Properties

Label 42T19
Order \(126\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times C_7:C_3$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $19$
Group :  $S_3\times C_7:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,41)(2,16,42)(3,18,37)(4,17,38)(5,14,39)(6,13,40)(7,27,24)(8,28,23)(9,29,20)(10,30,19)(11,26,21)(12,25,22)(31,33,35)(32,34,36), (1,9,13,21,25,33,37,4,8,16,19,27,32,39)(2,10,14,22,26,34,38,3,7,15,20,28,31,40)(5,12,17,23,29,36,42,6,11,18,24,30,35,41)
$|\Aut(F/K)|$:  $6$

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:  $C_7:C_3$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 7: $C_7:C_3$

Degree 14: $(C_7:C_3) \times C_2$

Degree 21: 21T11

Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

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, 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, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $7$ $3$ $( 7,14,26)( 8,13,25)( 9,16,27)(10,15,28)(11,17,29)(12,18,30)(19,37,32) (20,38,31)(21,39,33)(22,40,34)(23,41,36)(24,42,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $7$ $3$ $( 7,26,14)( 8,25,13)( 9,27,16)(10,28,15)(11,29,17)(12,30,18)(19,32,37) (20,31,38)(21,33,39)(22,34,40)(23,36,41)(24,35,42)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 5)( 4, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,23) (22,24)(25,26)(27,30)(28,29)(31,32)(33,36)(34,35)(37,38)(39,41)(40,42)$
$ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ $21$ $6$ $( 1, 2)( 3, 5)( 4, 6)( 7,13,26, 8,14,25)( 9,18,27,12,16,30)(10,17,28,11,15,29) (19,38,32,20,37,31)(21,41,33,23,39,36)(22,42,34,24,40,35)$
$ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ $21$ $6$ $( 1, 2)( 3, 5)( 4, 6)( 7,25,14, 8,26,13)( 9,30,16,12,27,18)(10,29,15,11,28,17) (19,31,37,20,32,38)(21,36,39,23,33,41)(22,35,40,24,34,42)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 3, 6)( 2, 4, 5)( 7, 9,11)( 8,10,12)(13,15,18)(14,16,17)(19,22,23) (20,21,24)(25,28,30)(26,27,29)(31,33,35)(32,34,36)(37,40,41)(38,39,42)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $14$ $3$ $( 1, 3, 6)( 2, 4, 5)( 7,16,29)( 8,15,30)( 9,17,26)(10,18,25)(11,14,27) (12,13,28)(19,40,36)(20,39,35)(21,42,31)(22,41,32)(23,37,34)(24,38,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $14$ $3$ $( 1, 3, 6)( 2, 4, 5)( 7,27,17)( 8,28,18)( 9,29,14)(10,30,13)(11,26,16) (12,25,15)(19,34,41)(20,33,42)(21,35,38)(22,36,37)(23,32,40)(24,31,39)$
$ 14, 14, 14 $ $9$ $14$ $( 1, 7,13,20,25,31,37, 2, 8,14,19,26,32,38)( 3,11,15,24,28,35,40, 5,10,17,22, 29,34,42)( 4,12,16,23,27,36,39, 6, 9,18,21,30,33,41)$
$ 7, 7, 7, 7, 7, 7 $ $3$ $7$ $( 1, 8,13,19,25,32,37)( 2, 7,14,20,26,31,38)( 3,10,15,22,28,34,40) ( 4, 9,16,21,27,33,39)( 5,11,17,24,29,35,42)( 6,12,18,23,30,36,41)$
$ 21, 21 $ $6$ $21$ $( 1,10,18,19,28,36,37, 3,12,13,22,30,32,40, 6, 8,15,23,25,34,41) ( 2, 9,17,20,27,35,38, 4,11,14,21,29,31,39, 5, 7,16,24,26,33,42)$
$ 7, 7, 7, 7, 7, 7 $ $3$ $7$ $( 1,19,37,13,32, 8,25)( 2,20,38,14,31, 7,26)( 3,22,40,15,34,10,28) ( 4,21,39,16,33, 9,27)( 5,24,42,17,35,11,29)( 6,23,41,18,36,12,30)$
$ 14, 14, 14 $ $9$ $14$ $( 1,20,37,14,32, 7,25, 2,19,38,13,31, 8,26)( 3,24,40,17,34,11,28, 5,22,42,15, 35,10,29)( 4,23,39,18,33,12,27, 6,21,41,16,36, 9,30)$
$ 21, 21 $ $6$ $21$ $( 1,22,41,13,34,12,25, 3,23,37,15,36, 8,28, 6,19,40,18,32,10,30) ( 2,21,42,14,33,11,26, 4,24,38,16,35, 7,27, 5,20,39,17,31, 9,29)$

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 14a  7a 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 14b  7b  7b  7a 14a  7a
     5P 1a 3b 3a 2a  6b  6a 3c  3e  3d 14b  7b 21b  7a 14a 21a
     7P 1a 3a 3b 2a  6a  6b 3c  3d  3e  2a  1a  3c  1a  2a  3c
    11P 1a 3b 3a 2a  6b  6a 3c  3e  3d 14a  7a 21a  7b 14b 21b
    13P 1a 3a 3b 2a  6a  6b 3c  3d  3e 14b  7b 21b  7a 14a 21a
    17P 1a 3b 3a 2a  6b  6a 3c  3e  3d 14b  7b 21b  7a 14a 21a
    19P 1a 3a 3b 2a  6a  6b 3c  3d  3e 14b  7b 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)-E(7)^2-E(7)^4
  = (1-Sqrt(-7))/2 = -b7
D = 2*E(7)^3+2*E(7)^5+2*E(7)^6
  = -1-Sqrt(-7) = -1-i7