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Properties

Label 18T28
Order \(72\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_9$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $28$
Group :  $F_9$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,14)(2,7,13)(3,9,16)(4,10,15)(5,12,18)(6,11,17), (1,12,3,18,6,13,9,7)(2,11,4,17,5,14,10,8)(15,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: None

Degree 9: $C_3^2:C_8$

Low degree siblings

9T15, 12T46

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 4, 4, 4, 4, 1, 1 $ $9$ $4$ $( 3, 9,17,11)( 4,10,18,12)( 5, 7,15,13)( 6, 8,16,14)$
$ 4, 4, 4, 4, 1, 1 $ $9$ $4$ $( 3,11,17, 9)( 4,12,18,10)( 5,13,15, 7)( 6,14,16, 8)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $9$ $2$ $( 3,17)( 4,18)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11)(10,12)$
$ 8, 8, 2 $ $9$ $8$ $( 1, 2)( 3, 5,11,13,17,15, 9, 7)( 4, 6,12,14,18,16,10, 8)$
$ 8, 8, 2 $ $9$ $8$ $( 1, 2)( 3, 7, 9,15,17,13,11, 5)( 4, 8,10,16,18,14,12, 6)$
$ 8, 8, 2 $ $9$ $8$ $( 1, 2)( 3,13, 9, 5,17, 7,11,15)( 4,14,10, 6,18, 8,12,16)$
$ 8, 8, 2 $ $9$ $8$ $( 1, 2)( 3,15,11, 7,17, 5, 9,13)( 4,16,12, 8,18, 6,10,14)$
$ 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1, 3,17)( 2, 4,18)( 5, 7,10)( 6, 8, 9)(11,14,16)(12,13,15)$

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 4a 4b 2a  8a  8b  8c  8d 3a
    2P 1a 2a 2a 1a  4b  4a  4a  4b 3a
    3P 1a 4b 4a 2a  8c  8d  8a  8b 1a
    5P 1a 4a 4b 2a  8d  8c  8b  8a 3a
    7P 1a 4b 4a 2a  8b  8a  8d  8c 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 -1  1   A  -A  -A   A  1
X.4     1 -1 -1  1  -A   A   A  -A  1
X.5     1  A -A -1   B  /B -/B  -B  1
X.6     1  A -A -1  -B -/B  /B   B  1
X.7     1 -A  A -1 -/B  -B   B  /B  1
X.8     1 -A  A -1  /B   B  -B -/B  1
X.9     8  .  .  .   .   .   .   . -1

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