Group action invariants
Degree $n$: | $36$ | |
Transitive number $t$: | $11$ | |
Group: | $C_2^2:C_9$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $36$ | |
Generators: | (1,8,11,4,5,10)(2,7,12,3,6,9)(13,19,22,14,20,21)(15,18,23,16,17,24)(25,32,36,28,30,34)(26,31,35,27,29,33), (1,27,13,5,31,20,11,33,22)(2,28,14,6,32,19,12,34,21)(3,26,15,7,29,17,9,35,23)(4,25,16,8,30,18,10,36,24) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $9$: $C_9$ $12$: $A_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 4: $A_4$
Degree 6: $A_4$
Degree 9: $C_9$
Degree 12: $A_4$
Degree 18: $C_2^2 : C_9$
Low degree siblings
18T7Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24) (22,23)(25,27)(26,28)(29,32)(30,31)(33,36)(34,35)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 5,11)( 2, 6,12)( 3, 7, 9)( 4, 8,10)(13,20,22)(14,19,21)(15,17,23) (16,18,24)(25,30,36)(26,29,35)(27,31,33)(28,32,34)$ |
$ 6, 6, 6, 6, 6, 6 $ | $3$ | $6$ | $( 1, 6,11, 2, 5,12)( 3, 8, 9, 4, 7,10)(13,17,22,15,20,23)(14,18,21,16,19,24) (25,31,36,27,30,33)(26,32,35,28,29,34)$ |
$ 6, 6, 6, 6, 6, 6 $ | $3$ | $6$ | $( 1, 9, 5, 3,11, 7)( 2,10, 6, 4,12, 8)(13,24,20,16,22,18)(14,23,19,15,21,17) (25,35,30,26,36,29)(27,34,31,28,33,32)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,11, 5)( 2,12, 6)( 3, 9, 7)( 4,10, 8)(13,22,20)(14,21,19)(15,23,17) (16,24,18)(25,36,30)(26,35,29)(27,33,31)(28,34,32)$ |
$ 9, 9, 9, 9 $ | $4$ | $9$ | $( 1,13,31,11,22,27, 5,20,33)( 2,14,32,12,21,28, 6,19,34)( 3,15,29, 9,23,26, 7, 17,35)( 4,16,30,10,24,25, 8,18,36)$ |
$ 9, 9, 9, 9 $ | $4$ | $9$ | $( 1,17,28,11,15,34, 5,23,32)( 2,18,27,12,16,33, 6,24,31)( 3,20,25, 9,13,36, 7, 22,30)( 4,19,26,10,14,35, 8,21,29)$ |
$ 9, 9, 9, 9 $ | $4$ | $9$ | $( 1,21,36,11,19,30, 5,14,25)( 2,22,35,12,20,29, 6,13,26)( 3,24,34, 9,18,32, 7, 16,28)( 4,23,33,10,17,31, 8,15,27)$ |
$ 9, 9, 9, 9 $ | $4$ | $9$ | $( 1,25,14, 5,30,19,11,36,21)( 2,26,13, 6,29,20,12,35,22)( 3,28,16, 7,32,18, 9, 34,24)( 4,27,15, 8,31,17,10,33,23)$ |
$ 9, 9, 9, 9 $ | $4$ | $9$ | $( 1,29,24, 5,35,16,11,26,18)( 2,30,23, 6,36,15,12,25,17)( 3,31,21, 7,33,14, 9, 27,19)( 4,32,22, 8,34,13,10,28,20)$ |
$ 9, 9, 9, 9 $ | $4$ | $9$ | $( 1,33,20, 5,27,22,11,31,13)( 2,34,19, 6,28,21,12,32,14)( 3,35,17, 7,26,23, 9, 29,15)( 4,36,18, 8,25,24,10,30,16)$ |
Group invariants
Order: | $36=2^{2} \cdot 3^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [36, 3] |
Character table: |
2 2 2 2 2 2 2 . . . . . . 3 2 1 2 1 1 2 2 2 2 2 2 2 1a 2a 3a 6a 6b 3b 9a 9b 9c 9d 9e 9f 2P 1a 1a 3b 3b 3a 3a 9e 9d 9f 9a 9c 9b 3P 1a 2a 1a 2a 2a 1a 3b 3b 3b 3a 3a 3a 5P 1a 2a 3b 6b 6a 3a 9d 9f 9e 9b 9a 9c 7P 1a 2a 3a 6a 6b 3b 9b 9c 9a 9f 9d 9e X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 A A A /A /A /A X.3 1 1 1 1 1 1 /A /A /A A A A X.4 1 1 A A /A /A C D E /E /D /C X.5 1 1 A A /A /A D E C /C /E /D X.6 1 1 A A /A /A E C D /D /C /E X.7 1 1 /A /A A A /C /D /E E D C X.8 1 1 /A /A A A /E /C /D D C E X.9 1 1 /A /A A A /D /E /C C E D X.10 3 -1 3 -1 -1 3 . . . . . . X.11 3 -1 B -/A -A /B . . . . . . X.12 3 -1 /B -A -/A B . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 3*E(3) = (-3+3*Sqrt(-3))/2 = 3b3 C = -E(9)^4-E(9)^7 D = E(9)^7 E = E(9)^4 |