Properties

Label 36T9
Order \(36\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_9:C_4$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $9$
Group :  $C_9:C_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,2,4)(5,34,6,33)(7,35,8,36)(9,32,10,31)(11,30,12,29)(13,28,14,27)(15,26,16,25)(17,23,18,24)(19,21,20,22), (1,34,32,28,24,19,16,12,7)(2,33,31,27,23,20,15,11,8)(3,35,29,25,21,17,14,10,6)(4,36,30,26,22,18,13,9,5)
$|\Aut(F/K)|$:  $36$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
18:  $D_{9}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $S_3$

Degree 9: $D_{9}$

Degree 12: $C_3 : C_4$

Degree 18: $D_9$

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

Group invariants

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

        1a 2a 4a 4b 9a 18a 18b 9b 6a 3a 9c 18c
     2P 1a 1a 2a 2a 9b  9b  9c 9c 3a 3a 9a  9a
     3P 1a 2a 4b 4a 3a  6a  6a 3a 2a 1a 3a  6a
     5P 1a 2a 4a 4b 9c 18c 18a 9a 6a 3a 9b 18b
     7P 1a 2a 4b 4a 9b 18b 18c 9c 6a 3a 9a 18a
    11P 1a 2a 4b 4a 9b 18b 18c 9c 6a 3a 9a 18a
    13P 1a 2a 4a 4b 9c 18c 18a 9a 6a 3a 9b 18b
    17P 1a 2a 4a 4b 9a 18a 18b 9b 6a 3a 9c 18c

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  A -A  1  -1  -1  1 -1  1  1  -1
X.4      1 -1 -A  A  1  -1  -1  1 -1  1  1  -1
X.5      2  2  .  . -1  -1  -1 -1  2  2 -1  -1
X.6      2 -2  .  . -1   1   1 -1 -2  2 -1   1
X.7      2  2  .  .  B   B   D  D -1 -1  C   C
X.8      2  2  .  .  C   C   B  B -1 -1  D   D
X.9      2  2  .  .  D   D   C  C -1 -1  B   B
X.10     2 -2  .  .  B  -B  -D  D  1 -1  C  -C
X.11     2 -2  .  .  C  -C  -B  B  1 -1  D  -D
X.12     2 -2  .  .  D  -D  -C  C  1 -1  B  -B

A = -E(4)
  = -Sqrt(-1) = -i
B = E(9)^2+E(9)^7
C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7
D = E(9)^4+E(9)^5