Properties

Label 36T26
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times C_3:D_4$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $26$
Group :  $C_3\times C_3:D_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,31,11,3,29,10)(2,32,12,4,30,9)(5,22,14,8,24,16)(6,21,13,7,23,15)(17,33,25,20,35,27)(18,34,26,19,36,28), (1,24,15,34,25,10,2,23,16,33,26,9)(3,21,13,35,27,12,4,22,14,36,28,11)(5,30,19,8,31,18,6,29,20,7,32,17)
$|\Aut(F/K)|$:  $6$

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: $(C_6\times C_2):C_2$, $D_4 \times C_3$

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 30]
Character table: Data not available.