Properties

Label 36T40
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 $t$ :  $40$
Group :  $C_2\times S_3^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,35,6,4,33,8)(2,36,5,3,34,7)(9,29,13,22,18,26)(10,30,14,21,17,25)(11,31,16,24,20,27)(12,32,15,23,19,28), (1,34)(2,33)(3,35)(4,36)(5,6)(7,8)(9,14)(10,13)(11,15)(12,16)(17,18)(19,20)(21,26)(22,25)(23,27)(24,28)(29,30)(31,32), (1,9)(2,10)(3,12)(4,11)(5,17)(6,18)(7,19)(8,20)(13,33)(14,34)(15,36)(16,35)(21,28)(22,27)(23,25)(24,26)(29,31)(30,32)
$|\Aut(F/K)|$:  $12$

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: $S_3$, $D_{6}$ x 5

Degree 9: $S_3^2$

Degree 12: $D_6$, $S_3 \times C_2^2$

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

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  1  2  1  2  2  2
      3  2  1  2  1  1  .  1  .  2  2  1  1  2  1  2  1  2  2

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

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