Properties

Label 38T24
Order \(8664\)
n \(38\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
8:  $D_{4}$
12:  $C_6\times C_2$
24:  $D_4 \times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: None

Low degree siblings

38T24

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 19, 19 $ $24$ $19$ $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$
$ 19, 19 $ $24$ $19$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$
$ 19, 19 $ $24$ $19$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$
$ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $19$ $(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$
$ 19, 19 $ $12$ $19$ $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,35,31,27,23,38, 34,30,26,22,37,33,29,25,21,36,32,28,24)$
$ 19, 19 $ $24$ $19$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,32,25,37,30,23, 35,28,21,33,26,38,31,24,36,29,22,34,27)$
$ 19, 19 $ $24$ $19$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$
$ 19, 19 $ $24$ $19$ $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,37,35,33,31,29, 27,25,23,21,38,36,34,32,30,28,26,24,22)$
$ 19, 19 $ $24$ $19$ $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$
$ 19, 19 $ $24$ $19$ $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$
$ 19, 19 $ $24$ $19$ $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,31,23,34,26,37, 29,21,32,24,35,27,38,30,22,33,25,36,28)$
$ 19, 19 $ $24$ $19$ $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,24,28,32,36,21, 25,29,33,37,22,26,30,34,38,23,27,31,35)$
$ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $19$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)$
$ 19, 19 $ $24$ $19$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38)$
$ 19, 19 $ $12$ $19$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,31,23,34,26,37, 29,21,32,24,35,27,38,30,22,33,25,36,28)$
$ 19, 19 $ $24$ $19$ $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$
$ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $19$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)$
$ 19, 19 $ $12$ $19$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,23,26,29,32,35, 38,22,25,28,31,34,37,21,24,27,30,33,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $361$ $3$ $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,27,31) (22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $361$ $3$ $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$
$ 6, 6, 6, 6, 6, 6, 1, 1 $ $361$ $6$ $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $361$ $2$ $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,38)(22,37) (23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$
$ 6, 6, 6, 6, 6, 6, 1, 1 $ $361$ $6$ $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,32,31,38,27,28) (22,25,23,37,34,36)(24,30,26,35,29,33)$
$ 12, 12, 12, 2 $ $722$ $12$ $( 1,26, 7,38,17,20, 2,28,15,35, 5,34)( 3,30, 4,32,12,29,19,24,18,22,10,25) ( 6,36, 9,23,14,33,16,37,13,31, 8,21)(11,27)$
$ 12, 12, 12, 2 $ $722$ $12$ $( 1,24, 6,37, 9,22, 7,32, 2,38,18,34)( 3,33,11,31,12,26, 5,23,16,25,15,30) ( 4,28)( 8,27,14,35,10,36,19,29,13,21,17,20)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $722$ $4$ $( 1,29, 9,34)( 2,32, 8,31)( 3,35, 7,28)( 4,38, 6,25)( 5,22)(10,37,19,26) (11,21,18,23)(12,24,17,20)(13,27,16,36)(14,30,15,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $38$ $2$ $(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$
$ 19, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $228$ $38$ $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,36)(21,35) (22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(37,38)$
$ 19, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $228$ $38$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)(34,38)(35,37)$
$ 19, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $228$ $38$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,27)(21,26) (22,25)(23,24)(28,38)(29,37)(30,36)(31,35)(32,34)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3, 1, 1 $ $722$ $6$ $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,32,31,38,27,28 )(22,25,23,37,34,36)(24,30,26,35,29,33)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3, 1, 1 $ $722$ $6$ $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,28,27,38,31,32 )(22,36,34,37,23,25)(24,33,29,35,26,30)$
$ 6, 6, 6, 6, 6, 6, 2 $ $722$ $6$ $( 1,26,16,37,10,25)( 2,28, 8,21,17,20)( 3,30,19,24, 5,34)( 4,32,11,27,12,29) ( 6,36,14,33, 7,38)( 9,23)(13,31,15,35,18,22)$
$ 6, 6, 6, 6, 6, 6, 2 $ $722$ $6$ $( 1,24,10,36,16,25)( 2,38,17,20, 8,27)( 3,33, 5,23,19,29)( 4,28,12,26,11,31) ( 6,37, 7,32,14,35)( 9,22)(13,21,18,34,15,30)$
$ 38 $ $228$ $38$ $( 1,29,15,33,10,37, 5,22,19,26,14,30, 9,34, 4,38,18,23,13,27, 8,31, 3,35,17, 20,12,24, 7,28, 2,32,16,36,11,21, 6,25)$
$ 38 $ $228$ $38$ $( 1,26,19,23,18,20,17,36,16,33,15,30,14,27,13,24,12,21,11,37,10,34, 9,31, 8, 28, 7,25, 6,22, 5,38, 4,35, 3,32, 2,29)$
$ 38 $ $228$ $38$ $( 1,28,10,36,19,25, 9,33,18,22, 8,30,17,38, 7,27,16,35, 6,24,15,32, 5,21,14, 29, 4,37,13,26, 3,34,12,23, 2,31,11,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $38$ $2$ $( 1,30)( 2,33)( 3,36)( 4,20)( 5,23)( 6,26)( 7,29)( 8,32)( 9,35)(10,38)(11,22) (12,25)(13,28)(14,31)(15,34)(16,37)(17,21)(18,24)(19,27)$

Group invariants

Order:  $8664=2^{3} \cdot 3 \cdot 19^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.