Properties

Label 36T29
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_6\times S_3$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $29$
Group :  $C_2\times C_6\times S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,29,15,7,25,17)(2,30,16,8,26,18)(3,31,13,6,28,19)(4,32,14,5,27,20)(9,33,24)(10,34,23)(11,36,22)(12,35,21), (1,5)(2,6)(3,8)(4,7)(9,12)(10,11)(13,18)(14,17)(15,20)(16,19)(21,24)(22,23)(25,32)(26,31)(27,29)(28,30)(33,35)(34,36), (1,16,25,2,15,26)(3,14,28,4,13,27)(5,10,32,34,20,23)(6,9,31,33,19,24)(7,11,29,36,17,22)(8,12,30,35,18,21)
$|\Aut(F/K)|$:  $12$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
3:  $C_3$
4:  $C_2^2$ x 7
6:  $S_3$, $C_6$ x 7
8:  $C_2^3$
12:  $D_{6}$ x 3
18:  $S_3\times C_3$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$, $S_3$

Degree 4: $C_2^2$

Degree 6: $C_6$ x 3, $D_{6}$ x 3

Degree 9: $S_3\times C_3$

Degree 12: $C_6\times C_2$, $S_3 \times C_2^2$

Degree 18: $S_3 \times C_6$ x 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 TypeSizeOrderRepresentative
$ 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 $ $3$ $2$ $( 5,33)( 6,34)( 7,35)( 8,36)( 9,20)(10,19)(11,18)(12,17)(21,29)(22,30)(23,31) (24,32)$
$ 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 $ $3$ $2$ $( 1, 2)( 3, 4)( 5,34)( 6,33)( 7,36)( 8,35)( 9,19)(10,20)(11,17)(12,18)(13,14) (15,16)(21,30)(22,29)(23,32)(24,31)(25,26)(27,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 3)( 2, 4)( 5,36)( 6,35)( 7,34)( 8,33)( 9,18)(10,17)(11,20)(12,19)(13,15) (14,16)(21,31)(22,32)(23,29)(24,30)(25,28)(26,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,27)(26,28)(29,32)(30,31)(33,35)(34,36)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2, 3)( 5,35)( 6,36)( 7,33)( 8,34)( 9,17)(10,18)(11,19)(12,20)(13,16) (14,15)(21,32)(22,31)(23,30)(24,29)(25,27)(26,28)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1, 5,35, 4, 7,33)( 2, 6,36, 3, 8,34)( 9,15,20,12,14,17)(10,16,19,11,13,18) (21,27,29,24,25,32)(22,28,30,23,26,31)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1, 6,35, 3, 7,34)( 2, 5,36, 4, 8,33)( 9,16,20,11,14,18)(10,15,19,12,13,17) (21,28,29,23,25,31)(22,27,30,24,26,32)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 7,35)( 2, 8,36)( 3, 6,34)( 4, 5,33)( 9,14,20)(10,13,19)(11,16,18) (12,15,17)(21,25,29)(22,26,30)(23,28,31)(24,27,32)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1, 8,35, 2, 7,36)( 3, 5,34, 4, 6,33)( 9,13,20,10,14,19)(11,15,18,12,16,17) (21,26,29,22,25,30)(23,27,31,24,28,32)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1, 9,29, 4,12,32)( 2,10,30, 3,11,31)( 5,15,24, 7,14,21)( 6,16,23, 8,13,22) (17,27,35,20,25,33)(18,28,36,19,26,34)$
$ 6, 6, 6, 6, 6, 6 $ $3$ $6$ $( 1, 9,25,33,15,24)( 2,10,26,34,16,23)( 3,11,28,36,13,22)( 4,12,27,35,14,21) ( 5,17,32, 7,20,29)( 6,18,31, 8,19,30)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,10,29, 3,12,31)( 2, 9,30, 4,11,32)( 5,16,24, 8,14,22)( 6,15,23, 7,13,21) (17,28,35,19,25,34)(18,27,36,20,26,33)$
$ 6, 6, 6, 6, 6, 6 $ $3$ $6$ $( 1,10,25,34,15,23)( 2, 9,26,33,16,24)( 3,12,28,35,13,21)( 4,11,27,36,14,22) ( 5,18,32, 8,20,30)( 6,17,31, 7,19,29)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,11,29, 2,12,30)( 3, 9,31, 4,10,32)( 5,13,24, 6,14,23)( 7,16,21, 8,15,22) (17,26,35,18,25,36)(19,27,34,20,28,33)$
