Group action invariants
| Degree $n$ : | $36$ | |
| Transitive number $t$ : | $8$ | |
| Group : | $C_2\times C_3:S_3$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,4)(2,3)(5,34)(6,33)(7,35)(8,36)(9,31)(10,32)(11,30)(12,29)(13,28)(14,27)(15,25)(16,26)(17,23)(18,24)(19,21)(20,22), (1,11)(2,12)(3,10)(4,9)(5,18)(6,17)(7,20)(8,19)(13,34)(14,33)(15,35)(16,36)(21,25)(22,26)(23,28)(24,27)(29,31)(30,32), (1,34,7)(2,33,8)(3,36,6)(4,35,5)(9,18,15)(10,17,16)(11,20,13)(12,19,14)(21,29,27)(22,30,28)(23,32,26)(24,31,25) | |
| $|\Aut(F/K)|$: | $36$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 6: $S_3$ x 4 12: $D_{6}$ x 4 18: $C_3^2:C_2$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$ x 3
Degree 3: $S_3$ x 4
Degree 4: $C_2^2$
Degree 6: $S_3$ x 4, $D_{6}$ x 8
Degree 9: $C_3^2:C_2$
Degree 12: $D_6$ x 4
Degree 18: $C_3^2 : C_2$, 18T12 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 $ | $9$ | $2$ | $( 1, 3)( 2, 4)( 5,33)( 6,34)( 7,36)( 8,35)( 9,32)(10,31)(11,29)(12,30)(13,27) (14,28)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 4)( 2, 3)( 5,34)( 6,33)( 7,35)( 8,36)( 9,31)(10,32)(11,30)(12,29)(13,28) (14,27)(15,25)(16,26)(17,23)(18,24)(19,21)(20,22)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 7,34)( 2, 8,33)( 3, 6,36)( 4, 5,35)( 9,15,18)(10,16,17)(11,13,20) (12,14,19)(21,27,29)(22,28,30)(23,26,32)(24,25,31)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 8,34, 2, 7,33)( 3, 5,36, 4, 6,35)( 9,16,18,10,15,17)(11,14,20,12,13,19) (21,28,29,22,27,30)(23,25,32,24,26,31)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 9,29, 2,10,30)( 3,12,31, 4,11,32)( 5,13,23, 6,14,24)( 7,15,21, 8,16,22) (17,28,34,18,27,33)(19,25,35,20,26,36)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,10,29)( 2, 9,30)( 3,11,31)( 4,12,32)( 5,14,23)( 6,13,24)( 7,16,21) ( 8,15,22)(17,27,34)(18,28,33)(19,26,35)(20,25,36)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,15,27, 2,16,28)( 3,14,25, 4,13,26)( 5,20,32, 6,19,31)( 7,18,29, 8,17,30) ( 9,21,33,10,22,34)(11,23,36,12,24,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,27)( 2,15,28)( 3,13,25)( 4,14,26)( 5,19,32)( 6,20,31)( 7,17,29) ( 8,18,30)( 9,22,33)(10,21,34)(11,24,36)(12,23,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,17,21)( 2,18,22)( 3,20,24)( 4,19,23)( 5,12,26)( 6,11,25)( 7,10,27) ( 8, 9,28)(13,31,36)(14,32,35)(15,30,33)(16,29,34)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,18,21, 2,17,22)( 3,19,24, 4,20,23)( 5,11,26, 6,12,25)( 7, 9,27, 8,10,28) (13,32,36,14,31,35)(15,29,33,16,30,34)$ |
Group invariants
| Order: | $36=2^{2} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [36, 13] |
| Character table: |
2 2 2 2 2 1 1 1 1 1 1 1 1
3 2 2 . . 2 2 2 2 2 2 2 2
1a 2a 2b 2c 3a 6a 6b 3b 6c 3c 3d 6d
2P 1a 1a 1a 1a 3a 3a 3b 3b 3c 3c 3d 3d
3P 1a 2a 2b 2c 1a 2a 2a 1a 2a 1a 1a 2a
5P 1a 2a 2b 2c 3a 6a 6b 3b 6c 3c 3d 6d
X.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
X.3 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
X.5 2 2 . . 2 2 -1 -1 -1 -1 -1 -1
X.6 2 -2 . . 2 -2 1 -1 1 -1 -1 1
X.7 2 2 . . -1 -1 2 2 -1 -1 -1 -1
X.8 2 -2 . . -1 1 -2 2 1 -1 -1 1
X.9 2 -2 . . -1 1 1 -1 -2 2 -1 1
X.10 2 -2 . . -1 1 1 -1 1 -1 2 -2
X.11 2 2 . . -1 -1 -1 -1 -1 -1 2 2
X.12 2 2 . . -1 -1 -1 -1 2 2 -1 -1
|