Show commands:
Magma
magma: G := TransitiveGroup(46, 5);
Group action invariants
| Degree $n$: | $46$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $5$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $F_{23}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,13,42,45,39,26,23,3,34,11,22,29,17,35,32,37,6,7,28,44,19,9)(2,14,41,46,40,25,24,4,33,12,21,30,18,36,31,38,5,8,27,43,20,10)(15,16), (1,22,36,32,19,30,14,12,6,34,25)(2,21,35,31,20,29,13,11,5,33,26)(3,27,7,40,44,9,45,16,18,24,41)(4,28,8,39,43,10,46,15,17,23,42) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $11$: $C_{11}$ $22$: 22T1 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 23: $F_{23}$
Low degree siblings
23T4Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Cycle Type | Size | Order | Representative |
| $ 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 $ | $1$ | $1$ | $()$ |
| $ 23, 23 $ | $22$ | $23$ | $( 1,42,36,30,23,17,12, 6,46,39,34,28,22,15,10, 4,43,38,32,25,19,14, 8) ( 2,41,35,29,24,18,11, 5,45,40,33,27,21,16, 9, 3,44,37,31,26,20,13, 7)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,18,37,13, 5,33,27,26, 9,20, 7)( 4,17,38,14, 6,34,28,25,10,19, 8) (11,35,44,16,21,24,40,29,41,45,31)(12,36,43,15,22,23,39,30,42,46,32)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,37, 5,27, 9, 7,18,13,33,26,20)( 4,38, 6,28,10, 8,17,14,34,25,19) (11,44,21,40,41,31,35,16,24,29,45)(12,43,22,39,42,32,36,15,23,30,46)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3, 5, 9,18,33,20,37,27, 7,13,26)( 4, 6,10,17,34,19,38,28, 8,14,25) (11,21,41,35,24,45,44,40,31,16,29)(12,22,42,36,23,46,43,39,32,15,30)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3, 9,33,37, 7,26, 5,18,20,27,13)( 4,10,34,38, 8,25, 6,17,19,28,14) (11,41,24,44,31,29,21,35,45,40,16)(12,42,23,43,32,30,22,36,46,39,15)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,33, 7, 5,20,13, 9,37,26,18,27)( 4,34, 8, 6,19,14,10,38,25,17,28) (11,24,31,21,45,16,41,44,29,35,40)(12,23,32,22,46,15,42,43,30,36,39)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3, 7,20, 9,26,27,33, 5,13,37,18)( 4, 8,19,10,25,28,34, 6,14,38,17) (11,31,45,41,29,40,24,21,16,44,35)(12,32,46,42,30,39,23,22,15,43,36)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,20,26,33,13,18, 7, 9,27, 5,37)( 4,19,25,34,14,17, 8,10,28, 6,38) (11,45,29,24,16,35,31,41,40,21,44)(12,46,30,23,15,36,32,42,39,22,43)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,26,13, 7,27,37,20,33,18, 9, 5)( 4,25,14, 8,28,38,19,34,17,10, 6) (11,29,16,31,40,44,45,24,35,41,21)(12,30,15,32,39,43,46,23,36,42,22)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,13,27,20,18, 5,26, 7,37,33, 9)( 4,14,28,19,17, 6,25, 8,38,34,10) (11,16,40,45,35,21,29,31,44,24,41)(12,15,39,46,36,22,30,32,43,23,42)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,27,18,26,37, 9,13,20, 5, 7,33)( 4,28,17,25,38,10,14,19, 6, 8,34) (11,40,35,29,44,41,16,45,21,31,24)(12,39,36,30,43,42,15,46,22,32,23)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,13,42,45,39,26,23, 3,34,11,22,29,17,35,32,37, 6, 7,28,44,19, 9) ( 2,14,41,46,40,25,24, 4,33,12,21,30,18,36,31,38, 5, 8,27,43,20,10)(15,16)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1, 5, 4,27,15,21,42,31,14,45,30,37,34,35,12,24,17,44, 8,26,39, 9) ( 2, 6, 3,28,16,22,41,32,13,46,29,38,33,36,11,23,18,43, 7,25,40,10)(19,20)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,33,43, 3,25,29,14,31, 6,18,15,24,38,27,22,45,42,11,39,20, 8, 9) ( 2,34,44, 4,26,30,13,32, 5,17,16,23,37,28,21,46,41,12,40,19, 7,10)(35,36)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,26,46,31,28,40, 4,20,17,24, 6,13,36,16,30,33,22,11,42,44,38, 9) ( 2,25,45,32,27,39, 3,19,18,23, 5,14,35,15,29,34,21,12,41,43,37,10)( 7, 8)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1, 3,38,18,46,16,12,35,30,20,34,41,39, 5,25,44,28,31, 8,13,23, 9) ( 2, 4,37,17,45,15,11,36,29,19,33,42,40, 6,26,43,27,32, 7,14,24,10)(21,22)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $23$ | $2$ | $( 1, 9)( 2,10)( 3, 8)( 4, 7)( 5, 6)(11,46)(12,45)(13,43)(14,44)(15,41)(16,42) (17,40)(18,39)(19,37)(20,38)(21,36)(22,35)(23,33)(24,34)(25,31)(26,32)(27,30) (28,29)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,37,14,29, 4, 5,19,26,22,40,28,35,46,24, 8,33,32,18,12,16,43, 9) ( 2,38,13,30, 3, 6,20,25,21,39,27,36,45,23, 7,34,31,17,11,15,44,10)(41,42)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,18,32, 3,14,40,34,45,22,24,19,27,12,44,25,16,36,41,30, 7, 6, 9) ( 2,17,31, 4,13,39,33,46,21,23,20,28,11,43,26,15,35,42,29, 8, 5,10)(37,38)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1, 7,38, 3,17,41,23,26,36,40,14,21,15,31,19, 5,28,45,43,33,30, 9) ( 2, 8,37, 4,18,42,24,25,35,39,13,22,16,32,20, 6,27,46,44,34,29,10)(11,12)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,27,23,13,12,29, 6,37,25,41,36,44,17,21,32,33,15,40, 8,20, 4, 9) ( 2,28,24,14,11,30, 5,38,26,42,35,43,18,22,31,34,16,39, 7,19, 3,10)(45,46)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,20,14,16,46,35,23,27,42,21,43, 5,34,40,38, 7,17,29,25,11,32, 9) ( 2,19,13,15,45,36,24,28,41,22,44, 6,33,39,37, 8,18,30,26,12,31,10)( 3, 4)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $506=2 \cdot 11 \cdot 23$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 506.1 | magma: IdentifyGroup(G);
|
| Character table: not available. |
magma: CharacterTable(G);