Properties

Label 36T23
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:S_4$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $23$
Group :  $C_3:S_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,30,10)(2,29,9)(3,31,12)(4,32,11)(5,23,14)(6,24,13)(7,21,15)(8,22,16)(17,33,28)(18,34,27)(19,35,25)(20,36,26), (1,21,18)(2,22,17)(3,24,19)(4,23,20)(5,26,12)(6,25,11)(7,28,9)(8,27,10)(13,36,32)(14,35,31)(15,33,30)(16,34,29), (1,3)(2,4)(5,34)(6,33)(7,35)(8,36)(9,32)(10,31)(11,29)(12,30)(13,28)(14,27)(15,25)(16,26)(17,24)(18,23)(19,21)(20,22)
$|\Aut(F/K)|$:  $12$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
18:  $C_3^2:C_2$
24:  $S_4$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 4: None

Degree 6: $S_3$ x 4, $S_4$, $S_4$

Degree 9: $C_3^2:C_2$

Degree 12: $S_4$

Degree 18: $C_3^2 : C_2$, 18T37, 18T40

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 43]
Character table:   
     2  3  3  2  2  .  .  2  2  .
     3  2  1  .  .  2  2  1  2  2

       1a 2a 2b 4a 3a 3b 6a 3c 3d
    2P 1a 1a 1a 2a 3a 3b 3c 3c 3d
    3P 1a 2a 2b 4a 1a 1a 2a 1a 1a
    5P 1a 2a 2b 4a 3a 3b 6a 3c 3d

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