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Properties

Label 21T19
Order \(294\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7:D_7:C_3$

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $19$
Group :  $C_7:D_7:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7)(2,6)(3,5)(8,13)(9,12)(10,11)(15,18)(16,17)(19,21), (1,14,21,6,8,17)(2,10,16,5,12,15)(3,13,18,4,9,20)(7,11,19)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
42:  $F_7$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T19, 42T58 x 2

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 $ $6$ $7$ $( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $49$ $2$ $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
$ 6, 6, 6, 3 $ $49$ $6$ $( 1, 8,15)( 2,11,17, 7,12,20)( 3,14,19, 6, 9,18)( 4,10,21, 5,13,16)$
$ 3, 3, 3, 3, 3, 3, 3 $ $49$ $3$ $( 1, 8,15)( 2,12,17)( 3, 9,19)( 4,13,21)( 5,10,16)( 6,14,18)( 7,11,20)$
$ 3, 3, 3, 3, 3, 3, 3 $ $49$ $3$ $( 1,15, 8)( 2,17,12)( 3,19, 9)( 4,21,13)( 5,16,10)( 6,18,14)( 7,20,11)$
$ 6, 6, 6, 3 $ $49$ $6$ $( 1,15, 8)( 2,20,12, 7,17,11)( 3,18, 9, 6,19,14)( 4,16,13, 5,21,10)$

Group invariants

Order:  $294=2 \cdot 3 \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [294, 14]
Character table:    
      2  1  .  .  .  1  .  .  .  .  .   1   1   1   1
      3  1  .  .  .  1  .  .  .  .  .   1   1   1   1
      7  2  2  2  2  .  2  2  2  2  2   .   .   .   .

        1a 7a 7b 7c 2a 7d 7e 7f 7g 7h  6a  3a  3b  6b
     2P 1a 7b 7c 7a 1a 7d 7e 7h 7f 7g  3b  3b  3a  3a
     3P 1a 7c 7a 7b 2a 7d 7e 7g 7h 7f  2a  1a  1a  2a
     5P 1a 7b 7c 7a 2a 7d 7e 7h 7f 7g  6b  3b  3a  6a
     7P 1a 1a 1a 1a 2a 1a 1a 1a 1a 1a  6a  3a  3b  6b

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      1  1  1  1 -1  1  1  1  1  1   G  -G -/G  /G
X.4      1  1  1  1 -1  1  1  1  1  1  /G -/G  -G   G
X.5      1  1  1  1  1  1  1  1  1  1 -/G -/G  -G  -G
X.6      1  1  1  1  1  1  1  1  1  1  -G  -G -/G -/G
X.7      6 -1 -1 -1  .  6 -1 -1 -1 -1   .   .   .   .
X.8      6 -1 -1 -1  . -1  6 -1 -1 -1   .   .   .   .
X.9      6  A  B  C  . -1 -1  E  D  F   .   .   .   .
X.10     6  B  C  A  . -1 -1  F  E  D   .   .   .   .
X.11     6  C  A  B  . -1 -1  D  F  E   .   .   .   .
X.12     6  D  E  F  . -1 -1  C  B  A   .   .   .   .
X.13     6  E  F  D  . -1 -1  A  C  B   .   .   .   .
X.14     6  F  D  E  . -1 -1  B  A  C   .   .   .   .

A = -2*E(7)-2*E(7)^2-2*E(7)^5-2*E(7)^6
B = -2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5
C = -2*E(7)-2*E(7)^3-2*E(7)^4-2*E(7)^6
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 = -E(3)
  = (1-Sqrt(-3))/2 = -b3