Properties

Label 36T43
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:D_{12}$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $43$
Group :  $C_3:D_{12}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,35,3,7,34,2,5,36,4,8,33)(9,13,17,11,16,20,10,14,18,12,15,19)(21,27,31,23,26,29,22,28,32,24,25,30), (1,25,13)(2,26,14)(3,27,16)(4,28,15)(5,29,18)(6,30,17)(7,31,20)(8,32,19)(9,33,24)(10,34,23)(11,35,21)(12,36,22), (1,14)(2,13)(3,16)(4,15)(5,9)(6,10)(7,11)(8,12)(17,34)(18,33)(19,36)(20,35)(21,31)(22,32)(23,30)(24,29)(25,26)
$|\Aut(F/K)|$:  $2$

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
8:  $D_{4}$
12:  $D_{6}$ x 4
18:  $C_3^2:C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 4: $D_{4}$

Degree 6: $D_{6}$ x 4

Degree 9: $C_3^2:C_2$

Degree 12: $D_{12}$ x 4

Degree 18: 18T12

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 33]
Character table: Data not available.