Properties

Label 36T35
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_3:S_3.C_2$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: $C_3^2:C_4$ x 2

Degree 9: $C_3^2:C_4$

Degree 12: 12T41 x 2

Degree 18: $C_3^2 : C_4$

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 45]
Character table:   
      2  3  3  3  3  3  3  3  3  1  1  1  1
      3  2  .  2  .  .  .  .  .  2  2  2  2

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

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

A = -E(4)
  = -Sqrt(-1) = -i