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Properties

Label 21T10
Order \(126\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{21}:C_3$

Related objects

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: $F_7$

Low degree siblings

42T18, 42T22

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

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

        1a 3a 3b  6a  6b 2a 3c  3d  3e 21a 7a 21b
     2P 1a 3b 3a  3a  3b 1a 3c  3e  3d 21b 7a 21a
     3P 1a 1a 1a  2a  2a 2a 1a  1a  1a  7a 7a  7a
     5P 1a 3b 3a  6b  6a 2a 3c  3e  3d 21a 7a 21b
     7P 1a 3a 3b  6a  6b 2a 3c  3d  3e  3c 1a  3c
    11P 1a 3b 3a  6b  6a 2a 3c  3e  3d 21b 7a 21a
    13P 1a 3a 3b  6a  6b 2a 3c  3d  3e 21b 7a 21a
    17P 1a 3b 3a  6b  6a 2a 3c  3e  3d 21a 7a 21b
    19P 1a 3a 3b  6a  6b 2a 3c  3d  3e 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 -/A  -A -1  1   A  /A   1  1   1
X.4      1 /A  A  -A -/A -1  1  /A   A   1  1   1
X.5      1  A /A  /A   A  1  1   A  /A   1  1   1
X.6      1 /A  A   A  /A  1  1  /A   A   1  1   1
X.7      2  2  2   .   .  . -1  -1  -1  -1  2  -1
X.8      2  B /B   .   .  . -1  -A -/A  -1  2  -1
X.9      2 /B  B   .   .  . -1 -/A  -A  -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