# Properties

 Label 36T20 Order $$72$$ n $$36$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_3\times S_4$

## Group action invariants

 Degree $n$ : $36$ Transitive number $t$ : $20$ Group : $C_3\times S_4$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,12,28,36,15,23,2,11,27,35,16,24)(3,10,25,34,14,22,4,9,26,33,13,21)(5,17,31,7,20,30)(6,18,32,8,19,29), (1,4)(2,3)(5,34)(6,33)(7,36)(8,35)(9,20)(10,19)(11,17)(12,18)(13,15)(14,16)(21,31)(22,32)(23,29)(24,30)(25,27)(26,28) $|\Aut(F/K)|$: $12$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
24:  $S_4$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: None

Degree 6: $C_6$, $S_3$, $S_3\times C_3$, $S_4$, $S_4$

Degree 9: $S_3\times C_3$

Degree 12: $S_4$

Degree 18: $S_3 \times C_3$, 18T30, 18T33

## 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(17,18)(19,20)(21,22)(23,24)(29,30)(31,32)(33,34) (35,36)$ $4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2$ $6$ $4$ $( 1, 3)( 2, 4)( 5,33, 6,34)( 7,35, 8,36)( 9,20,10,19)(11,17,12,18)(13,16) (14,15)(21,31,22,32)(23,29,24,30)(25,28)(26,27)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $6$ $2$ $( 1, 4)( 2, 3)( 5,33)( 6,34)( 7,35)( 8,36)( 9,19)(10,20)(11,18)(12,17)(13,15) (14,16)(21,32)(22,31)(23,30)(24,29)(25,27)(26,28)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $8$ $3$ $( 1, 7,33)( 2, 8,34)( 3, 5,35)( 4, 6,36)( 9,16,18)(10,15,17)(11,13,19) (12,14,20)(21,28,29)(22,27,30)(23,26,31)(24,25,32)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $8$ $3$ $( 1, 9,29)( 2,10,30)( 3,11,32)( 4,12,31)( 5,13,23)( 6,14,24)( 7,16,22) ( 8,15,21)(17,28,33)(18,27,34)(19,26,36)(20,25,35)$ $12, 12, 6, 6$ $6$ $12$ $( 1,11,28,35,15,24, 2,12,27,36,16,23)( 3, 9,25,33,14,21, 4,10,26,34,13,22) ( 5,17,31, 7,20,30)( 6,18,32, 8,19,29)$ $6, 6, 6, 6, 6, 6$ $6$ $6$ $( 1,11,27,36,15,24)( 2,12,28,35,16,23)( 3, 9,26,34,14,21)( 4,10,25,33,13,22) ( 5,18,31, 8,20,29)( 6,17,32, 7,19,30)$ $6, 6, 6, 6, 3, 3, 3, 3$ $3$ $6$ $( 1,15,27)( 2,16,28)( 3,14,26)( 4,13,25)( 5,19,31, 6,20,32)( 7,18,30, 8,17,29) ( 9,22,34,10,21,33)(11,23,36,12,24,35)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1,15,27)( 2,16,28)( 3,14,26)( 4,13,25)( 5,20,31)( 6,19,32)( 7,17,30) ( 8,18,29)( 9,21,34)(10,22,33)(11,24,36)(12,23,35)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $8$ $3$ $( 1,21,18)( 2,22,17)( 3,24,19)( 4,23,20)( 5,25,12)( 6,26,11)( 7,28,10) ( 8,27, 9)(13,35,31)(14,36,32)(15,34,29)(16,33,30)$ $6, 6, 6, 6, 6, 6$ $6$ $6$ $( 1,23,15,35,27,12)( 2,24,16,36,28,11)( 3,22,14,33,26,10)( 4,21,13,34,25, 9) ( 5,29,20, 8,31,18)( 6,30,19, 7,32,17)$ $12, 12, 6, 6$ $6$ $12$ $( 1,23,16,36,27,12, 2,24,15,35,28,11)( 3,22,13,34,26,10, 4,21,14,33,25, 9) ( 5,30,20, 7,31,17)( 6,29,19, 8,32,18)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1,27,15)( 2,28,16)( 3,26,14)( 4,25,13)( 5,31,20)( 6,32,19)( 7,30,17) ( 8,29,18)( 9,34,21)(10,33,22)(11,36,24)(12,35,23)$ $6, 6, 6, 6, 3, 3, 3, 3$ $3$ $6$ $( 1,27,15)( 2,28,16)( 3,26,14)( 4,25,13)( 5,32,20, 6,31,19)( 7,29,17, 8,30,18) ( 9,33,21,10,34,22)(11,35,24,12,36,23)$

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [72, 42]
 Character table:  2 3 3 2 2 . . 2 2 3 3 . 2 2 3 3 3 2 1 1 1 2 2 1 1 1 2 2 1 1 2 1 1a 2a 4a 2b 3a 3b 12a 6a 6b 3c 3d 6c 12b 3e 6d 2P 1a 1a 2a 1a 3a 3d 6d 3e 3e 3e 3b 3c 6b 3c 3c 3P 1a 2a 4a 2b 1a 1a 4a 2b 2a 1a 1a 2b 4a 1a 2a 5P 1a 2a 4a 2b 3a 3d 12b 6c 6d 3e 3b 6a 12a 3c 6b 7P 1a 2a 4a 2b 3a 3b 12a 6a 6b 3c 3d 6c 12b 3e 6d 11P 1a 2a 4a 2b 3a 3d 12b 6c 6d 3e 3b 6a 12a 3c 6b X.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 X.3 1 1 -1 -1 1 A -A -A A A /A -/A -/A /A /A X.4 1 1 -1 -1 1 /A -/A -/A /A /A A -A -A A A X.5 1 1 1 1 1 A A A A A /A /A /A /A /A X.6 1 1 1 1 1 /A /A /A /A /A A A A A A X.7 2 2 . . -1 -1 . . 2 2 -1 . . 2 2 X.8 2 2 . . -1 -A . . B B -/A . . /B /B X.9 2 2 . . -1 -/A . . /B /B -A . . B B X.10 3 -1 -1 1 . . -1 1 -1 3 . 1 -1 3 -1 X.11 3 -1 1 -1 . . 1 -1 -1 3 . -1 1 3 -1 X.12 3 -1 -1 1 . . -/A /A -/A C . A -A /C -A X.13 3 -1 -1 1 . . -A A -A /C . /A -/A C -/A X.14 3 -1 1 -1 . . A -A -A /C . -/A /A C -/A X.15 3 -1 1 -1 . . /A -/A -/A C . -A A /C -A A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 C = 3*E(3) = (-3+3*Sqrt(-3))/2 = 3b3