Properties

Label 42T23
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 $t$ :  $23$
Group :  $S_3\times C_7:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,20)(2,8,21)(3,7,19)(4,35,12)(5,36,10)(6,34,11)(13,31,27)(14,32,26)(15,33,25)(16,17,18)(22,29,40)(23,30,42)(24,28,41), (1,18,15,24,37,36)(2,16,14,22,38,34)(3,17,13,23,39,35)(4,8,41,26,28,19)(5,7,40,27,29,20)(6,9,42,25,30,21)(10,31,12,32,11,33)
$|\Aut(F/K)|$:  $3$

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: None

Degree 6: $S_3\times C_3$

Degree 7: $C_7:C_3$

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

Degree 21: None

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

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  2  2  2   .   1   1   1   1   .   1   1  1
      7  1   .   .  1  .  .   1   .   .   1   1   1   1   1  1

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

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

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