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Properties

Label 18T367
Order \(2304\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $367$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,5,4,6)(7,11)(8,12)(13,17,14,18)(15,16), (1,18,7,2,17,8)(3,13,9,4,14,10)(5,16,12,6,15,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
12:  $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$
24:  $S_4$
36:  $C_6\times S_3$
48:  $S_4\times C_2$
72:  12T45
144:  18T61
288:  $A_4\wr C_2$
576:  12T158
1152:  12T205

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: None

Degree 9: $S_3\times C_3$

Low degree siblings

18T367

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 3, 4)( 5, 6)(13,14)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(15,16)(17,18)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $(13,14)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 7, 8)( 9,10)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $128$ $3$ $( 1, 6, 4)( 2, 5, 3)( 7,11,10)( 8,12, 9)(13,18,15)(14,17,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $24$ $2$ $( 1, 5)( 2, 6)( 7,10)( 8, 9)(13,16)(14,15)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $72$ $4$ $( 1, 6, 2, 5)( 3, 4)( 7,10)( 8, 9)(13,16,14,15)(17,18)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $72$ $4$ $( 1, 5)( 2, 6)( 7,10)( 8, 9)(13,16,14,15)(17,18)$
$ 4, 4, 4, 2, 2, 2 $ $24$ $4$ $( 1, 6, 2, 5)( 3, 4)( 7, 9, 8,10)(11,12)(13,15,14,16)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $32$ $3$ $( 1, 7,17)( 2, 8,18)( 3, 9,14)( 4,10,13)( 5,12,15)( 6,11,16)$
$ 6, 6, 3, 3 $ $96$ $6$ $( 1, 7,18, 2, 8,17)( 3, 9,13)( 4,10,14)( 5,12,15, 6,11,16)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1,10,15, 2, 9,16)( 3,12,18)( 4,11,17)( 5, 8,13, 6, 7,14)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,10,16)( 2, 9,15)( 3,12,18)( 4,11,17)( 5, 8,14)( 6, 7,13)$
$ 12, 6 $ $96$ $12$ $( 1,10,16, 2, 9,15)( 3, 8,18, 6,11,13, 4, 7,17, 5,12,14)$
$ 6, 6, 3, 3 $ $96$ $6$ $( 1,10,15)( 2, 9,16)( 3, 8,17, 5,12,14)( 4, 7,18, 6,11,13)$
$ 3, 3, 3, 3, 3, 3 $ $32$ $3$ $( 1,17, 7)( 2,18, 8)( 3,14, 9)( 4,13,10)( 5,15,12)( 6,16,11)$
$ 6, 6, 3, 3 $ $96$ $6$ $( 1,18, 8, 2,17, 7)( 3,13,10)( 4,14, 9)( 5,15,12, 6,16,11)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1,16,10)( 2,15, 9)( 3,17,11, 4,18,12)( 5,13, 7, 6,14, 8)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,16,10)( 2,15, 9)( 3,18,12)( 4,17,11)( 5,14, 8)( 6,13, 7)$
$ 12, 6 $ $96$ $12$ $( 1,17,10, 4,16,11, 2,18, 9, 3,15,12)( 5,14, 8, 6,13, 7)$
$ 6, 6, 3, 3 $ $96$ $6$ $( 1,18, 9, 3,15,12)( 2,17,10, 4,16,11)( 5,13, 7)( 6,14, 8)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 5, 6)(11,12)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 3, 4)( 7, 8)( 9,10)(11,12)(15,16)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 5, 6)( 7, 8)(13,14)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 5, 6)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 3, 4)(11,12)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 6, 6, 6 $ $128$ $6$ $( 1, 6, 3, 2, 5, 4)( 7,11, 9, 8,12,10)(13,18,16,14,17,15)$
$ 4, 2, 2, 2, 2, 2, 2, 1, 1 $ $72$ $4$ $( 1, 5, 2, 6)( 7,10)( 8, 9)(11,12)(13,16)(14,15)(17,18)$
$ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ $24$ $4$ $( 1, 5, 2, 6)( 7, 9, 8,10)(13,15,14,16)$
$ 4, 4, 2, 2, 2, 1, 1, 1, 1 $ $72$ $4$ $( 1, 5, 2, 6)( 7,10)( 8, 9)(11,12)(13,16,14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1, 6)( 2, 5)( 3, 4)( 7,10)( 8, 9)(11,12)(13,16)(14,15)(17,18)$
$ 6, 3, 3, 3, 3 $ $96$ $6$ $( 1, 7,17, 2, 8,18)( 3, 9,14)( 4,10,13)( 5,11,15)( 6,12,16)$
$ 6, 6, 6 $ $32$ $6$ $( 1, 7,17, 2, 8,18)( 3,10,14, 4, 9,13)( 5,12,15, 6,11,16)$
$ 6, 3, 3, 3, 3 $ $48$ $6$ $( 1,10,15, 2, 9,16)( 3,12,17)( 4,11,18)( 5, 7,14)( 6, 8,13)$
$ 6, 6, 6 $ $16$ $6$ $( 1,10,15, 2, 9,16)( 3,12,18, 4,11,17)( 5, 7,13, 6, 8,14)$
$ 6, 6, 6 $ $96$ $6$ $( 1,10,16, 2, 9,15)( 3, 8,18, 5,11,14)( 4, 7,17, 6,12,13)$
$ 12, 3, 3 $ $96$ $12$ $( 1,10,15)( 2, 9,16)( 3, 8,17, 6,12,13, 4, 7,18, 5,11,14)$
$ 6, 3, 3, 3, 3 $ $96$ $6$ $( 1,17, 8, 2,18, 7)( 3,14, 9)( 4,13,10)( 5,16,11)( 6,15,12)$
$ 6, 6, 6 $ $32$ $6$ $( 1,17, 8, 2,18, 7)( 3,13, 9, 4,14,10)( 5,15,11, 6,16,12)$
$ 6, 3, 3, 3, 3 $ $48$ $6$ $( 1,16,10)( 2,15, 9)( 3,17,12, 4,18,11)( 5,14, 8)( 6,13, 7)$
$ 6, 6, 6 $ $16$ $6$ $( 1,15, 9, 2,16,10)( 3,17,12, 4,18,11)( 5,13, 7, 6,14, 8)$
$ 12, 3, 3 $ $96$ $12$ $( 1,17, 9, 3,15,12, 2,18,10, 4,16,11)( 5,13, 7)( 6,14, 8)$
$ 6, 6, 6 $ $96$ $6$ $( 1,18,10, 4,16,11)( 2,17, 9, 3,15,12)( 5,14, 8, 6,13, 7)$

Group invariants

Order:  $2304=2^{8} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.