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Properties

Label 21T26
Order \(882\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $26$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,21,2,18,4,19)(3,15,6,20,5,16)(7,17)(8,12,14)(10,13,11), (1,11,15)(2,10,16)(3,9,17)(4,8,18)(5,14,19)(6,13,20)(7,12,21)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

14T26, 21T25, 42T143, 42T144, 42T152, 42T153, 42T154, 42T155

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 $ $18$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $49$ $3$ $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $49$ $3$ $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$
$ 3, 3, 3, 3, 3, 3, 3 $ $98$ $3$ $( 1,21, 9)( 2,15, 8)( 3,16,14)( 4,17,13)( 5,18,12)( 6,19,11)( 7,20,10)$
$ 21 $ $42$ $21$ $( 1,20,12, 2,15, 8, 3,17,11, 4,19,14, 5,21,10, 6,16,13, 7,18, 9)$
$ 21 $ $42$ $21$ $( 1,20,10, 4,19,12, 7,18,14, 3,17, 9, 6,16,11, 2,15,13, 5,21, 8)$
$ 3, 3, 3, 3, 3, 3, 3 $ $14$ $3$ $( 1,20,13)( 2,15, 9)( 3,17,12)( 4,19, 8)( 5,21,11)( 6,16,14)( 7,18,10)$
$ 21 $ $42$ $21$ $( 1,18,10, 4,16,11, 7,21,12, 3,19,13, 6,17,14, 2,15, 8, 5,20, 9)$
$ 21 $ $42$ $21$ $( 1,18, 8, 3,19,11, 5,20,14, 7,21,10, 2,15,13, 4,16, 9, 6,17,12)$
$ 3, 3, 3, 3, 3, 3, 3 $ $14$ $3$ $( 1,18,11)( 2,15, 9)( 3,19,14)( 4,16,12)( 5,20,10)( 6,17, 8)( 7,21,13)$
$ 6, 6, 3, 3, 2, 1 $ $147$ $6$ $( 2, 5, 3)( 4, 6, 7)( 8,18,13,19, 9,21)(10,17)(11,20,12,16,14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,16)(14,15)$
$ 14, 7 $ $63$ $14$ $( 1, 5, 2, 6, 3, 7, 4)( 8,21,12,17, 9,20,13,16,10,19,14,15,11,18)$
$ 14, 7 $ $63$ $14$ $( 1, 6, 4, 2, 7, 5, 3)( 8,21,13,16,11,18, 9,20,14,15,12,17,10,19)$
$ 6, 6, 3, 3, 2, 1 $ $147$ $6$ $( 2, 3, 5)( 4, 7, 6)( 8,20,10,16,11,21)( 9,18,14,15,13,17)(12,19)$

Group invariants

Order:  $882=2 \cdot 3^{2} \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [882, 34]
Character table:    
      2  1  .  .  1  1  .  1  1  .   .   .   .   .   .   .   1  1   1   1   1
      3  2  .  1  .  .  1  2  2  2   1   1   2   1   1   2   1  1   .   .   1
      7  2  2  2  2  2  2  .  .  .   1   1   1   1   1   1   .  1   1   1   .

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

X.1      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  -1  -1  -1
X.3      1  1  1  1  1  1  E /E  1   E   E   E  /E  /E  /E -/E -1  -1  -1  -E
X.4      1  1  1  1  1  1 /E  E  1  /E  /E  /E   E   E   E  -E -1  -1  -1 -/E
X.5      1  1  1  1  1  1  E /E  1   E   E   E  /E  /E  /E  /E  1   1   1   E
X.6      1  1  1  1  1  1 /E  E  1  /E  /E  /E   E   E   E   E  1   1   1  /E
X.7      2  2  2  2  2  2  2  2 -1  -1  -1  -1  -1  -1  -1   .  .   .   .   .
X.8      2  2  2  2  2  2  F /F -1  -E  -E  -E -/E -/E -/E   .  .   .   .   .
X.9      2  2  2  2  2  2 /F  F -1 -/E -/E -/E  -E  -E  -E   .  .   .   .   .
X.10     6 -1  A  C /C /A  .  .  .   G  /G   3  /G   G   3   .  .   .   .   .
X.11     6 -1 /A /C  C  A  .  .  .  /G   G   3   G  /G   3   .  .   .   .   .
X.12     6 -1  A  C /C /A  .  .  .   H  /I   J  /H   I  /J   .  .   .   .   .
X.13     6 -1  A  C /C /A  .  .  .   I  /H  /J  /I   H   J   .  .   .   .   .
X.14     6 -1 /A /C  C  A  .  .  .  /I   H   J   I  /H  /J   .  .   .   .   .
X.15     6 -1 /A /C  C  A  .  .  .  /H   I  /J   H  /I   J   .  .   .   .   .
X.16     9  2  B  D /D /B  .  .  .   .   .   .   .   .   .   . -3  -G -/G   .
X.17     9  2 /B /D  D  B  .  .  .   .   .   .   .   .   .   . -3 -/G  -G   .
X.18     9  2  B  D /D /B  .  .  .   .   .   .   .   .   .   .  3   G  /G   .
X.19     9  2 /B /D  D  B  .  .  .   .   .   .   .   .   .   .  3  /G   G   .
X.20    18 -3 -3  4  4 -3  .  .  .   .   .   .   .   .   .   .  .   .   .   .

A = -2*E(7)-2*E(7)^2-3*E(7)^3-2*E(7)^4-3*E(7)^5-3*E(7)^6
  = (5+Sqrt(-7))/2 = 3+b7
B = 3*E(7)^3+3*E(7)^5+3*E(7)^6
  = (-3-3*Sqrt(-7))/2 = -3-3b7
C = 2*E(7)^3+2*E(7)^5+2*E(7)^6
  = -1-Sqrt(-7) = -1-i7
D = E(7)+E(7)^2+2*E(7)^3+E(7)^4+2*E(7)^5+2*E(7)^6
  = (-3-Sqrt(-7))/2 = -2-b7
E = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
F = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3
G = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
H = E(21)^2+E(21)^8+E(21)^11
I = E(21)+E(21)^4+E(21)^16
J = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3