Properties

Label 21T12
Order \(147\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7:(C_7:C_3)$

Related objects

Learn more about

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T12

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 $ $1$ $1$ $()$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $3$ $7$ $( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $3$ $7$ $( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $3$ $7$ $( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $3$ $7$ $( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $3$ $7$ $( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $3$ $7$ $( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,18,21,17,20,16,19)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $3$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$
$ 3, 3, 3, 3, 3, 3, 3 $ $49$ $3$ $( 1, 8,15)( 2, 9,21)( 3,10,20)( 4,11,19)( 5,12,18)( 6,13,17)( 7,14,16)$
$ 3, 3, 3, 3, 3, 3, 3 $ $49$ $3$ $( 1,15, 8)( 2,21, 9)( 3,20,10)( 4,19,11)( 5,18,12)( 6,17,13)( 7,16,14)$

Group invariants

Order:  $147=3 \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [147, 5]
Character table:   
      3  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  1  1
      7  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  .  .

        1a 7a 7b 7c 7d 7e 7f 7g 7h 7i 7j 7k 7l 7m 7n 7o 7p 3a 3b
     2P 1a 7b 7d 7f 7a 7c 7e 7k 7h 7l 7j 7o 7p 7m 7n 7g 7i 3b 3a
     3P 1a 7c 7f 7b 7e 7a 7d 7i 7n 7k 7m 7l 7o 7j 7h 7p 7g 1a 1a
     5P 1a 7e 7c 7a 7f 7d 7b 7p 7n 7g 7m 7i 7k 7j 7h 7l 7o 3b 3a
     7P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 3a 3b

X.1      1  1  1  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  1  1  H /H
X.3      1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 /H  H
X.4      3  A  A /A  A /A /A /A  3  A  A /A  A /A  3 /A  A  .  .
X.5      3 /A /A  A /A  A  A  A  3 /A /A  A /A  A  3  A /A  .  .
X.6      3  A  A /A  A /A /A  A /A /A  3  A /A  3  A  A /A  .  .
X.7      3 /A /A  A /A  A  A /A  A  A  3 /A  A  3 /A /A  A  .  .
X.8      3  B  D  C  C  D  B  E /A /G /A  F /E  A  A  G /F  .  .
X.9      3  C  B  D  D  B  C  G /A /F /A  E /G  A  A  F /E  .  .
X.10     3  D  C  B  B  C  D  F /A /E /A  G /F  A  A  E /G  .  .
X.11     3  B  D  C  C  D  B /E  A  G  A /F  E /A /A /G  F  .  .
X.12     3  C  B  D  D  B  C /G  A  F  A /E  G /A /A /F  E  .  .
X.13     3  D  C  B  B  C  D /F  A  E  A /G  F /A /A /E  G  .  .
X.14     3  E  F /G  G /F /E  C /A  D  A  B  C /A  A  D  B  .  .
X.15     3  F  G /E  E /G /F  B /A  C  A  D  B /A  A  C  D  .  .
X.16     3  G  E /F  F /E /G  D /A  B  A  C  D /A  A  B  C  .  .
X.17     3 /F /G  E /E  G  F  B  A  C /A  D  B  A /A  C  D  .  .
X.18     3 /E /F  G /G  F  E  C  A  D /A  B  C  A /A  D  B  .  .
X.19     3 /G /E  F /F  E  G  D  A  B /A  C  D  A /A  B  C  .  .

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
B = -E(7)^2-E(7)^3-E(7)^4-E(7)^5
C = -E(7)-E(7)^2-E(7)^5-E(7)^6
D = -E(7)-E(7)^3-E(7)^4-E(7)^6
E = 2*E(7)+E(7)^5
F = 2*E(7)^2+E(7)^3
G = 2*E(7)^4+E(7)^6
H = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3