Properties

Label 46T25
Order \(518144\)
n \(46\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $46$
Transitive number $t$ :  $25$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,29,36,44,23,28,17,19,37,16)(2,14,30,35,43,24,27,18,20,38,15)(3,32,8,22,9,39,34,26,45,41,6)(4,31,7,21,10,40,33,25,46,42,5), (1,10,22,40,44,3,36,37,18,33,11)(2,9,21,39,43,4,35,38,17,34,12)(5,16,7,42,23,20,14,28,25,46,30)(6,15,8,41,24,19,13,27,26,45,29)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
11:  $C_{11}$
253:  $C_{23}:C_{11}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 23: $C_{23}:C_{11}$

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, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $253$ $2$ $( 1, 2)( 9,10)(17,18)(19,20)(21,22)(31,32)(33,34)(35,36)$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $253$ $2$ $( 1, 2)( 7, 8)( 9,10)(11,12)(17,18)(25,26)(35,36)(43,44)$
$ 2, 2, 2, 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, 1, 1 $ $253$ $2$ $( 1, 2)( 7, 8)(13,14)(17,18)(21,22)(25,26)(31,32)(33,34)(35,36)(37,38)(39,40) (43,44)$
$ 2, 2, 2, 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, 1, 1 $ $253$ $2$ $( 1, 2)( 9,10)(15,16)(17,18)(19,20)(23,24)(29,30)(31,32)(35,36)(37,38)(39,40) (45,46)$
$ 2, 2, 2, 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, 1, 1 $ $253$ $2$ $( 1, 2)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(29,30)(33,34)(35,36) (45,46)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $253$ $2$ $( 1, 2)( 7, 8)( 9,10)(11,12)(15,16)(17,18)(21,22)(23,24)(25,26)(29,30)(33,34) (35,36)(37,38)(39,40)(43,44)(45,46)$
$ 2, 2, 2, 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, 1, 1 $ $23$ $2$ $( 1, 2)( 7, 8)( 9,10)(13,14)(17,18)(19,20)(21,22)(23,24)(25,26)(35,36)(39,40) (45,46)$
$ 2, 2, 2, 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, 1, 1 $ $253$ $2$ $( 1, 2)( 7, 8)(11,12)(17,18)(19,20)(23,24)(25,26)(31,32)(33,34)(35,36)(37,38) (45,46)$
$ 2, 2, 2, 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, 1, 1 $ $253$ $2$ $( 1, 2)( 7, 8)(11,12)(15,16)(17,18)(19,20)(21,22)(25,26)(29,30)(31,32)(35,36) (39,40)$
$ 23, 23 $ $22528$ $23$ $( 1,43,40,36,31,28,23,19,15,12, 7, 3,45,41,38,34,30,26,22,17,14,10, 5) ( 2,44,39,35,32,27,24,20,16,11, 8, 4,46,42,37,33,29,25,21,18,13, 9, 6)$
$ 23, 23 $ $22528$ $23$ $( 1,28, 7,34,14,40,19,45,26, 5,31,12,38,17,43,23, 3,30,10,36,15,41,22) ( 2,27, 8,33,13,39,20,46,25, 6,32,11,37,18,44,24, 4,29, 9,35,16,42,21)$
$ 11, 11, 11, 11, 1, 1 $ $23552$ $11$ $( 3,19,25,34,14,18, 8, 9,27, 6,38)( 4,20,26,33,13,17, 7,10,28, 5,37) (11,46,29,23,15,36,31,41,39,21,43)(12,45,30,24,16,35,32,42,40,22,44)$
$ 22, 11, 11, 2 $ $23552$ $22$ $( 1, 2)( 3,20,26,34,14,17, 7, 9,27, 6,38)( 4,19,25,33,13,18, 8,10,28, 5,37) (11,46,29,23,15,35,31,41,39,22,44,12,45,30,24,16,36,32,42,40,21,43)$
$ 11, 11, 11, 11, 1, 1 $ $23552$ $11$ $( 3,25,14, 8,27,38,19,34,18, 9, 6)( 4,26,13, 7,28,37,20,33,17,10, 5) (11,29,15,31,39,43,46,23,36,41,21)(12,30,16,32,40,44,45,24,35,42,22)$
