Properties

Label 36T28
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times D_{12}$

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: $D_{4}$

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

Degree 9: $S_3\times C_3$

Degree 12: $D_{12}$, $D_4 \times C_3$

Degree 18: $S_3 \times C_6$

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 28]
Character table: Data not available.