Group action invariants
| Degree $n$ : | $36$ | |
| Transitive number $t$ : | $35$ | |
| Group : | $C_2\times C_3:S_3.C_2$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,13,28,2,14,27)(3,15,25,4,16,26)(5,20,32,6,19,31)(7,17,30,8,18,29)(9,21,36,10,22,35)(11,24,33,12,23,34), (1,25,33,9)(2,26,34,10)(3,27,36,12)(4,28,35,11)(5,7,6,8)(13,17,24,30)(14,18,23,29)(15,19,21,31)(16,20,22,32) | |
| $|\Aut(F/K)|$: | $4$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_4$ x 2, $C_2^2$ 8: $C_4\times C_2$ 36: $C_3^2:C_4$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $C_4$
Degree 6: $C_3^2:C_4$ x 2
Degree 9: $C_3^2:C_4$
Degree 12: 12T41 x 2
Degree 18: $C_3^2 : C_4$
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 $ | $9$ | $2$ | $( 5,34)( 6,33)( 7,35)( 8,36)( 9,29)(10,30)(11,31)(12,32)(13,27)(14,28)(15,26) (16,25)(17,22)(18,21)(19,24)(20,23)$ |
| $ 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 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5,33)( 6,34)( 7,36)( 8,35)( 9,30)(10,29)(11,32)(12,31)(13,28) (14,27)(15,25)(16,26)(17,21)(18,22)(19,23)(20,24)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $9$ | $4$ | $( 1, 3, 2, 4)( 5,18,33,22)( 6,17,34,21)( 7,20,36,24)( 8,19,35,23)( 9,14,30,27) (10,13,29,28)(11,15,32,25)(12,16,31,26)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $9$ | $4$ | $( 1, 3, 2, 4)( 5,21,33,17)( 6,22,34,18)( 7,23,36,19)( 8,24,35,20)( 9,28,30,13) (10,27,29,14)(11,26,32,16)(12,25,31,15)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $9$ | $4$ | $( 1, 4, 2, 3)( 5,17,33,21)( 6,18,34,22)( 7,19,36,23)( 8,20,35,24)( 9,13,30,28) (10,14,29,27)(11,16,32,26)(12,15,31,25)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $9$ | $4$ | $( 1, 4, 2, 3)( 5,22,33,18)( 6,21,34,17)( 7,24,36,20)( 8,23,35,19)( 9,27,30,14) (10,28,29,13)(11,25,32,15)(12,26,31,16)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5,34)( 2, 6,33)( 3, 7,35)( 4, 8,36)( 9,15,17)(10,16,18)(11,13,20) (12,14,19)(21,25,30)(22,26,29)(23,27,31)(24,28,32)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $4$ | $6$ | $( 1, 6,34, 2, 5,33)( 3, 8,35, 4, 7,36)( 9,16,17,10,15,18)(11,14,20,12,13,19) (21,26,30,22,25,29)(23,28,31,24,27,32)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $4$ | $6$ | $( 1,11,32, 2,12,31)( 3, 9,30, 4,10,29)( 5,13,24, 6,14,23)( 7,15,21, 8,16,22) (17,25,36,18,26,35)(19,27,34,20,28,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,12,32)( 2,11,31)( 3,10,30)( 4, 9,29)( 5,14,24)( 6,13,23)( 7,16,21) ( 8,15,22)(17,26,36)(18,25,35)(19,28,34)(20,27,33)$ |
Group invariants
| Order: | $72=2^{3} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [72, 45] |
| Character table: |
2 3 3 3 3 3 3 3 3 1 1 1 1
3 2 . 2 . . . . . 2 2 2 2
1a 2a 2b 2c 4a 4b 4c 4d 3a 6a 6b 3b
2P 1a 1a 1a 1a 2c 2c 2c 2c 3a 3a 3b 3b
3P 1a 2a 2b 2c 4d 4c 4b 4a 1a 2b 2b 1a
5P 1a 2a 2b 2c 4a 4b 4c 4d 3a 6a 6b 3b
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 1 -1 1 -1 A -A A -A 1 1 1 1
X.6 1 -1 1 -1 -A A -A A 1 1 1 1
X.7 1 1 -1 -1 A A -A -A 1 -1 -1 1
X.8 1 1 -1 -1 -A -A A A 1 -1 -1 1
X.9 4 . -4 . . . . . -2 2 -1 1
X.10 4 . -4 . . . . . 1 -1 2 -2
X.11 4 . 4 . . . . . -2 -2 1 1
X.12 4 . 4 . . . . . 1 1 -2 -2
A = -E(4)
= -Sqrt(-1) = -i
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