Properties

Label 36T44
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times C_3:S_3$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $44$
Group :  $C_2^2\times C_3:S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,33,7,3,36,5)(2,34,8,4,35,6)(9,20,16,12,18,13)(10,19,15,11,17,14)(21,29,26,24,32,27)(22,30,25,23,31,28), (1,22)(2,21)(3,23)(4,24)(5,30)(6,29)(7,31)(8,32)(9,15)(10,16)(11,13)(12,14)(17,18)(19,20)(25,36)(26,35)(27,34)(28,33), (1,16,26,4,14,28)(2,15,25,3,13,27)(5,20,29,8,17,31)(6,19,30,7,18,32)(9,21,34,11,23,36)(10,22,33,12,24,35)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$ x 4

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 12

Degree 9: $C_3^2:C_2$

Degree 12: $S_3 \times C_2^2$ x 4

Degree 18: 18T12 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, 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 $ $9$ $2$ $( 1, 2)( 3, 4)( 5,34)( 6,33)( 7,35)( 8,36)( 9,29)(10,30)(11,31)(12,32)(13,26) (14,25)(15,28)(16,27)(17,23)(18,24)(19,22)(20,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 3)( 2, 4)( 5,36)( 6,35)( 7,33)( 8,34)( 9,31)(10,32)(11,29)(12,30)(13,28) (14,27)(15,26)(16,25)(17,21)(18,22)(19,24)(20,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,23) (22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 4)( 2, 3)( 5,35)( 6,36)( 7,34)( 8,33)( 9,32)(10,31)(11,30)(12,29)(13,27) (14,28)(15,25)(16,26)(17,22)(18,21)(19,23)(20,24)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1, 5,36, 3, 7,33)( 2, 6,35, 4, 8,34)( 9,13,18,12,16,20)(10,14,17,11,15,19) (21,27,32,24,26,29)(22,28,31,23,25,30)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1, 6,36, 4, 7,34)( 2, 5,35, 3, 8,33)( 9,14,18,11,16,19)(10,13,17,12,15,20) (21,28,32,23,26,30)(22,27,31,24,25,29)$
$ 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 $ $2$ $6$ $( 1, 8,36, 2, 7,35)( 3, 6,33, 4, 5,34)( 9,15,18,10,16,17)(11,13,19,12,14,20) (21,25,32,22,26,31)(23,27,30,24,28,29)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1, 9,32, 4,11,30)( 2,10,31, 3,12,29)( 5,13,24, 8,15,22)( 6,14,23, 7,16,21) (17,25,33,20,27,35)(18,26,34,19,28,36)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,10,32, 3,11,29)( 2, 9,31, 4,12,30)( 5,14,24, 7,15,21)( 6,13,23, 8,16,22) (17,26,33,19,27,36)(18,25,34,20,28,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,11,32)( 2,12,31)( 3,10,29)( 4, 9,30)( 5,15,24)( 6,16,23)( 7,14,21) ( 8,13,22)(17,27,33)(18,28,34)(19,26,36)(20,25,35)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,12,32, 2,11,31)( 3, 9,29, 4,10,30)( 5,16,24, 6,15,23)( 7,13,21, 8,14,22) (17,28,33,18,27,34)(19,25,36,20,26,35)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,13,26, 2,14,25)( 3,16,27, 4,15,28)( 5,18,29, 6,17,30)( 7,20,32, 8,19,31) ( 9,24,34,10,23,33)(11,22,36,12,21,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,14,26)( 2,13,25)( 3,15,27)( 4,16,28)( 5,17,29)( 6,18,30)( 7,19,32) ( 8,20,31)( 9,23,34)(10,24,33)(11,21,36)(12,22,35)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,15,26, 3,14,27)( 2,16,25, 4,13,28)( 5,19,29, 7,17,32)( 6,20,30, 8,18,31) ( 9,22,34,12,23,35)(10,21,33,11,24,36)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,16,26, 4,14,28)( 2,15,25, 3,13,27)( 5,20,29, 8,17,31)( 6,19,30, 7,18,32) ( 9,21,34,11,23,36)(10,22,33,12,24,35)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,17,21, 3,19,24)( 2,18,22, 4,20,23)( 5,11,27, 7,10,26)( 6,12,28, 8, 9,25) (13,30,35,16,31,34)(14,29,36,15,32,33)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,18,21, 4,19,23)( 2,17,22, 3,20,24)( 5,12,27, 8,10,25)( 6,11,28, 7, 9,26) (13,29,35,15,31,33)(14,30,36,16,32,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,19,21)( 2,20,22)( 3,17,24)( 4,18,23)( 5,10,27)( 6, 9,28)( 7,11,26) ( 8,12,25)(13,31,35)(14,32,36)(15,29,33)(16,30,34)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,20,21, 2,19,22)( 3,18,24, 4,17,23)( 5, 9,27, 6,10,28)( 7,12,26, 8,11,25) (13,32,35,14,31,36)(15,30,33,16,29,34)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 49]
Character table: Data not available.