Properties

 Label 18T11 Degree $18$ Order $36$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3^2$

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Group action invariants

 Degree $n$: $18$ Transitive number $t$: $11$ Group: $S_3^2$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $6$ Generators: (1,11,8,17,13,6)(2,12,7,18,14,5)(3,15,10)(4,16,9), (1,4)(2,3)(5,11)(6,12)(7,15)(8,16)(9,13)(10,14)(17,18), (1,15,6)(2,16,5)(3,11,8)(4,12,7)(9,18,14)(10,17,13)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$12$:  $D_{6}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 2

Degree 6: $S_3$, $D_{6}$

Degree 9: $S_3^2$

Low degree siblings

6T9, 9T8, 12T16, 18T9, 18T11

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

Group invariants

 Order: $36=2^{2} \cdot 3^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [36, 10]
 Character table:  2 2 2 2 2 1 1 . 1 1 3 2 1 1 . 2 1 2 1 2 1a 2a 2b 2c 3a 6a 3b 6b 3c 2P 1a 1a 1a 1a 3a 3a 3b 3c 3c 3P 1a 2a 2b 2c 1a 2b 1a 2a 1a 5P 1a 2a 2b 2c 3a 6a 3b 6b 3c 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 1 1 1 -1 1 X.4 1 1 -1 -1 1 -1 1 1 1 X.5 2 -2 . . 2 . -1 1 -1 X.6 2 2 . . 2 . -1 -1 -1 X.7 2 . -2 . -1 1 -1 . 2 X.8 2 . 2 . -1 -1 -1 . 2 X.9 4 . . . -2 . 1 . -2