Properties

Label 36T49
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_9$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $49$
Group :  $F_9$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,2,4)(5,13,18,29,34,25,22,10)(6,14,17,30,33,26,21,9)(7,16,20,31,36,28,24,12)(8,15,19,32,35,27,23,11), (1,22,27,6)(2,21,28,5)(3,24,25,8)(4,23,26,7)(9,35,19,29)(10,36,20,30)(11,34,18,31)(12,33,17,32)(13,14)(15,16)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Degree 9: $C_3^2:C_8$

Degree 12: 12T46

Degree 18: 18T28

Low degree siblings

9T15

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 39]
Character table:   
     2  3  3  3  3   3   3   3   3  .
     3  2  .  .  .   .   .   .   .  2

       1a 2a 4a 4b  8a  8b  8c  8d 3a
    2P 1a 1a 2a 2a  4a  4a  4b  4b 3a
    3P 1a 2a 4b 4a  8d  8c  8b  8a 1a
    5P 1a 2a 4a 4b  8b  8a  8d  8c 3a
    7P 1a 2a 4b 4a  8c  8d  8a  8b 3a

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

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)^3