Properties

Label 36T41
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4\times C_3:S_3$

Learn more about

Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $41$
Group :  $C_4\times C_3:S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7)(2,8)(3,5)(4,6)(9,23)(10,24)(11,21)(12,22)(13,31)(14,32)(15,29)(16,30)(17,27)(18,28)(19,26)(20,25), (1,19,21)(2,20,22)(3,18,23)(4,17,24)(5,9,28)(6,10,27)(7,11,26)(8,12,25)(13,31,35)(14,32,36)(15,29,34)(16,30,33), (1,3,2,4)(5,35,6,36)(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)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$ x 4
8:  $C_4\times C_2$
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: $C_4$

Degree 6: $D_{6}$ x 4

Degree 9: $C_3^2:C_2$

Degree 12: $S_3 \times C_4$ 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, 1, 1, 1, 1 $ $9$ $2$ $( 5,33)( 6,34)( 7,36)( 8,35)( 9,30)(10,29)(11,32)(12,31)(13,25)(14,26)(15,27) (16,28)(17,24)(18,23)(19,21)(20,22)$
$ 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 $ $9$ $2$ $( 1, 2)( 3, 4)( 5,34)( 6,33)( 7,35)( 8,36)( 9,29)(10,30)(11,31)(12,32)(13,26) (14,25)(15,28)(16,27)(17,23)(18,24)(19,22)(20,21)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,19,18,20)(21,23,22,24) (25,27,26,28)(29,32,30,31)(33,35,34,36)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $9$ $4$ $( 1, 3, 2, 4)( 5,35, 6,36)( 7,33, 8,34)( 9,31,10,32)(11,30,12,29)(13,27,14,28) (15,26,16,25)(17,21,18,22)(19,23,20,24)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,31,30,32)(33,36,34,35)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $9$ $4$ $( 1, 4, 2, 3)( 5,36, 6,35)( 7,34, 8,33)( 9,32,10,31)(11,29,12,30)(13,28,14,27) (15,25,16,26)(17,22,18,21)(19,24,20,23)$
$ 12, 12, 12 $ $2$ $12$ $( 1, 5,35, 4, 7,33, 2, 6,36, 3, 8,34)( 9,13,17,11,16,20,10,14,18,12,15,19) (21,28,31,24,26,30,22,27,32,23,25,29)$
$ 12, 12, 12 $ $2$ $12$ $( 1, 6,35, 3, 7,34, 2, 5,36, 4, 8,33)( 9,14,17,12,16,19,10,13,18,11,15,20) (21,27,31,23,26,29,22,28,32,24,25,30)$
$ 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 $ $2$ $12$ $( 1, 9,31, 4,11,30, 2,10,32, 3,12,29)( 5,13,24, 7,16,22, 6,14,23, 8,15,21) (17,26,33,20,27,36,18,25,34,19,28,35)$
$ 12, 12, 12 $ $2$ $12$ $( 1,10,31, 3,11,29, 2, 9,32, 4,12,30)( 5,14,24, 8,16,21, 6,13,23, 7,15,22) (17,25,33,19,27,35,18,26,34,20,28,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,11,32)( 2,12,31)( 3, 9,30)( 4,10,29)( 5,16,23)( 6,15,24)( 7,14,21) ( 8,13,22)(17,27,34)(18,28,33)(19,26,36)(20,25,35)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,12,32, 2,11,31)( 3,10,30, 4, 9,29)( 5,15,23, 6,16,24)( 7,13,21, 8,14,22) (17,28,34,18,27,33)(19,25,36,20,26,35)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,13,26, 2,14,25)( 3,15,28, 4,16,27)( 5,17,30, 6,18,29)( 7,20,32, 8,19,31) ( 9,24,33,10,23,34)(11,22,36,12,21,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,14,26)( 2,13,25)( 3,16,28)( 4,15,27)( 5,18,30)( 6,17,29)( 7,19,32) ( 8,20,31)( 9,23,33)(10,24,34)(11,21,36)(12,22,35)$
$ 12, 12, 12 $ $2$ $12$ $( 1,15,25, 3,14,27, 2,16,26, 4,13,28)( 5,19,29, 8,18,32, 6,20,30, 7,17,31) ( 9,21,34,12,23,36,10,22,33,11,24,35)$
$ 12, 12, 12 $ $2$ $12$ $( 1,16,25, 4,14,28, 2,15,26, 3,13,27)( 5,20,29, 7,18,31, 6,19,30, 8,17,32) ( 9,22,34,11,23,35,10,21,33,12,24,36)$
$ 12, 12, 12 $ $2$ $12$ $( 1,17,22, 3,19,24, 2,18,21, 4,20,23)( 5,11,27, 8, 9,26, 6,12,28, 7,10,25) (13,30,36,15,31,33,14,29,35,16,32,34)$
$ 12, 12, 12 $ $2$ $12$ $( 1,18,22, 4,19,23, 2,17,21, 3,20,24)( 5,12,27, 7, 9,25, 6,11,28, 8,10,26) (13,29,36,16,31,34,14,30,35,15,32,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,19,21)( 2,20,22)( 3,18,23)( 4,17,24)( 5, 9,28)( 6,10,27)( 7,11,26) ( 8,12,25)(13,31,35)(14,32,36)(15,29,34)(16,30,33)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,20,21, 2,19,22)( 3,17,23, 4,18,24)( 5,10,28, 6, 9,27)( 7,12,26, 8,11,25) (13,32,35,14,31,36)(15,30,34,16,29,33)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 32]
Character table: Data not available.