Group action invariants
| Degree $n$ : | $36$ | |
| Transitive number $t$ : | $6$ | |
| Group : | $C_6\times S_3$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,29,10,2,30,9)(3,32,12,4,31,11)(5,23,13,6,24,14)(7,21,15,8,22,16)(17,34,27,18,33,28)(19,35,26,20,36,25), (1,25,15,3,27,14)(2,26,16,4,28,13)(5,21,19,34,32,9)(6,22,20,33,31,10)(7,23,17,35,30,12)(8,24,18,36,29,11) | |
| $|\Aut(F/K)|$: | $36$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $S_3$, $C_6$ x 3 12: $D_{6}$ 18: $S_3\times C_3$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 6: $C_6$ x 3, $S_3$, $D_{6}$ x 2, $S_3\times C_3$
Degree 9: $S_3\times C_3$
Degree 12: $C_6\times C_2$, $D_6$, $C_6\times S_3$
Degree 18: $S_3 \times C_3$, $S_3 \times C_6$ x 2
Low degree siblings
There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.
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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 3)( 2, 4)( 5,34)( 6,33)( 7,35)( 8,36)( 9,19)(10,20)(11,18)(12,17)(13,16) (14,15)(21,32)(22,31)(23,30)(24,29)(25,27)(26,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2, 3)( 5,33)( 6,34)( 7,36)( 8,35)( 9,20)(10,19)(11,17)(12,18)(13,15) (14,16)(21,31)(22,32)(23,29)(24,30)(25,28)(26,27)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 7,33)( 2, 8,34)( 3, 6,35)( 4, 5,36)( 9,16,18)(10,15,17)(11,13,19) (12,14,20)(21,28,29)(22,27,30)(23,25,31)(24,26,32)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 8,33, 2, 7,34)( 3, 5,35, 4, 6,36)( 9,15,18,10,16,17)(11,14,19,12,13,20) (21,27,29,22,28,30)(23,26,31,24,25,32)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 9,30, 2,10,29)( 3,11,31, 4,12,32)( 5,14,24, 6,13,23)( 7,16,22, 8,15,21) (17,28,33,18,27,34)(19,25,36,20,26,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,10,30)( 2, 9,29)( 3,12,31)( 4,11,32)( 5,13,24)( 6,14,23)( 7,15,22) ( 8,16,21)(17,27,33)(18,28,34)(19,26,36)(20,25,35)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $3$ | $6$ | $( 1,11,27,36,15,24)( 2,12,28,35,16,23)( 3, 9,25,34,14,21)( 4,10,26,33,13,22) ( 5,17,32, 7,19,30)( 6,18,31, 8,20,29)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $3$ | $6$ | $( 1,12,27,35,15,23)( 2,11,28,36,16,24)( 3,10,25,33,14,22)( 4, 9,26,34,13,21) ( 5,18,32, 8,19,29)( 6,17,31, 7,20,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,15,27)( 2,16,28)( 3,14,25)( 4,13,26)( 5,19,32)( 6,20,31)( 7,17,30) ( 8,18,29)( 9,21,34)(10,22,33)(11,24,36)(12,23,35)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,16,27, 2,15,28)( 3,13,25, 4,14,26)( 5,20,32, 6,19,31)( 7,18,30, 8,17,29) ( 9,22,34,10,21,33)(11,23,36,12,24,35)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,21,17, 2,22,18)( 3,24,20, 4,23,19)( 5,25,11, 6,26,12)( 7,28,10, 8,27, 9) (13,35,32,14,36,31)(15,34,30,16,33,29)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,22,17)( 2,21,18)( 3,23,20)( 4,24,19)( 5,26,11)( 6,25,12)( 7,27,10) ( 8,28, 9)(13,36,32)(14,35,31)(15,33,30)(16,34,29)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $3$ | $6$ | $( 1,23,15,35,27,12)( 2,24,16,36,28,11)( 3,22,14,33,25,10)( 4,21,13,34,26, 9) ( 5,29,19, 8,32,18)( 6,30,20, 7,31,17)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $3$ | $6$ | $( 1,24,15,36,27,11)( 2,23,16,35,28,12)( 3,21,14,34,25, 9)( 4,22,13,33,26,10) ( 5,30,19, 7,32,17)( 6,29,20, 8,31,18)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,27,15)( 2,28,16)( 3,25,14)( 4,26,13)( 5,32,19)( 6,31,20)( 7,30,17) ( 8,29,18)( 9,34,21)(10,33,22)(11,36,24)(12,35,23)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,28,15, 2,27,16)( 3,26,14, 4,25,13)( 5,31,19, 6,32,20)( 7,29,17, 8,30,18) ( 9,33,21,10,34,22)(11,35,24,12,36,23)$ |
Group invariants
| Order: | $36=2^{2} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [36, 12] |
| Character table: |
2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 2 2 2 2
3 2 2 1 1 2 2 2 2 1 1 2 2 2 2 1 1 2 2
1a 2a 2b 2c 3a 6a 6b 3b 6c 6d 3c 6e 6f 3d 6g 6h 3e 6i
2P 1a 1a 1a 1a 3a 3a 3d 3d 3e 3e 3e 3e 3b 3b 3c 3c 3c 3c
3P 1a 2a 2b 2c 1a 2a 2a 1a 2c 2b 1a 2a 2a 1a 2b 2c 1a 2a
5P 1a 2a 2b 2c 3a 6a 6f 3d 6h 6g 3e 6i 6b 3b 6d 6c 3c 6e
X.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
X.3 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
X.5 1 -1 -1 1 1 -1 A -A -A A -A A /A -/A /A -/A -/A /A
X.6 1 -1 -1 1 1 -1 /A -/A -/A /A -/A /A A -A A -A -A A
X.7 1 -1 1 -1 1 -1 A -A A -A -A A /A -/A -/A /A -/A /A
X.8 1 -1 1 -1 1 -1 /A -/A /A -/A -/A /A A -A -A A -A A
X.9 1 1 -1 -1 1 1 -/A -/A /A /A -/A -/A -A -A A A -A -A
X.10 1 1 -1 -1 1 1 -A -A A A -A -A -/A -/A /A /A -/A -/A
X.11 1 1 1 1 1 1 -/A -/A -/A -/A -/A -/A -A -A -A -A -A -A
X.12 1 1 1 1 1 1 -A -A -A -A -A -A -/A -/A -/A -/A -/A -/A
X.13 2 -2 . . -1 1 1 -1 . . 2 -2 1 -1 . . 2 -2
X.14 2 2 . . -1 -1 -1 -1 . . 2 2 -1 -1 . . 2 2
X.15 2 -2 . . -1 1 -/A /A . . B -B -A A . . /B -/B
X.16 2 -2 . . -1 1 -A A . . /B -/B -/A /A . . B -B
X.17 2 2 . . -1 -1 /A /A . . B B A A . . /B /B
X.18 2 2 . . -1 -1 A A . . /B /B /A /A . . B B
A = -E(3)
= (1-Sqrt(-3))/2 = -b3
B = 2*E(3)^2
= -1-Sqrt(-3) = -1-i3
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