Properties

Label 36T42
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_6^2:C_2$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $42$
Group :  $C_6^2:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,33,2,34)(3,35,4,36)(5,8,6,7)(9,25,10,26)(11,27,12,28)(13,23,14,24)(15,21,16,22)(17,32,18,31)(19,29,20,30), (1,24,20,4,22,18)(2,23,19,3,21,17)(5,26,10,8,28,11)(6,25,9,7,27,12)(13,34,31,16,36,29)(14,33,32,15,35,30), (1,12)(2,11)(3,9)(4,10)(5,18)(6,17)(7,20)(8,19)(13,36)(14,35)(15,34)(16,33)(21,26)(22,25)(23,27)(24,28)(29,30)
$|\Aut(F/K)|$:  $2$

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: $(C_6\times C_2):C_2$ 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 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $18$ $4$ $( 1, 3, 2, 4)( 5,35, 6,36)( 7,33, 8,34)( 9,31,10,32)(11,29,12,30)(13,28,14,27) (15,26,16,25)(17,21,18,22)(19,24,20,23)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1, 5,36, 3, 7,33)( 2, 6,35, 4, 8,34)( 9,14,18,11,16,19)(10,13,17,12,15,20) (21,27,32,24,26,29)(22,28,31,23,25,30)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1, 6,36, 4, 7,34)( 2, 5,35, 3, 8,33)( 9,13,18,12,16,20)(10,14,17,11,15,19) (21,28,32,23,26,30)(22,27,31,24,25,29)$
$ 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 $ $2$ $6$ $( 1, 9,31, 4,12,29)( 2,10,32, 3,11,30)( 5,14,23, 8,15,21)( 6,13,24, 7,16,22) (17,26,33,19,28,35)(18,25,34,20,27,36)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,10,31, 3,12,30)( 2, 9,32, 4,11,29)( 5,13,23, 7,15,22)( 6,14,24, 8,16,21) (17,25,33,20,28,36)(18,26,34,19,27,35)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,11,31, 2,12,32)( 3, 9,30, 4,10,29)( 5,16,23, 6,15,24)( 7,14,22, 8,13,21) (17,27,33,18,28,34)(19,25,35,20,26,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,12,31)( 2,11,32)( 3,10,30)( 4, 9,29)( 5,15,23)( 6,16,24)( 7,13,22) ( 8,14,21)(17,28,33)(18,27,34)(19,26,35)(20,25,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,13,25)( 2,14,26)( 3,15,28)( 4,16,27)( 5,17,30)( 6,18,29)( 7,20,31) ( 8,19,32)( 9,24,34)(10,23,33)(11,21,35)(12,22,36)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,14,25, 2,13,26)( 3,16,28, 4,15,27)( 5,18,30, 6,17,29)( 7,19,31, 8,20,32) ( 9,23,34,10,24,33)(11,22,35,12,21,36)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,15,25, 3,13,28)( 2,16,26, 4,14,27)( 5,20,30, 7,17,31)( 6,19,29, 8,18,32) ( 9,21,34,11,24,35)(10,22,33,12,23,36)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,16,25, 4,13,27)( 2,15,26, 3,14,28)( 5,19,30, 8,17,32)( 6,20,29, 7,18,31) ( 9,22,34,12,24,36)(10,21,33,11,23,35)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,17,22, 3,20,23)( 2,18,21, 4,19,24)( 5,12,28, 7,10,25)( 6,11,27, 8, 9,26) (13,30,36,15,31,33)(14,29,35,16,32,34)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,18,22, 4,20,24)( 2,17,21, 3,19,23)( 5,11,28, 8,10,26)( 6,12,27, 7, 9,25) (13,29,36,16,31,34)(14,30,35,15,32,33)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,19,22, 2,20,21)( 3,18,23, 4,17,24)( 5, 9,28, 6,10,27)( 7,11,25, 8,12,26) (13,32,36,14,31,35)(15,29,33,16,30,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,20,22)( 2,19,21)( 3,17,23)( 4,18,24)( 5,10,28)( 6, 9,27)( 7,12,25) ( 8,11,26)(13,31,36)(14,32,35)(15,30,33)(16,29,34)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 35]
Character table: Data not available.