$ 22, 11, 11, 2 $ $23552$ $22$ $( 1, 2)( 3,25,14, 8,27,38,20,34,17, 9, 6)( 4,26,13, 7,28,37,19,33,18,10, 5) (11,29,15,32,40,44,45,24,36,41,22,12,30,16,31,39,43,46,23,35,42,21)$
$ 11, 11, 11, 11, 1, 1 $ $23552$ $11$ $( 3,14,27,19,18, 6,25, 8,38,34, 9)( 4,13,28,20,17, 5,26, 7,37,33,10) (11,15,39,46,36,21,29,31,43,23,41)(12,16,40,45,35,22,30,32,44,24,42)$
$ 22, 11, 11, 2 $ $23552$ $22$ $( 1, 2)( 3,14,27,20,18, 6,25, 8,38,33, 9)( 4,13,28,19,17, 5,26, 7,37,34,10) (11,15,39,46,35,21,29,32,44,24,42,12,16,40,45,36,22,30,31,43,23,41)$
$ 11, 11, 11, 11, 1, 1 $ $23552$ $11$ $( 3,27,18,25,38, 9,14,19, 6, 8,34)( 4,28,17,26,37,10,13,20, 5, 7,33) (11,39,36,29,43,41,15,46,21,31,23)(12,40,35,30,44,42,16,45,22,32,24)$
$ 22, 11, 11, 2 $ $23552$ $22$ $( 1, 2)( 3,27,17,26,37, 9,14,20, 5, 7,34)( 4,28,18,25,38,10,13,19, 6, 8,33) (11,39,35,30,44,42,16,45,21,32,24,12,40,36,29,43,41,15,46,22,31,23)$
$ 11, 11, 11, 11, 1, 1 $ $23552$ $11$ $( 3,18,38,14, 6,34,27,25, 9,19, 8)( 4,17,37,13, 5,33,28,26,10,20, 7) (11,36,43,15,21,23,39,29,41,46,31)(12,35,44,16,22,24,40,30,42,45,32)$
$ 22, 11, 11, 2 $ $23552$ $22$ $( 1, 2)( 3,17,37,13, 5,34,27,25,10,19, 8)( 4,18,38,14, 6,33,28,26, 9,20, 7) (11,35,44,16,21,23,39,29,41,46,32,12,36,43,15,22,24,40,30,42,45,31)$
$ 11, 11, 11, 11, 1, 1 $ $23552$ $11$ $( 3,38, 6,27, 9, 8,18,14,34,25,19)( 4,37, 5,28,10, 7,17,13,33,26,20) (11,43,21,39,41,31,36,15,23,29,46)(12,44,22,40,42,32,35,16,24,30,45)$
$ 22, 11, 11, 2 $ $23552$ $22$ $( 1, 2)( 3,38, 6,27,10, 7,18,14,33,26,19)( 4,37, 5,28, 9, 8,17,13,34,25,20) (11,43,22,40,42,31,35,16,24,30,45,12,44,21,39,41,32,36,15,23,29,46)$
$ 11, 11, 11, 11, 1, 1 $ $23552$ $11$ $( 3, 6, 9,18,34,19,38,27, 8,14,25)( 4, 5,10,17,33,20,37,28, 7,13,26) (11,21,41,36,23,46,43,39,31,15,29)(12,22,42,35,24,45,44,40,32,16,30)$
$ 22, 11, 11, 2 $ $23552$ $22$ $( 1, 2)( 3, 6,10,18,33,19,38,27, 8,14,25)( 4, 5, 9,17,34,20,37,28, 7,13,26) (11,22,42,36,23,46,43,39,32,16,30,12,21,41,35,24,45,44,40,31,15,29)$
$ 11, 11, 11, 11, 1, 1 $ $23552$ $11$ $( 3, 9,34,38, 8,25, 6,18,19,27,14)( 4,10,33,37, 7,26, 5,17,20,28,13) (11,41,23,43,31,29,21,36,46,39,15)(12,42,24,44,32,30,22,35,45,40,16)$
$ 22, 11, 11, 2 $ $23552$ $22$ $( 1, 2)( 3,10,34,38, 8,25, 6,17,19,27,14)( 4, 9,33,37, 7,26, 5,18,20,28,13) (11,41,23,43,32,30,21,35,45,40,16,12,42,24,44,31,29,22,36,46,39,15)$
$ 11, 11, 11, 11, 1, 1 $ $23552$ $11$ $( 3,34, 8, 6,19,14, 9,38,25,18,27)( 4,33, 7, 5,20,13,10,37,26,17,28) (11,23,31,21,46,15,41,43,29,36,39)(12,24,32,22,45,16,42,44,30,35,40)$
$ 22, 11, 11, 2 $ $23552$ $22$ $( 1, 2)( 3,33, 7, 5,19,14,10,37,26,18,27)( 4,34, 8, 6,20,13, 9,38,25,17,28) (11,23,32,21,46,15,41,43,29,35,40,12,24,31,22,45,16,42,44,30,36,39)$
$ 11, 11, 11, 11, 1, 1 $ $23552$ $11$ $( 3, 8,19, 9,25,27,34, 6,14,38,18)( 4, 7,20,10,26,28,33, 5,13,37,17) (11,31,46,41,29,39,23,21,15,43,36)(12,32,45,42,30,40,24,22,16,44,35)$
$ 22, 11, 11, 2 $ $23552$ $22$ $( 1, 2)( 3, 8,20, 9,25,27,33, 5,13,37,18)( 4, 7,19,10,26,28,34, 6,14,38,17) (11,32,45,42,30,40,24,21,15,43,35,12,31,46,41,29,39,23,22,16,44,36)$

Group invariants

Order:  $518144=2^{11} \cdot 11 \cdot 23$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.