Properties

Label 21T23
Order \(588\)
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 $t$ :  $23$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,3)(5,7)(8,20,11,15,14,17,10,19,13,21,9,16,12,18), (1,10,6,9,4,8,2,14,7,13,5,12,3,11)(15,19,16,20,17,21,18)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

14T25, 21T23, 28T78, 42T110 x 2, 42T111 x 2, 42T112 x 2, 42T122

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

Group invariants

Order:  $588=2^{2} \cdot 3 \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [588, 35]
Character table:   
      2  2  1  1  1  1  1  .  1  1   1   1  2   1  2  1  2   1   1   1
      3  1  .  .  .  .  .  .  .  1   .   .  .   .  1  1  .   .   .   .
      7  2  2  2  2  2  2  2  2  .   1   1  1   1  .  .  1   1   1   1

        1a 7a 7b 7c 7d 7e 7f 7g 3a 14a 14b 2a 14c 2b 6a 2c 14d 14e 14f
     2P 1a 7c 7a 7b 7g 7d 7f 7e 3a  7c  7a 1a  7b 1a 3a 1a  7d  7e  7g
     3P 1a 7b 7c 7a 7e 7g 7f 7d 1a 14b 14c 2a 14a 2b 2b 2c 14e 14f 14d
     5P 1a 7c 7a 7b 7g 7d 7f 7e 3a 14c 14a 2a 14b 2b 6a 2c 14f 14d 14e
     7P 1a 1a 1a 1a 1a 1a 1a 1a 3a  2a  2a 2a  2a 2b 6a 2c  2c  2c  2c
    11P 1a 7b 7c 7a 7e 7g 7f 7d 3a 14b 14c 2a 14a 2b 6a 2c 14e 14f 14d
    13P 1a 7a 7b 7c 7d 7e 7f 7g 3a 14a 14b 2a 14c 2b 6a 2c 14d 14e 14f

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

A = -2*E(7)-2*E(7)^2-2*E(7)^5-2*E(7)^6
B = -2*E(7)-2*E(7)^3-2*E(7)^4-2*E(7)^6
C = -2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5
D = 2*E(7)^2+E(7)^3+E(7)^4+2*E(7)^5
E = E(7)+2*E(7)^3+2*E(7)^4+E(7)^6
F = 2*E(7)+E(7)^2+E(7)^5+2*E(7)^6
G = -E(7)^3-E(7)^4
H = -E(7)-E(7)^6
I = -E(7)^2-E(7)^5