Properties

Label 38T7
Order \(342\)
n \(38\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_{19}:C_3.C_3$

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Group action invariants

Degree $n$ :  $38$
Transitive number $t$ :  $7$
Group :  $C_2\times C_{19}:C_3.C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,29,27,20,26,12,31,36,14,2,30,28,19,25,11,32,35,13)(3,38,21,34,5,7,16,10,23,4,37,22,33,6,8,15,9,24)(17,18), (1,37,15,2,38,16)(3,22,29,4,21,30)(5,6)(7,28,19,8,27,20)(9,11,33,10,12,34)(13,17,23,14,18,24)(25,35,32,26,36,31)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
9:  $C_9$
18:  $C_{18}$
171:  $C_{19}:C_{9}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: $C_{19}:C_{9}$

Low degree siblings

There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 9, 9, 9, 9, 1, 1 $ $19$ $9$ $( 3, 9,33,16,20,36,23,13,12)( 4,10,34,15,19,35,24,14,11)( 5,18,28,29,37,32, 8, 25,21)( 6,17,27,30,38,31, 7,26,22)$
$ 9, 9, 9, 9, 1, 1 $ $19$ $9$ $( 3,12,13,23,36,20,16,33, 9)( 4,11,14,24,35,19,15,34,10)( 5,21,25, 8,32,37,29, 28,18)( 6,22,26, 7,31,38,30,27,17)$
$ 9, 9, 9, 9, 1, 1 $ $19$ $9$ $( 3,13,36,16, 9,12,23,20,33)( 4,14,35,15,10,11,24,19,34)( 5,25,32,29,18,21, 8, 37,28)( 6,26,31,30,17,22, 7,38,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $19$ $3$ $( 3,16,23)( 4,15,24)( 5,29, 8)( 6,30, 7)( 9,20,13)(10,19,14)(11,34,35) (12,33,36)(17,38,26)(18,37,25)(21,28,32)(22,27,31)$
$ 9, 9, 9, 9, 1, 1 $ $19$ $9$ $( 3,20,12,16,13,33,23, 9,36)( 4,19,11,15,14,34,24,10,35)( 5,37,21,29,25,28, 8, 18,32)( 6,38,22,30,26,27, 7,17,31)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $19$ $3$ $( 3,23,16)( 4,24,15)( 5, 8,29)( 6, 7,30)( 9,13,20)(10,14,19)(11,35,34) (12,36,33)(17,26,38)(18,25,37)(21,32,28)(22,31,27)$
$ 9, 9, 9, 9, 1, 1 $ $19$ $9$ $( 3,33,20,23,12, 9,16,36,13)( 4,34,19,24,11,10,15,35,14)( 5,28,37, 8,21,18,29, 32,25)( 6,27,38, 7,22,17,30,31,26)$
$ 9, 9, 9, 9, 1, 1 $ $19$ $9$ $( 3,36, 9,23,33,13,16,12,20)( 4,35,10,24,34,14,15,11,19)( 5,32,18, 8,28,25,29, 21,37)( 6,31,17, 7,27,26,30,22,38)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)$
$ 18, 18, 2 $ $19$ $18$ $( 1, 2)( 3,10,33,15,20,35,23,14,12, 4, 9,34,16,19,36,24,13,11)( 5,17,28,30,37, 31, 8,26,21, 6,18,27,29,38,32, 7,25,22)$
$ 18, 18, 2 $ $19$ $18$ $( 1, 2)( 3,11,13,24,36,19,16,34, 9, 4,12,14,23,35,20,15,33,10)( 5,22,25, 7,32, 38,29,27,18, 6,21,26, 8,31,37,30,28,17)$
$ 18, 18, 2 $ $19$ $18$ $( 1, 2)( 3,14,36,15, 9,11,23,19,33, 4,13,35,16,10,12,24,20,34)( 5,26,32,30,18, 22, 8,38,28, 6,25,31,29,17,21, 7,37,27)$
$ 6, 6, 6, 6, 6, 6, 2 $ $19$ $6$ $( 1, 2)( 3,15,23, 4,16,24)( 5,30, 8, 6,29, 7)( 9,19,13,10,20,14) (11,33,35,12,34,36)(17,37,26,18,38,25)(21,27,32,22,28,31)$
$ 18, 18, 2 $ $19$ $18$ $( 1, 2)( 3,19,12,15,13,34,23,10,36, 4,20,11,16,14,33,24, 9,35)( 5,38,21,30,25, 27, 8,17,32, 6,37,22,29,26,28, 7,18,31)$
$ 6, 6, 6, 6, 6, 6, 2 $ $19$ $6$ $( 1, 2)( 3,24,16, 4,23,15)( 5, 7,29, 6, 8,30)( 9,14,20,10,13,19) (11,36,34,12,35,33)(17,25,38,18,26,37)(21,31,28,22,32,27)$
$ 18, 18, 2 $ $19$ $18$ $( 1, 2)( 3,34,20,24,12,10,16,35,13, 4,33,19,23,11, 9,15,36,14)( 5,27,37, 7,21, 17,29,31,25, 6,28,38, 8,22,18,30,32,26)$
$ 18, 18, 2 $ $19$ $18$ $( 1, 2)( 3,35, 9,24,33,14,16,11,20, 4,36,10,23,34,13,15,12,19)( 5,31,18, 7,28, 26,29,22,37, 6,32,17, 8,27,25,30,21,38)$
$ 38 $ $9$ $38$ $( 1, 3, 6, 8,10,12,14,16,17,20,22,23,26,28,30,32,34,36,38, 2, 4, 5, 7, 9,11, 13,15,18,19,21,24,25,27,29,31,33,35,37)$
$ 19, 19 $ $9$ $19$ $( 1, 4, 6, 7,10,11,14,15,17,19,22,24,26,27,30,31,34,35,38)( 2, 3, 5, 8, 9,12, 13,16,18,20,21,23,25,28,29,32,33,36,37)$
$ 38 $ $9$ $38$ $( 1, 5,10,13,17,21,26,29,34,37, 4, 8,11,16,19,23,27,32,35, 2, 6, 9,14,18,22, 25,30,33,38, 3, 7,12,15,20,24,28,31,36)$
$ 19, 19 $ $9$ $19$ $( 1, 6,10,14,17,22,26,30,34,38, 4, 7,11,15,19,24,27,31,35)( 2, 5, 9,13,18,21, 25,29,33,37, 3, 8,12,16,20,23,28,32,36)$

Group invariants

Order:  $342=2 \cdot 3^{2} \cdot 19$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [342, 8]
Character table: Data not available.