Properties

Label 36T8
Order \(36\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_3:S_3$

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Group action invariants

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

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
12:  $D_{6}$ x 4
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: $S_3$ x 4, $D_{6}$ x 8

Degree 9: $C_3^2:C_2$

Degree 12: $D_6$ x 4

Degree 18: $C_3^2 : C_2$, 18T12 x 2

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 13]
Character table:   
      2  2  2  2  2  1  1  1  1  1  1  1  1
      3  2  2  .  .  2  2  2  2  2  2  2  2

        1a 2a 2b 2c 3a 6a 6b 3b 6c 3c 3d 6d
     2P 1a 1a 1a 1a 3a 3a 3b 3b 3c 3c 3d 3d
     3P 1a 2a 2b 2c 1a 2a 2a 1a 2a 1a 1a 2a
     5P 1a 2a 2b 2c 3a 6a 6b 3b 6c 3c 3d 6d

X.1      1  1  1  1  1  1  1  1  1  1  1  1
X.2      1 -1 -1  1  1 -1 -1  1 -1  1  1 -1
X.3      1 -1  1 -1  1 -1 -1  1 -1  1  1 -1
X.4      1  1 -1 -1  1  1  1  1  1  1  1  1
X.5      2  2  .  .  2  2 -1 -1 -1 -1 -1 -1
X.6      2 -2  .  .  2 -2  1 -1  1 -1 -1  1
X.7      2  2  .  . -1 -1  2  2 -1 -1 -1 -1
X.8      2 -2  .  . -1  1 -2  2  1 -1 -1  1
X.9      2 -2  .  . -1  1  1 -1 -2  2 -1  1
X.10     2 -2  .  . -1  1  1 -1  1 -1  2 -2
X.11     2  2  .  . -1 -1 -1 -1 -1 -1  2  2
X.12     2  2  .  . -1 -1 -1 -1  2  2 -1 -1