Properties

Label 21T17
Order \(294\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7^2:S_3$

Related objects

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Group action invariants

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

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, 21T18, 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 $ $3$ $7$ $( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $3$ $7$ $( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $3$ $7$ $( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $3$ $7$ $( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $3$ $7$ $( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $3$ $7$ $( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$
$ 14, 2, 2, 2, 1 $ $21$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,15, 9,16,10,17,11,18,12,19,13,20,14,21)$
$ 14, 2, 2, 2, 1 $ $21$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,16,11,19,14,15,10,18,13,21, 9,17,12,20)$
$ 14, 2, 2, 2, 1 $ $21$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,17,13,15,11,20, 9,18,14,16,12,21,10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $21$ $2$ $( 2, 7)( 3, 6)( 4, 5)( 8,18)( 9,19)(10,20)(11,21)(12,15)(13,16)(14,17)$
$ 14, 2, 2, 2, 1 $ $21$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,19,10,21,12,16,14,18, 9,20,11,15,13,17)$
$ 14, 2, 2, 2, 1 $ $21$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,20,12,17, 9,21,13,18,10,15,14,19,11,16)$
$ 14, 2, 2, 2, 1 $ $21$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,21,14,20,13,19,12,18,11,17,10,16, 9,15)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$
$ 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 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
$ 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  2  2  2   1   1   1  1   1   1   1  2  2  2  2  2  .

        1a 7a 7b 7c 7d 7e 7f 14a 14b 14c 2a 14d 14e 14f 7g 7h 7i 7j 7k 3a
     2P 1a 7b 7d 7f 7a 7c 7e  7a  7c  7e 1a  7b  7d  7f 7k 7h 7g 7j 7i 3a
     3P 1a 7c 7f 7b 7e 7a 7d 14b 14d 14a 2a 14f 14c 14e 7i 7j 7k 7h 7g 1a
     5P 1a 7e 7c 7a 7f 7d 7b 14c 14a 14e 2a 14b 14f 14d 7k 7j 7g 7h 7i 3a
     7P 1a 1a 1a 1a 1a 1a 1a  2a  2a  2a 2a  2a  2a  2a 1a 1a 1a 1a 1a 3a
    11P 1a 7d 7a 7e 7b 7f 7c 14e 14c 14f 2a 14a 14d 14b 7i 7h 7k 7j 7g 3a
    13P 1a 7f 7e 7d 7c 7b 7a 14f 14e 14d 2a 14c 14b 14a 7g 7j 7i 7h 7k 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 /B /C  C  B /A   H  /J   I -1  /I   J  /H  K  Q  M /Q  L  .
X.5      3  B /C  A /A  C /B   I   H   J -1  /J  /H  /I  L /Q  K  Q  M  .
X.6      3  C  A  B /B /A /C   J   I  /H -1   H  /I  /J  M  Q  L /Q  K  .
X.7      3 /A  B  C /C /B  A  /H   J  /I -1   I  /J   H  K /Q  M  Q  L  .
X.8      3 /B  C /A  A /C  B  /I  /H  /J -1   J   H   I  L  Q  K /Q  M  .
X.9      3 /C /A /B  B  A  C  /J  /I   H -1  /H   I   J  M /Q  L  Q  K  .
X.10     3  A /B /C  C  B /A  -H -/J  -I  1 -/I  -J -/H  K  Q  M /Q  L  .
X.11     3  B /C  A /A  C /B  -I  -H  -J  1 -/J -/H -/I  L /Q  K  Q  M  .
X.12     3  C  A  B /B /A /C  -J  -I -/H  1  -H -/I -/J  M  Q  L /Q  K  .
X.13     3 /A  B  C /C /B  A -/H  -J -/I  1  -I -/J  -H  K /Q  M  Q  L  .
X.14     3 /B  C /A  A /C  B -/I -/H -/J  1  -J  -H  -I  L  Q  K /Q  M  .
X.15     3 /C /A /B  B  A  C -/J -/I  -H  1 -/H  -I  -J  M /Q  L  Q  K  .
X.16     6  D  E  F  F  E  D   .   .   .  .   .   .   .  N -1  P -1  O  .
X.17     6  E  F  D  D  F  E   .   .   .  .   .   .   .  O -1  N -1  P  .
X.18     6  F  D  E  E  D  F   .   .   .  .   .   .   .  P -1  O -1  N  .
X.19     6  G  G /G  G /G /G   .   .   .  .   .   .   . -1  R -1 /R -1  .
X.20     6 /G /G  G /G  G  G   .   .   .  .   .   .   . -1 /R -1  R -1  .

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