Group action invariants
| Degree $n$ : | $36$ | |
| Transitive number $t$ : | $50$ | |
| Group : | $S_3\times A_4$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,10)(2,9)(3,12)(4,11)(13,18)(14,17)(15,20)(16,19)(21,24)(22,23)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35), (1,24,29,7,18,36)(2,23,30,8,17,35)(3,21,31,6,19,34)(4,22,32,5,20,33)(9,16,26,10,15,25)(11,13,28,12,14,27) | |
| $|\Aut(F/K)|$: | $4$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $S_3$, $C_6$ 12: $A_4$ 18: $S_3\times C_3$ 24: $A_4\times C_2$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: None
Degree 4: None
Degree 6: $A_4$, $A_4\times C_2$
Degree 9: $S_3\times C_3$
Degree 12: $A_4\times C_2$
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, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5,12)( 6,11)( 7, 9)( 8,10)(13,23)(14,24)(15,21)(16,22)(17,20)(18,19)(25,34) (26,33)(27,36)(28,35)(29,32)(30,31)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5,11)( 6,12)( 7,10)( 8, 9)(13,21)(14,22)(15,23)(16,24)(17,18) (19,20)(25,36)(26,35)(27,34)(28,33)(29,30)(31,32)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $6$ | $6$ | $( 1, 5,10, 3, 8,12)( 2, 6, 9, 4, 7,11)(13,20,22,14,19,21)(15,18,24,16,17,23) (25,32,35,28,29,34)(26,31,36,27,30,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 8,10)( 2, 7, 9)( 3, 5,12)( 4, 6,11)(13,19,22)(14,20,21)(15,17,24) (16,18,23)(25,29,35)(26,30,36)(27,31,33)(28,32,34)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1,13,34)( 2,14,33)( 3,16,36)( 4,15,35)( 5,18,26)( 6,17,25)( 7,20,27) ( 8,19,28)( 9,21,31)(10,22,32)(11,24,29)(12,23,30)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $12$ | $6$ | $( 1,13,30, 9,20,25)( 2,14,29,10,19,26)( 3,16,32,11,17,27)( 4,15,31,12,18,28) ( 5,23,34, 6,24,33)( 7,21,35, 8,22,36)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,17,31)( 2,18,32)( 3,20,29)( 4,19,30)( 5,21,35)( 6,22,36)( 7,23,34) ( 8,24,33)( 9,16,28)(10,15,27)(11,13,26)(12,14,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1,25,23)( 2,26,24)( 3,27,22)( 4,28,21)( 5,31,13)( 6,32,14)( 7,30,15) ( 8,29,16)( 9,36,17)(10,35,18)(11,34,20)(12,33,19)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $12$ | $6$ | $( 1,25,20, 9,30,13)( 2,26,19,10,29,14)( 3,27,17,11,32,16)( 4,28,18,12,31,15) ( 5,33,24, 6,34,23)( 7,36,22, 8,35,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,29,18)( 2,30,17)( 3,31,19)( 4,32,20)( 5,33,22)( 6,34,21)( 7,36,24) ( 8,35,23)( 9,26,15)(10,25,16)(11,28,14)(12,27,13)$ |
Group invariants
| Order: | $72=2^{3} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [72, 44] |
| Character table: |
2 3 3 3 3 2 2 . 1 1 . 1 1
3 2 . 1 1 1 2 2 1 2 2 1 2
1a 2a 2b 2c 6a 3a 3b 6b 3c 3d 6c 3e
2P 1a 1a 1a 1a 3a 3a 3d 3e 3e 3b 3c 3c
3P 1a 2a 2b 2c 2b 1a 1a 2c 1a 1a 2c 1a
5P 1a 2a 2b 2c 6a 3a 3d 6c 3e 3b 6b 3c
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 A -A A /A -/A /A
X.4 1 -1 1 -1 1 1 /A -/A /A A -A A
X.5 1 1 1 1 1 1 A A A /A /A /A
X.6 1 1 1 1 1 1 /A /A /A A A A
X.7 2 . 2 . -1 -1 -1 . 2 -1 . 2
X.8 2 . 2 . -1 -1 -A . B -/A . /B
X.9 2 . 2 . -1 -1 -/A . /B -A . B
X.10 3 -1 -1 3 -1 3 . . . . . .
X.11 3 1 -1 -3 -1 3 . . . . . .
X.12 6 . -2 . 1 -3 . . . . . .
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)^2
= -1-Sqrt(-3) = -1-i3
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