Properties

Label 36T34
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_3^2$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$ x 2
8:  $C_2^3$
12:  $D_{6}$ x 6
36:  $S_3^2$

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

Degree 9: $S_3^2$

Degree 12: $S_3 \times C_2^2$ x 2, 12T37

Degree 18: $S_3^2$, 18T29 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, 1, 1, 1, 1 $ $9$ $2$ $( 5,36)( 6,35)( 7,33)( 8,34)( 9,32)(10,31)(11,29)(12,30)(13,27)(14,28)(15,25) (16,26)(17,23)(18,24)(19,22)(20,21)$
$ 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 $ $9$ $2$ $( 1, 2)( 3, 4)( 5,35)( 6,36)( 7,34)( 8,33)( 9,31)(10,32)(11,30)(12,29)(13,28) (14,27)(15,26)(16,25)(17,24)(18,23)(19,21)(20,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,21)(10,22)(11,23)(12,24)(13,26)(14,25)(15,28) (16,27)(17,29)(18,30)(19,31)(20,32)(33,36)(34,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 3)( 2, 4)( 5,33)( 6,34)( 7,36)( 8,35)( 9,20)(10,19)(11,17)(12,18)(13,16) (14,15)(21,32)(22,31)(23,29)(24,30)(25,28)(26,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,22)(10,21)(11,24)(12,23)(13,25)(14,26)(15,27) (16,28)(17,30)(18,29)(19,32)(20,31)(33,35)(34,36)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2, 3)( 5,34)( 6,33)( 7,35)( 8,36)( 9,19)(10,20)(11,18)(12,17)(13,15) (14,16)(21,31)(22,32)(23,30)(24,29)(25,27)(26,28)$
$ 6, 6, 6, 6, 6, 6 $ $6$ $6$ $( 1, 5,33, 3, 7,36)( 2, 6,34, 4, 8,35)( 9,27,18,21,16,30)(10,28,17,22,15,29) (11,25,19,23,14,31)(12,26,20,24,13,32)$
$ 6, 6, 6, 6, 6, 6 $ $6$ $6$ $( 1, 6,33, 4, 7,35)( 2, 5,34, 3, 8,36)( 9,28,18,22,16,29)(10,27,17,21,15,30) (11,26,19,24,14,32)(12,25,20,23,13,31)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 7,33)( 2, 8,34)( 3, 5,36)( 4, 6,35)( 9,16,18)(10,15,17)(11,14,19) (12,13,20)(21,27,30)(22,28,29)(23,25,31)(24,26,32)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1, 8,33, 2, 7,34)( 3, 6,36, 4, 5,35)( 9,15,18,10,16,17)(11,13,19,12,14,20) (21,28,30,22,27,29)(23,26,31,24,25,32)$
$ 6, 6, 6, 6, 6, 6 $ $6$ $6$ $( 1, 9,27,36,13,24)( 2,10,28,35,14,23)( 3,12,26,33,16,21)( 4,11,25,34,15,22) ( 5,20,32, 7,18,30)( 6,19,31, 8,17,29)$
$ 6, 6, 6, 6, 6, 6 $ $6$ $6$ $( 1,10,27,35,13,23)( 2, 9,28,36,14,24)( 3,11,26,34,16,22)( 4,12,25,33,15,21) ( 5,19,32, 8,18,29)( 6,20,31, 7,17,30)$
$ 6, 6, 6, 6, 6, 6 $ $4$ $6$ $( 1,11,30, 2,12,29)( 3,10,32, 4, 9,31)( 5,15,24, 6,16,23)( 7,14,21, 8,13,22) (17,26,35,18,25,36)(19,27,34,20,28,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,12,30)( 2,11,29)( 3, 9,32)( 4,10,31)( 5,16,24)( 6,15,23)( 7,13,21) ( 8,14,22)(17,25,35)(18,26,36)(19,28,34)(20,27,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,13,27)( 2,14,28)( 3,16,26)( 4,15,25)( 5,18,32)( 6,17,31)( 7,20,30) ( 8,19,29)( 9,24,36)(10,23,35)(11,22,34)(12,21,33)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,14,27, 2,13,28)( 3,15,26, 4,16,25)( 5,17,32, 6,18,31)( 7,19,30, 8,20,29) ( 9,23,36,10,24,35)(11,21,34,12,22,33)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 46]
Character table:   
      2  3  3  3  3  3  3  3  3  2  2  2  2  2  2  1  1  2  2
      3  2  .  2  .  1  1  1  1  1  1  2  2  1  1  2  2  2  2

        1a 2a 2b 2c 2d 2e 2f 2g 6a 6b 3a 6c 6d 6e 6f 3b 3c 6g
     2P 1a 1a 1a 1a 1a 1a 1a 1a 3a 3a 3a 3a 3c 3c 3b 3b 3c 3c
     3P 1a 2a 2b 2c 2d 2e 2f 2g 2d 2f 1a 2b 2e 2g 2b 1a 1a 2b
     5P 1a 2a 2b 2c 2d 2e 2f 2g 6a 6b 3a 6c 6d 6e 6f 3b 3c 6g

X.1      1  1  1  1  1  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  1 -1 -1  1  1 -1
X.3      1 -1 -1  1  1 -1 -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  1  1  1  1  1  1
X.5      1 -1  1 -1  1 -1  1 -1  1  1  1  1 -1 -1  1  1  1  1
X.6      1  1 -1 -1 -1 -1  1  1 -1  1  1 -1 -1  1 -1  1  1 -1
X.7      1  1 -1 -1  1  1 -1 -1  1 -1  1 -1  1 -1 -1  1  1 -1
X.8      1  1  1  1 -1 -1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1
X.9      2  . -2  .  . -2  .  2  .  .  2 -2  1 -1  1 -1 -1  1
X.10     2  . -2  .  .  2  . -2  .  .  2 -2 -1  1  1 -1 -1  1
X.11     2  .  2  .  . -2  . -2  .  .  2  2  1  1 -1 -1 -1 -1
X.12     2  .  2  .  .  2  .  2  .  .  2  2 -1 -1 -1 -1 -1 -1
X.13     2  . -2  . -2  .  2  .  1 -1 -1  1  .  .  1 -1  2 -2
X.14     2  . -2  .  2  . -2  . -1  1 -1  1  .  .  1 -1  2 -2
X.15     2  .  2  . -2  . -2  .  1  1 -1 -1  .  . -1 -1  2  2
X.16     2  .  2  .  2  .  2  . -1 -1 -1 -1  .  . -1 -1  2  2
X.17     4  .  4  .  .  .  .  .  .  . -2 -2  .  .  1  1 -2 -2
X.18     4  . -4  .  .  .  .  .  .  . -2  2  .  . -1  1 -2  2