Properties

Label 21T18
Order \(294\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7^2:S_3$

Learn more about

Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $18$
Group :  $C_7^2:S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17,3,15,5,20,7,18,2,16,4,21,6,19)(8,13,11,9,14,12,10), (1,8,3,10,5,12,7,14,2,9,4,11,6,13)(15,17,19,21,16,18,20)
$|\Aut(F/K)|$:  $7$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

14T15, 21T17, 42T56, 42T57, 42T62

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 $ $1$ $1$ $()$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 8,15)( 9,21)(10,20)(11,19)(12,18)(13,17)(14,16)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,18,21,17,20,16,19)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$
$ 14, 7 $ $21$ $14$ $( 1, 2, 3, 4, 5, 6, 7)( 8,15,14,16,13,17,12,18,11,19,10,20, 9,21)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$
$ 14, 7 $ $21$ $14$ $( 1, 3, 5, 7, 2, 4, 6)( 8,15,13,17,11,19, 9,21,14,16,12,18,10,20)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
$ 14, 7 $ $21$ $14$ $( 1, 4, 7, 3, 6, 2, 5)( 8,15,12,18, 9,21,13,17,10,20,14,16,11,19)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$
$ 14, 7 $ $21$ $14$ $( 1, 5, 2, 6, 3, 7, 4)( 8,15,11,19,14,16,10,20,13,17, 9,21,12,18)$
$ 14, 7 $ $21$ $14$ $( 1, 6, 4, 2, 7, 5, 3)( 8,15,10,20,12,18,14,16, 9,21,11,19,13,17)$
$ 14, 7 $ $21$ $14$ $( 1, 7, 6, 5, 4, 3, 2)( 8,15, 9,21,10,20,11,19,12,18,13,17,14,16)$
$ 3, 3, 3, 3, 3, 3, 3 $ $98$ $3$ $( 1, 8,15)( 2, 9,21)( 3,10,20)( 4,11,19)( 5,12,18)( 6,13,17)( 7,14,16)$

Group invariants

Order:  $294=2 \cdot 3 \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [294, 7]
Character table:   
      2  1  .  .  .  1  1  .  1   1  1  1   1  .   1  1  1   1   1   1  .
      3  1  .  .  .  .  .  .  .   .  .  .   .  .   .  .  .   .   .   .  1
      7  2  2  2  2  1  2  2  2   1  2  2   1  2   1  2  2   1   1   1  .

        1a 7a 7b 7c 2a 7d 7e 7f 14a 7g 7h 14b 7i 14c 7j 7k 14d 14e 14f 3a
     2P 1a 7b 7c 7a 1a 7g 7e 7h  7h 7j 7k  7k 7i  7j 7d 7f  7f  7g  7d 3a
     3P 1a 7c 7a 7b 2a 7f 7i 7g 14c 7h 7j 14f 7e 14b 7k 7d 14e 14a 14d 1a
     5P 1a 7b 7c 7a 2a 7k 7i 7d 14e 7f 7g 14c 7e 14a 7h 7j 14f 14d 14b 3a
     7P 1a 1a 1a 1a 2a 1a 1a 1a  2a 1a 1a  2a 1a  2a 1a 1a  2a  2a  2a 3a
    11P 1a 7c 7a 7b 2a 7j 7e 7k 14d 7d 7f 14a 7i 14e 7g 7h 14b 14f 14c 3a
    13P 1a 7a 7b 7c 2a 7h 7i 7j 14f 7k 7d 14e 7e 14d 7f 7g 14c 14b 14a 3a

X.1      1  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  -1  1
X.3      2  2  2  2  .  2  2  2   .  2  2   .  2   .  2  2   .   .   . -1
X.4      3  A  C  B -1  G  N  H   P /I /G  /R /N   Q /H  I  /Q   R  /P  .
X.5      3  A  C  B -1 /G /N /H  /P  I  G   R  N  /Q  H /I   Q  /R   P  .
X.6      3  B  A  C -1  H /N /I   Q /G /H  /P  N  /R  I  G   R   P  /Q  .
X.7      3  B  A  C -1 /H  N  I  /Q  G  H   P /N   R /I /G  /R  /P   Q  .
X.8      3  C  B  A -1  I /N  G   R  H /I   Q  N   P /G /H  /P  /Q  /R  .
X.9      3  C  B  A -1 /I  N /G  /R /H  I  /Q /N  /P  G  H   P   Q   R  .
X.10     3  A  C  B  1  G  N  H  -P /I /G -/R /N  -Q /H  I -/Q  -R -/P  .
X.11     3  A  C  B  1 /G /N /H -/P  I  G  -R  N -/Q  H /I  -Q -/R  -P  .
X.12     3  B  A  C  1  H /N /I  -Q /G /H -/P  N -/R  I  G  -R  -P -/Q  .
X.13     3  B  A  C  1 /H  N  I -/Q  G  H  -P /N  -R /I /G -/R -/P  -Q  .
X.14     3  C  B  A  1  I /N  G  -R  H /I  -Q  N  -P /G /H -/P -/Q -/R  .
X.15     3  C  B  A  1 /I  N /G -/R /H  I -/Q /N -/P  G  H  -P  -Q  -R  .
X.16     6  D  E  F  .  J -1  L   .  K  J   . -1   .  L  K   .   .   .  .
X.17     6  E  F  D  .  K -1  J   .  L  K   . -1   .  J  L   .   .   .  .
X.18     6  F  D  E  .  L -1  K   .  J  L   . -1   .  K  J   .   .   .  .
X.19     6 -1 -1 -1  .  M  O /M   .  M /M   . /O   .  M /M   .   .   .  .
X.20     6 -1 -1 -1  . /M /O  M   . /M  M   .  O   . /M  M   .   .   .  .

A = -E(7)-E(7)^2-E(7)^5-E(7)^6
B = -E(7)-E(7)^3-E(7)^4-E(7)^6
C = -E(7)^2-E(7)^3-E(7)^4-E(7)^5
D = 2*E(7)+E(7)^2+E(7)^5+2*E(7)^6
E = 2*E(7)^2+E(7)^3+E(7)^4+2*E(7)^5
F = E(7)+2*E(7)^3+2*E(7)^4+E(7)^6
G = 2*E(7)^4+E(7)^6
H = E(7)^4+2*E(7)^5
I = E(7)^2+2*E(7)^6
J = -2*E(7)-2*E(7)^2-2*E(7)^5-2*E(7)^6
K = -2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5
L = -2*E(7)-2*E(7)^3-2*E(7)^4-2*E(7)^6
M = 2*E(7)^3+2*E(7)^5+2*E(7)^6
  = -1-Sqrt(-7) = -1-i7
N = E(7)+E(7)^2+E(7)^4
  = (-1+Sqrt(-7))/2 = b7
O = -2*E(7)-2*E(7)^2-3*E(7)^3-2*E(7)^4-3*E(7)^5-3*E(7)^6
  = (5+Sqrt(-7))/2 = 3+b7
P = -E(7)^4
Q = -E(7)^5
R = -E(7)^6