Group action invariants
| Degree $n$ : | $36$ | |
| Transitive number $t$ : | $5$ | |
| Group : | $C_3\times C_3:C_4$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,19,27,6,16,31,2,20,28,5,15,32)(3,17,26,7,13,29,4,18,25,8,14,30)(9,24,33,12,22,35,10,23,34,11,21,36), (1,17,22)(2,18,21)(3,20,23)(4,19,24)(5,11,26)(6,12,25)(7,10,27)(8,9,28)(13,32,36)(14,31,35)(15,30,33)(16,29,34) | |
| $|\Aut(F/K)|$: | $36$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 4: $C_4$ 6: $S_3$, $C_6$ 18: $S_3\times C_3$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 6: $C_6$, $S_3$, $S_3\times C_3$
Degree 9: $S_3\times C_3$
Degree 12: $C_{12}$, $C_3 : C_4$, $C_3\times (C_3 : C_4)$
Degree 18: $S_3 \times C_3$
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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 3, 2, 4)( 5,34, 6,33)( 7,35, 8,36)( 9,20,10,19)(11,17,12,18)(13,15,14,16) (21,31,22,32)(23,30,24,29)(25,27,26,28)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 4, 2, 3)( 5,33, 6,34)( 7,36, 8,35)( 9,19,10,20)(11,18,12,17)(13,16,14,15) (21,32,22,31)(23,29,24,30)(25,28,26,27)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 7,34, 2, 8,33)( 3, 5,36, 4, 6,35)( 9,15,17,10,16,18)(11,13,19,12,14,20) (21,28,30,22,27,29)(23,26,32,24,25,31)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 8,34)( 2, 7,33)( 3, 6,36)( 4, 5,35)( 9,16,17)(10,15,18)(11,14,19) (12,13,20)(21,27,30)(22,28,29)(23,25,32)(24,26,31)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 9,29)( 2,10,30)( 3,12,32)( 4,11,31)( 5,14,24)( 6,13,23)( 7,15,21) ( 8,16,22)(17,28,34)(18,27,33)(19,26,35)(20,25,36)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,10,29, 2, 9,30)( 3,11,32, 4,12,31)( 5,13,24, 6,14,23)( 7,16,21, 8,15,22) (17,27,34,18,28,33)(19,25,35,20,26,36)$ |
| $ 12, 12, 12 $ | $3$ | $12$ | $( 1,11,27,36,16,24, 2,12,28,35,15,23)( 3, 9,26,33,13,22, 4,10,25,34,14,21) ( 5,18,32, 8,19,30, 6,17,31, 7,20,29)$ |
| $ 12, 12, 12 $ | $3$ | $12$ | $( 1,12,27,35,16,23, 2,11,28,36,15,24)( 3,10,26,34,13,21, 4, 9,25,33,14,22) ( 5,17,32, 7,19,29, 6,18,31, 8,20,30)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,15,28, 2,16,27)( 3,14,25, 4,13,26)( 5,20,31, 6,19,32)( 7,17,30, 8,18,29) ( 9,21,34,10,22,33)(11,23,35,12,24,36)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,16,28)( 2,15,27)( 3,13,25)( 4,14,26)( 5,19,31)( 6,20,32)( 7,18,30) ( 8,17,29)( 9,22,34)(10,21,33)(11,24,35)(12,23,36)$ |
| $ 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 $ | $2$ | $3$ | $( 1,22,17)( 2,21,18)( 3,23,20)( 4,24,19)( 5,26,11)( 6,25,12)( 7,27,10) ( 8,28, 9)(13,36,32)(14,35,31)(15,33,30)(16,34,29)$ |
| $ 12, 12, 12 $ | $3$ | $12$ | $( 1,23,15,35,28,12, 2,24,16,36,27,11)( 3,21,14,34,25,10, 4,22,13,33,26, 9) ( 5,29,20, 7,31,17, 6,30,19, 8,32,18)$ |
| $ 12, 12, 12 $ | $3$ | $12$ | $( 1,24,15,36,28,11, 2,23,16,35,27,12)( 3,22,14,33,25, 9, 4,21,13,34,26,10) ( 5,30,20, 8,31,18, 6,29,19, 7,32,17)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $1$ | $6$ | $( 1,27,16, 2,28,15)( 3,26,13, 4,25,14)( 5,32,19, 6,31,20)( 7,29,18, 8,30,17) ( 9,33,22,10,34,21)(11,36,24,12,35,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,28,16)( 2,27,15)( 3,25,13)( 4,26,14)( 5,31,19)( 6,32,20)( 7,30,18) ( 8,29,17)( 9,34,22)(10,33,21)(11,35,24)(12,36,23)$ |
Group invariants
| Order: | $36=2^{2} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [36, 6] |
| 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 4a 4b 6a 3a 3b 6b 12a 12b 6c 3c 6d 3d 12c 12d 6e 3e
2P 1a 1a 2a 2a 3a 3a 3d 3d 6e 6e 3e 3e 3b 3b 6c 6c 3c 3c
3P 1a 2a 4b 4a 2a 1a 1a 2a 4a 4b 2a 1a 2a 1a 4b 4a 2a 1a
5P 1a 2a 4a 4b 6a 3a 3d 6d 12d 12c 6e 3e 6b 3b 12b 12a 6c 3c
7P 1a 2a 4b 4a 6a 3a 3b 6b 12b 12a 6c 3c 6d 3d 12d 12c 6e 3e
11P 1a 2a 4b 4a 6a 3a 3d 6d 12c 12d 6e 3e 6b 3b 12a 12b 6c 3c
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 A -A -1 1 1 -1 -A A -1 1 -1 1 A -A -1 1
X.4 1 -1 -A A -1 1 1 -1 A -A -1 1 -1 1 -A A -1 1
X.5 1 -1 A -A -1 1 B -B C -C -B B -/B /B /C -/C -/B /B
X.6 1 -1 A -A -1 1 /B -/B -/C /C -/B /B -B B -C C -B B
X.7 1 -1 -A A -1 1 B -B -C C -B B -/B /B -/C /C -/B /B
X.8 1 -1 -A A -1 1 /B -/B /C -/C -/B /B -B B C -C -B B
X.9 1 1 -1 -1 1 1 B B -B -B B B /B /B -/B -/B /B /B
X.10 1 1 -1 -1 1 1 /B /B -/B -/B /B /B B B -B -B B B
X.11 1 1 1 1 1 1 B B B B B B /B /B /B /B /B /B
X.12 1 1 1 1 1 1 /B /B /B /B /B /B B B B B B B
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 -B B . . D -D /B -/B . . /D -/D
X.16 2 -2 . . 1 -1 -/B /B . . /D -/D B -B . . D -D
X.17 2 2 . . -1 -1 -B -B . . -D -D -/B -/B . . -/D -/D
X.18 2 2 . . -1 -1 -/B -/B . . -/D -/D -B -B . . -D -D
A = -E(4)
= -Sqrt(-1) = -i
B = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
C = E(12)^11
D = -2*E(3)^2
= 1+Sqrt(-3) = 1+i3
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