$ 6, 6, 6, 6, 6, 6 $ $3$ $6$ $( 1,11,25,36,15,22)( 2,12,26,35,16,21)( 3, 9,28,33,13,24)( 4,10,27,34,14,23) ( 5,19,32, 6,20,31)( 7,18,29, 8,17,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,12,29)( 2,11,30)( 3,10,31)( 4, 9,32)( 5,14,24)( 6,13,23)( 7,15,21) ( 8,16,22)(17,25,35)(18,26,36)(19,28,34)(20,27,33)$
$ 6, 6, 6, 6, 3, 3, 3, 3 $ $3$ $6$ $( 1,12,25,35,15,21)( 2,11,26,36,16,22)( 3,10,28,34,13,23)( 4, 9,27,33,14,24) ( 5,20,32)( 6,19,31)( 7,17,29)( 8,18,30)$
$ 6, 6, 6, 6, 6, 6 $ $1$ $6$ $( 1,13,25, 3,15,28)( 2,14,26, 4,16,27)( 5,18,32, 8,20,30)( 6,17,31, 7,19,29) ( 9,22,33,11,24,36)(10,21,34,12,23,35)$
$ 6, 6, 6, 6, 6, 6 $ $1$ $6$ $( 1,14,25, 4,15,27)( 2,13,26, 3,16,28)( 5,17,32, 7,20,29)( 6,18,31, 8,19,30) ( 9,21,33,12,24,35)(10,22,34,11,23,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,15,25)( 2,16,26)( 3,13,28)( 4,14,27)( 5,20,32)( 6,19,31)( 7,17,29) ( 8,18,30)( 9,24,33)(10,23,34)(11,22,36)(12,21,35)$
$ 6, 6, 6, 6, 6, 6 $ $1$ $6$ $( 1,16,25, 2,15,26)( 3,14,28, 4,13,27)( 5,19,32, 6,20,31)( 7,18,29, 8,17,30) ( 9,23,33,10,24,34)(11,21,36,12,22,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,21,17)( 2,22,18)( 3,23,19)( 4,24,20)( 5,27, 9)( 6,28,10)( 7,25,12) ( 8,26,11)(13,34,31)(14,33,32)(15,35,29)(16,36,30)$
$ 6, 6, 6, 6, 3, 3, 3, 3 $ $3$ $6$ $( 1,21,15,35,25,12)( 2,22,16,36,26,11)( 3,23,13,34,28,10)( 4,24,14,33,27, 9) ( 5,32,20)( 6,31,19)( 7,29,17)( 8,30,18)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,22,17, 2,21,18)( 3,24,19, 4,23,20)( 5,28, 9, 6,27,10)( 7,26,12, 8,25,11) (13,33,31,14,34,32)(15,36,29,16,35,30)$
$ 6, 6, 6, 6, 6, 6 $ $3$ $6$ $( 1,22,15,36,25,11)( 2,21,16,35,26,12)( 3,24,13,33,28, 9)( 4,23,14,34,27,10) ( 5,31,20, 6,32,19)( 7,30,17, 8,29,18)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,23,17, 3,21,19)( 2,24,18, 4,22,20)( 5,26, 9, 8,27,11)( 6,25,10, 7,28,12) (13,35,31,15,34,29)(14,36,32,16,33,30)$
$ 6, 6, 6, 6, 6, 6 $ $3$ $6$ $( 1,23,15,34,25,10)( 2,24,16,33,26, 9)( 3,21,13,35,28,12)( 4,22,14,36,27,11) ( 5,30,20, 8,32,18)( 6,29,19, 7,31,17)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,24,17, 4,21,20)( 2,23,18, 3,22,19)( 5,25, 9, 7,27,12)( 6,26,10, 8,28,11) (13,36,31,16,34,30)(14,35,32,15,33,29)$
$ 6, 6, 6, 6, 6, 6 $ $3$ $6$ $( 1,24,15,33,25, 9)( 2,23,16,34,26,10)( 3,22,13,36,28,11)( 4,21,14,35,27,12) ( 5,29,20, 7,32,17)( 6,30,19, 8,31,18)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,25,15)( 2,26,16)( 3,28,13)( 4,27,14)( 5,32,20)( 6,31,19)( 7,29,17) ( 8,30,18)( 9,33,24)(10,34,23)(11,36,22)(12,35,21)$
$ 6, 6, 6, 6, 6, 6 $ $1$ $6$ $( 1,26,15, 2,25,16)( 3,27,13, 4,28,14)( 5,31,20, 6,32,19)( 7,30,17, 8,29,18) ( 9,34,24,10,33,23)(11,35,22,12,36,21)$
$ 6, 6, 6, 6, 6, 6 $ $1$ $6$ $( 1,27,15, 4,25,14)( 2,28,16, 3,26,13)( 5,29,20, 7,32,17)( 6,30,19, 8,31,18) ( 9,35,24,12,33,21)(10,36,23,11,34,22)$
$ 6, 6, 6, 6, 6, 6 $ $1$ $6$ $( 1,28,15, 3,25,13)( 2,27,16, 4,26,14)( 5,30,20, 8,32,18)( 6,29,19, 7,31,17) ( 9,36,24,11,33,22)(10,35,23,12,34,21)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 48]
Character table: Data not available.