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Properties

Label 21T11
Order \(126\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times C_7:C_3$

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $11$
Group :  $S_3\times C_7:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,5)(2,10,6)(3,11,4)(7,15,17)(8,13,18)(9,14,16)(19,21,20), (1,6,16,3,4,18)(2,5,17)(7,9)(10,21,13,12,19,15)(11,20,14)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
21:  $C_7:C_3$
42:  $(C_7:C_3) \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: $C_7:C_3$

Low degree siblings

42T19, 42T23

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

Group invariants

Order:  $126=2 \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [126, 8]
Character table:    
      2  1  1  1  1   1   1  .   .   .  1   1   .  1   1   .
      3  2  2  2  1   1   1  2   2   2  1   .   1  1   .   1
      7  1  .  .  1   .   .  1   .   .  1   1   1  1   1   1

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

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3
C = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
D = 2*E(7)^3+2*E(7)^5+2*E(7)^6
  = -1-Sqrt(-7) = -1-i7