Properties

Label 38T27
Order \(12996\)
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 $t$ :  $27$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,38,18,23,13,33,10,20,12,35,17,25)(2,36,11,37,5,30,9,22,19,21,6,28)(3,34,4,32,16,27,8,24,7,26,14,31)(15,29), (1,19,12)(2,7,4)(3,14,15)(5,9,18)(6,16,10)(8,11,13)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $S_3$, $C_6$
12:  $C_{12}$, $C_3 : C_4$
18:  $S_3\times C_3$
36:  $C_3\times (C_3 : C_4)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: None

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 $ $1$ $1$ $()$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $19$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)$
$ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $19$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)$
$ 19, 19 $ $36$ $19$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$
$ 19, 19 $ $36$ $19$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$
$ 19, 19 $ $36$ $19$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$
$ 19, 19 $ $36$ $19$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$
$ 19, 19 $ $36$ $19$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$
$ 19, 19 $ $36$ $19$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$
$ 19, 19 $ $36$ $19$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$
$ 19, 19 $ $36$ $19$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$
$ 19, 19 $ $36$ $19$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,34,29,24,38,33, 28,23,37,32,27,22,36,31,26,21,35,30,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $361$ $3$ $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $361$ $3$ $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,27,31) (22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$
$ 6, 6, 6, 6, 6, 6, 1, 1 $ $361$ $6$ $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,32,31,38,27,28) (22,25,23,37,34,36)(24,30,26,35,29,33)$
$ 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 $ $361$ $6$ $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $38$ $3$ $(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$
$ 19, 3, 3, 3, 3, 3, 3, 1 $ $228$ $57$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(21,27,31)(22,34,23) (24,29,26)(25,36,37)(28,38,32)(30,33,35)$
$ 19, 3, 3, 3, 3, 3, 3, 1 $ $228$ $57$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(21,27,31)(22,34,23) (24,29,26)(25,36,37)(28,38,32)(30,33,35)$
$ 19, 3, 3, 3, 3, 3, 3, 1 $ $228$ $57$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(21,27,31)(22,34,23) (24,29,26)(25,36,37)(28,38,32)(30,33,35)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $38$ $3$ $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)$
$ 19, 3, 3, 3, 3, 3, 3, 1 $ $228$ $57$ $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(20,33,27,21,34, 28,22,35,29,23,36,30,24,37,31,25,38,32,26)$
$ 19, 3, 3, 3, 3, 3, 3, 1 $ $228$ $57$ $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(20,27,34,22,29, 36,24,31,38,26,33,21,28,35,23,30,37,25,32)$
$ 19, 3, 3, 3, 3, 3, 3, 1 $ $228$ $57$ $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(20,34,29,24,38, 33,28,23,37,32,27,22,36,31,26,21,35,30,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $722$ $3$ $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$
$ 6, 6, 6, 6, 6, 6, 1, 1 $ $722$ $6$ $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$
$ 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $722$ $6$ $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11) (21,32,31,38,27,28)(22,25,23,37,34,36)(24,30,26,35,29,33)$
$ 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $722$ $6$ $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,38)(22,37)(23,36) (24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$
$ 12, 12, 12, 2 $ $1083$ $12$ $( 1,38,18,23,13,33,10,20,12,35,17,25)( 2,36,11,37, 5,30, 9,22,19,21, 6,28) ( 3,34, 4,32,16,27, 8,24, 7,26,14,31)(15,29)$
$ 12, 12, 12, 2 $ $1083$ $12$ $( 1,28,12,33, 5,35, 6,32,14,27, 2,25)( 3,22, 9,23,19,31, 4,38,17,37, 7,29) ( 8,26,11,36,16,21,18,34,15,24,10,20)(13,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $1083$ $4$ $( 1,32,11,25)( 2,37,10,20)( 3,23, 9,34)( 4,28, 8,29)( 5,33, 7,24)( 6,38) (12,30,19,27)(13,35,18,22)(14,21,17,36)(15,26,16,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $1083$ $4$ $( 1,27, 9,25)( 2,22, 8,30)( 3,36, 7,35)( 4,31, 6,21)( 5,26)(10,20,19,32) (11,34,18,37)(12,29,17,23)(13,24,16,28)(14,38,15,33)$
$ 12, 12, 12, 2 $ $1083$ $12$ $( 1,21,15,30,12,24,14,28,19,38, 3,25)( 2,23, 8,35, 4,27,13,26, 7,33,11,22) ( 5,29, 6,31,18,36,10,20, 9,37,16,32)(17,34)$
$ 12, 12, 12, 2 $ $1083$ $12$ $( 1,31,17,22,12,26,10,20,13,29,18,25)( 2,34, 6,27,19,28, 9,36, 5,24,11,23) ( 3,37,14,32, 7,30, 8,33,16,38, 4,21)(15,35)$

Group invariants

Order:  $12996=2^{2} \cdot 3^{2} \cdot 19^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.