Properties

Label 36T2
Order \(36\)
n \(36\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_{18}$

Related objects

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $2$
Group :  $C_2\times C_{18}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)(29,32)(30,31)(33,35)(34,36), (1,22,6,28,11,32,13,36,19,2,21,5,27,12,31,14,35,20)(3,23,8,26,10,30,16,33,17,4,24,7,25,9,29,15,34,18)
$|\Aut(F/K)|$:  $36$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
9:  $C_9$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$

Degree 4: $C_2^2$

Degree 6: $C_6$ x 3

Degree 9: $C_9$

Degree 12: $C_6\times C_2$

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  Yes
Solvable:  Yes
GAP id:  [36, 5]
Character table: Data not available.