Properties

Label 42T22
Order \(126\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{21}:C_3$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
42:  $F_7$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $S_3\times C_3$

Degree 7: $F_7$

Degree 14: $F_7$

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

Group invariants

Order:  $126=2 \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [126, 9]
Character table:   
      2  1   .   .  .  1  1  1   1   1   .  .   .
      3  2   2   2  2  2  2  1   1   1   1  1   1
      7  1   .   .  1  .  .  .   .   .   1  1   1

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

X.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
X.3      1   A  /A  1 /A  A -1 -/A  -A   1  1   1
X.4      1  /A   A  1  A /A -1  -A -/A   1  1   1
X.5      1   A  /A  1 /A  A  1  /A   A   1  1   1
X.6      1  /A   A  1  A /A  1   A  /A   1  1   1
X.7      2  -1  -1 -1  2  2  .   .   .  -1  2  -1
X.8      2 -/A  -A -1  B /B  .   .   .  -1  2  -1
X.9      2  -A -/A -1 /B  B  .   .   .  -1  2  -1
X.10     6   .   .  6  .  .  .   .   .  -1 -1  -1
X.11     6   .   . -3  .  .  .   .   .   C -1  *C
X.12     6   .   . -3  .  .  .   .   .  *C -1   C

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3
C = E(21)^2+E(21)^8+E(21)^10+E(21)^11+E(21)^13+E(21)^19
  = (1-Sqrt(21))/2 = -b21