1, 3, 2)( 4, 6, 5)( 7,13, 8,14, 9,15)(10,12,11)(16,19,17,20,18,21)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 7,14)( 8,15)( 9,13)(16,20)(17,21)(18,19)$
$ 6, 6, 3, 3, 3 $ $21$ $6$ $( 1, 2, 3)( 4, 5, 6)( 7,15, 9,14, 8,13)(10,11,12)(16,21,18,20,17,19)$
$ 12, 6, 3 $ $42$ $12$ $( 1, 2, 3)( 4, 9,11,21, 5, 7,12,19, 6, 8,10,20)(13,16,15,18,14,17)$
$ 12, 6, 3 $ $42$ $12$ $( 1, 3, 2)( 4, 7,10,21, 6, 9,12,20, 5, 8,11,19)(13,17,14,18,15,16)$
$ 4, 4, 4, 2, 2, 2, 1, 1, 1 $ $42$ $4$ $( 4, 8,12,21)( 5, 9,10,19)( 6, 7,11,20)(13,18)(14,16)(15,17)$
$ 3, 3, 3, 3, 3, 3, 3 $ $56$ $3$ $( 1, 2, 3)( 4, 9,14)( 5, 7,15)( 6, 8,13)(10,20,17)(11,21,18)(12,19,16)$
$ 3, 3, 3, 3, 3, 3, 3 $ $56$ $3$ $( 1, 3, 2)( 4, 7,13)( 5, 8,14)( 6, 9,15)(10,21,16)(11,19,17)(12,20,18)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $56$ $3$ $( 4, 8,15)( 5, 9,13)( 6, 7,14)(10,19,18)(11,20,16)(12,21,17)$
$ 7, 7, 7 $ $24$ $7$ $( 1, 4, 8,12,15,17,21)( 2, 5, 9,10,13,18,19)( 3, 6, 7,11,14,16,20)$
$ 21 $ $24$ $21$ $( 1, 5, 7,12,13,16,21, 2, 6, 8,10,14,17,19, 3, 4, 9,11,15,18,20)$
$ 21 $ $24$ $21$ $( 1, 6, 9,12,14,18,21, 3, 5, 8,11,13,17,20, 2, 4, 7,10,15,16,19)$
$ 7, 7, 7 $ $24$ $7$ $( 1, 4, 8,21,17,12,15)( 2, 5, 9,19,18,10,13)( 3, 6, 7,20,16,11,14)$
$ 21 $ $24$ $21$ $( 1, 5, 7,21,18,11,15, 2, 6, 8,19,16,12,13, 3, 4, 9,20,17,10,14)$
$ 21 $ $24$ $21$ $( 1, 6, 9,21,16,10,15, 3, 5, 8,20,18,12,14, 2, 4, 7,19,17,11,13)$

Group invariants

Order:  $504=2^{3} \cdot 3^{2} \cdot 7$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  [504, 157]
Character table:    
      2  3  3  3   3  3   3   2   2  2   .   .  .  .   .   .  .   .   .
      3  2  2  2   1  1   1   1   1  1   2   2  2  1   1   1  1   1   1
      7  1  1  1   .  .   .   .   .  .   .   .  .  1   1   1  1   1   1

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

X.1      1  1  1   1  1   1   1   1  1   1   1  1  1   1   1  1   1   1
X.2      1  A /A  /A  1   A   A  /A  1   A  /A  1  1   A  /A  1   A  /A
X.3      1 /A  A   A  1  /A  /A   A  1  /A   A  1  1  /A   A  1  /A   A
X.4      3  3  3  -1 -1  -1   1   1  1   .   .  .  G   G   G /G  /G  /G
X.5      3  3  3  -1 -1  -1   1   1  1   .   .  . /G  /G  /G  G   G   G
X.6      3  B /B -/A -1  -A   A  /A  1   .   .  .  G   H  /I /G   I  /H
X.7      3  B /B -/A -1  -A   A  /A  1   .   .  . /G   I  /H  G   H  /I
X.8      3 /B  B  -A -1 -/A  /A   A  1   .   .  .  G  /I   H /G  /H   I
X.9      3 /B  B  -A -1 -/A  /A   A  1   .   .  . /G  /H   I  G  /I   H
X.10     6  6  6   2  2   2   .   .  .   .   .  . -1  -1  -1 -1  -1  -1
X.11     6  C /C   F  2  /F   .   .  .   .   .  . -1  -A -/A -1  -A -/A
X.12     6 /C  C  /F  2   F   .   .  .   .   .  . -1 -/A  -A -1 -/A  -A
X.13     7  7  7  -1 -1  -1  -1  -1 -1   1   1  1  .   .   .  .   .   .
X.14     7  D /D -/A -1  -A  -A -/A -1   A  /A  1  .   .   .  .   .   .
X.15     7 /D  D  -A -1 -/A -/A  -A -1  /A   A  1  .   .   .  .   .   .
X.16     8  8  8   .  .   .   .   .  .  -1  -1 -1  1   1   1  1   1   1
X.17     8  E /E   .  .   .   .   .  .  -A -/A -1  1   A  /A  1   A  /A
X.18     8 /E  E   .  .   .   .   .  . -/A  -A -1  1  /A   A  1  /A   A

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
C = 6*E(3)^2
  = -3-3*Sqrt(-3) = -3-3i3
D = 7*E(3)^2
  = (-7-7*Sqrt(-3))/2 = -7-7b3
E = 8*E(3)^2
  = -4-4*Sqrt(-3) = -4-4i3
F = 2*E(3)
  = -1+Sqrt(-3) = 2b3
G = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
H = E(21)^2+E(21)^8+E(21)^11
I = E(21)^5+E(21)^17+E(21)^20