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Properties

Label 21T7
Degree $21$
Order $63$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times C_7:C_3$

Related objects

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$21$:  $C_7:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: $C_7:C_3$

Low degree siblings

21T7 x 2

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

Group invariants

Order:  $63=3^{2} \cdot 7$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [63, 3]
Character table:    
      3  2  2  2  2  2  2  2  2  2  1   1   1  1   1   1
      7  1  .  .  1  .  .  1  .  .  1   1   1  1   1   1

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

X.1      1  1  1  1  1  1  1  1  1  1   1   1  1   1   1
X.2      1  1  1  A  A  A /A /A /A  1   A  /A  1   A  /A
X.3      1  1  1 /A /A /A  A  A  A  1  /A   A  1  /A   A
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  A /A  1 /A  1  A  1   A  /A  1   A  /A
X.7      1 /A  A /A  A  1  A  1 /A  1  /A   A  1  /A   A
X.8      1  A /A /A  1  A  A /A  1  1  /A   A  1  /A   A
X.9      1 /A  A  A  1 /A /A  A  1  1   A  /A  1   A  /A
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  .  .  B  .  . /B  .  .  C   D   E /C  /E  /D
X.13     3  .  . /B  .  .  B  .  .  C   E   D /C  /D  /E
X.14     3  .  .  B  .  . /B  .  . /C  /E  /D  C   D   E
X.15     3  .  . /B  .  .  B  .  . /C  /D  /E  C   E   D

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
C = E(7)+E(7)^2+E(7)^4
  = (-1+Sqrt(-7))/2 = b7
D = E(21)^5+E(21)^17+E(21)^20
E = E(21)^10+E(21)^13+E(21)^19