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Properties

Label 18T156
Order \(432\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times (C_3\times A_4):S_3$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $156$
Group :  $C_2\times (C_3\times A_4):S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,6,2,3,5)(7,11,9,8,12,10)(13,14)(15,16)(17,18), (1,16,4,17,6,13,2,15,3,18,5,14)(7,12,9)(8,11,10), (1,9)(2,10)(3,12)(4,11)(5,8)(6,7)(13,14)(15,16)(17,18)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 4
12:  $D_{6}$ x 4
18:  $C_3^2:C_2$
24:  $S_4$
36:  18T12
48:  $S_4\times C_2$
54:  $(C_3^2:C_3):C_2$
72:  12T44
108:  18T52
144:  18T66
216:  18T107

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4\times C_2$

Degree 9: $(C_3^2:C_3):C_2$

Low degree siblings

18T156

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

Group invariants

Order:  $432=2^{4} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [432, 538]
Character table: Data not available.