Properties

Label 36T13
Order \(36\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3^2$

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$ x 2

Degree 4: $C_2^2$

Degree 6: $S_3$ x 2, $D_{6}$ x 4, $S_3^2$

Degree 9: $S_3^2$

Degree 12: $D_6$ x 2, $S_3^2$

Degree 18: $S_3^2$, $S_3^2$ x 2

Low degree siblings

6T9, 9T8

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 10]
Character table:   
     2  2  2  2  2  1  1  1  .  1
     3  2  .  1  1  1  2  1  2  2

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

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