Group action invariants
Degree $n$: | $21$ | |
Transitive number $t$: | $22$ | |
Group: | $C_3\times \PSL(2,7)$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $3$ | |
Generators: | (1,2,3)(4,9,6,8,5,7)(10,20,12,19,11,21)(13,14,15)(16,17,18), (1,8,17,15,4,21,12)(2,9,18,13,5,19,10)(3,7,16,14,6,20,11) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $168$: $\GL(3,2)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: $\GL(3,2)$
Low degree siblings
21T22, 24T1355 x 2, 24T1356, 42T96 x 2, 42T103 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
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 $ | $1$ | $1$ | $()$ |
$ 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 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$ |
$ 6, 6, 3, 3, 3 $ | $21$ | $6$ | $( 1, 3, 2)( 4, 6, 5)( 7,13, 8,14, 9,15)(10,12,11)(16,19,17,20,18,21)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 7,14)( 8,15)( 9,13)(16,20)(17,21)(18,19)$ |
$ 6, 6, 3, 3, 3 $ | $21$ | $6$ | $( 1, 2, 3)( 4, 5, 6)( 7,15, 9,14, 8,13)(10,11,12)(16,21,18,20,17,19)$ |
$ 12, 6, 3 $ | $42$ | $12$ | $( 1, 2, 3)( 4, 9,11,21, 5, 7,12,19, 6, 8,10,20)(13,16,15,18,14,17)$ |
$ 12, 6, 3 $ | $42$ | $12$ | $( 1, 3, 2)( 4, 7,10,21, 6, 9,12,20, 5, 8,11,19)(13,17,14,18,15,16)$ |
$ 4, 4, 4, 2, 2, 2, 1, 1, 1 $ | $42$ | $4$ | $( 4, 8,12,21)( 5, 9,10,19)( 6, 7,11,20)(13,18)(14,16)(15,17)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $56$ | $3$ | $( 1, 2, 3)( 4, 9,14)( 5, 7,15)( 6, 8,13)(10,20,17)(11,21,18)(12,19,16)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $56$ | $3$ | $( 1, 3, 2)( 4, 7,13)( 5, 8,14)( 6, 9,15)(10,21,16)(11,19,17)(12,20,18)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $56$ | $3$ | $( 4, 8,15)( 5, 9,13)( 6, 7,14)(10,19,18)(11,20,16)(12,21,17)$ |
$ 7, 7, 7 $ | $24$ | $7$ | $( 1, 4, 8,12,15,17,21)( 2, 5, 9,10,13,18,19)( 3, 6, 7,11,14,16,20)$ |
$ 21 $ | $24$ | $21$ | $( 1, 5, 7,12,13,16,21, 2, 6, 8,10,14,17,19, 3, 4, 9,11,15,18,20)$ |
$ 21 $ | $24$ | $21$ | $( 1, 6, 9,12,14,18,21, 3, 5, 8,11,13,17,20, 2, 4, 7,10,15,16,19)$ |
$ 7, 7, 7 $ | $24$ | $7$ | $( 1, 4, 8,21,17,12,15)( 2, 5, 9,19,18,10,13)( 3, 6, 7,20,16,11,14)$ |
$ 21 $ | $24$ | $21$ | $( 1, 5, 7,21,18,11,15, 2, 6, 8,19,16,12,13, 3, 4, 9,20,17,10,14)$ |
$ 21 $ | $24$ | $21$ | $( 1, 6, 9,21,16,10,15, 3, 5, 8,20,18,12,14, 2, 4, 7,19,17,11,13)$ |
Group invariants
Order: | $504=2^{3} \cdot 3^{2} \cdot 7$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | no | |
GAP id: | [504, 157] |
Character table: |
2 3 3 3 3 3 3 2 2 2 . . . . . . . . . 3 2 2 2 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 7 1 1 1 . . . . . . . . . 1 1 1 1 1 1 1a 3a 3b 6a 2a 6b 12a 12b 4a 3c 3d 3e 7a 21a 21b 7b 21c 21d 2P 1a 3b 3a 3a 1a 3b 6a 6b 2a 3d 3c 3e 7a 21b 21a 7b 21d 21c 3P 1a 1a 1a 2a 2a 2a 4a 4a 4a 1a 1a 1a 7b 7b 7b 7a 7a 7a 5P 1a 3b 3a 6b 2a 6a 12b 12a 4a 3d 3c 3e 7b 21d 21c 7a 21b 21a 7P 1a 3a 3b 6a 2a 6b 12a 12b 4a 3c 3d 3e 1a 3a 3b 1a 3a 3b 11P 1a 3b 3a 6b 2a 6a 12b 12a 4a 3d 3c 3e 7a 21b 21a 7b 21d 21c 13P 1a 3a 3b 6a 2a 6b 12a 12b 4a 3c 3d 3e 7b 21c 21d 7a 21a 21b 17P 1a 3b 3a 6b 2a 6a 12b 12a 4a 3d 3c 3e 7b 21d 21c 7a 21b 21a 19P 1a 3a 3b 6a 2a 6b 12a 12b 4a 3c 3d 3e 7b 21c 21d 7a 21a 21b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 A /A /A 1 A A /A 1 A /A 1 1 A /A 1 A /A X.3 1 /A A A 1 /A /A A 1 /A A 1 1 /A A 1 /A A X.4 3 3 3 -1 -1 -1 1 1 1 . . . G G G /G /G /G X.5 3 3 3 -1 -1 -1 1 1 1 . . . /G /G /G G G G X.6 3 B /B -/A -1 -A A /A 1 . . . G H /I /G I /H X.7 3 B /B -/A -1 -A A /A 1 . . . /G I /H G H /I X.8 3 /B B -A -1 -/A /A A 1 . . . G /I H /G /H I X.9 3 /B B -A -1 -/A /A A 1 . . . /G /H I G /I H X.10 6 6 6 2 2 2 . . . . . . -1 -1 -1 -1 -1 -1 X.11 6 C /C F 2 /F . . . . . . -1 -A -/A -1 -A -/A X.12 6 /C C /F 2 F . . . . . . -1 -/A -A -1 -/A -A X.13 7 7 7 -1 -1 -1 -1 -1 -1 1 1 1 . . . . . . X.14 7 D /D -/A -1 -A -A -/A -1 A /A 1 . . . . . . X.15 7 /D D -A -1 -/A -/A -A -1 /A A 1 . . . . . . X.16 8 8 8 . . . . . . -1 -1 -1 1 1 1 1 1 1 X.17 8 E /E . . . . . . -A -/A -1 1 A /A 1 A /A X.18 8 /E E . . . . . . -/A -A -1 1 /A A 1 /A A A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 C = 6*E(3)^2 = -3-3*Sqrt(-3) = -3-3i3 D = 7*E(3)^2 = (-7-7*Sqrt(-3))/2 = -7-7b3 E = 8*E(3)^2 = -4-4*Sqrt(-3) = -4-4i3 F = 2*E(3) = -1+Sqrt(-3) = 2b3 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)^5+E(21)^17+E(21)^20 |