Group action invariants
Degree $n$: | $21$ | |
Transitive number $t$: | $23$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
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) |
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, 42T122Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 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 |