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Magma
magma: G := TransitiveGroup(46, 12);
Group action invariants
| Degree $n$: | $46$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,26,9,41,13,37,15,35,16,34,5,45,11,39,14,36,4,46,22,28,8,42)(2,25,21,29,19,31,18,32,6,44,23,27,20,30,7,43,12,38,3,24,10,40)(17,33), (1,8,4,3,20,7,21,13,11,22,19)(2,14,17,12,5,9,10,16,6,15,23)(24,31,27,26,43,30,44,36,34,45,42)(25,37,40,35,28,32,33,39,29,38,46) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $11$: $C_{11}$ $22$: 22T1 $253$: $C_{23}:C_{11}$ $506$: $F_{23}$, 46T4 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 23: None
Low degree siblings
46T12 x 10Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | 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 $ | $22$ | $23$ | $( 1, 8,15,22, 6,13,20, 4,11,18, 2, 9,16,23, 7,14,21, 5,12,19, 3,10,17) (24,34,44,31,41,28,38,25,35,45,32,42,29,39,26,36,46,33,43,30,40,27,37)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1,13, 2,14, 3,15, 4,16, 5,17, 6,18, 7,19, 8,20, 9,21,10,22,11,23,12) (24,28,32,36,40,44,25,29,33,37,41,45,26,30,34,38,42,46,27,31,35,39,43)$ | |
| $ 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $22$ | $23$ | $(24,34,44,31,41,28,38,25,35,45,32,42,29,39,26,36,46,33,43,30,40,27,37)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1, 8,15,22, 6,13,20, 4,11,18, 2, 9,16,23, 7,14,21, 5,12,19, 3,10,17) (24,44,41,38,35,32,29,26,46,43,40,37,34,31,28,25,45,42,39,36,33,30,27)$ | |
| $ 23, 23 $ | $11$ | $23$ | $( 1,13, 2,14, 3,15, 4,16, 5,17, 6,18, 7,19, 8,20, 9,21,10,22,11,23,12) (24,38,29,43,34,25,39,30,44,35,26,40,31,45,36,27,41,32,46,37,28,42,33)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1,15, 6,20,11, 2,16, 7,21,12, 3,17, 8,22,13, 4,18, 9,23,14, 5,19,10) (24,31,38,45,29,36,43,27,34,41,25,32,39,46,30,37,44,28,35,42,26,33,40)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23) (24,42,37,32,27,45,40,35,30,25,43,38,33,28,46,41,36,31,26,44,39,34,29)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1, 6,11,16,21, 3, 8,13,18,23, 5,10,15,20, 2, 7,12,17,22, 4, 9,14,19) (24,28,32,36,40,44,25,29,33,37,41,45,26,30,34,38,42,46,27,31,35,39,43)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23, 2, 4, 6, 8,10,12,14,16,18,20,22) (24,27,30,33,36,39,42,45,25,28,31,34,37,40,43,46,26,29,32,35,38,41,44)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1, 5, 9,13,17,21, 2, 6,10,14,18,22, 3, 7,11,15,19,23, 4, 8,12,16,20) (24,43,39,35,31,27,46,42,38,34,30,26,45,41,37,33,29,25,44,40,36,32,28)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1,21,18,15,12, 9, 6, 3,23,20,17,14,11, 8, 5, 2,22,19,16,13,10, 7, 4) (24,33,42,28,37,46,32,41,27,36,45,31,40,26,35,44,30,39,25,34,43,29,38)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1, 9,17, 2,10,18, 3,11,19, 4,12,20, 5,13,21, 6,14,22, 7,15,23, 8,16) (24,29,34,39,44,26,31,36,41,46,28,33,38,43,25,30,35,40,45,27,32,37,42)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1,12,23,11,22,10,21, 9,20, 8,19, 7,18, 6,17, 5,16, 4,15, 3,14, 2,13) (24,30,36,42,25,31,37,43,26,32,38,44,27,33,39,45,28,34,40,46,29,35,41)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1,23,22,21,20,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2) (24,26,28,30,32,34,36,38,40,42,44,46,25,27,29,31,33,35,37,39,41,43,45)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1,22,20,18,16,14,12,10, 8, 6, 4, 2,23,21,19,17,15,13,11, 9, 7, 5, 3) (24,41,35,29,46,40,34,28,45,39,33,27,44,38,32,26,43,37,31,25,42,36,30)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1,14, 4,17, 7,20,10,23,13, 3,16, 6,19, 9,22,12, 2,15, 5,18, 8,21,11) (24,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1,20,16,12, 8, 4,23,19,15,11, 7, 3,22,18,14,10, 6, 2,21,17,13, 9, 5) (24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1, 4, 7,10,13,16,19,22, 2, 5, 8,11,14,17,20,23, 3, 6, 9,12,15,18,21) (24,35,46,34,45,33,44,32,43,31,42,30,41,29,40,28,39,27,38,26,37,25,36)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1,16, 8,23,15, 7,22,14, 6,21,13, 5,20,12, 4,19,11, 3,18,10, 2,17, 9) (24,39,31,46,38,30,45,37,29,44,36,28,43,35,27,42,34,26,41,33,25,40,32)$ | |
| $ 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $22$ | $23$ | $(24,28,32,36,40,44,25,29,33,37,41,45,26,30,34,38,42,46,27,31,35,39,43)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1, 8,15,22, 6,13,20, 4,11,18, 2, 9,16,23, 7,14,21, 5,12,19, 3,10,17) (24,38,29,43,34,25,39,30,44,35,26,40,31,45,36,27,41,32,46,37,28,42,33)$ | |
| $ 23, 23 $ | $11$ | $23$ | $( 1,15, 6,20,11, 2,16, 7,21,12, 3,17, 8,22,13, 4,18, 9,23,14, 5,19,10) (24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23) (24,36,25,37,26,38,27,39,28,40,29,41,30,42,31,43,32,44,33,45,34,46,35)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23, 2, 4, 6, 8,10,12,14,16,18,20,22) (24,44,41,38,35,32,29,26,46,43,40,37,34,31,28,25,45,42,39,36,33,30,27)$ | |
| $ 23, 23 $ | $22$ | $23$ | $( 1,11,21, 8,18, 5,15, 2,12,22, 9,19, 6,16, 3,13,23,10,20, 7,17, 4,14) (24,39,31,46,38,30,45,37,29,44,36,28,43,35,27,42,34,26,41,33,25,40,32)$ | |
| $ 11, 11, 11, 11, 1, 1 $ | $529$ | $11$ | $( 2,13, 7, 4,14,19,10,17, 9, 5, 3)( 6,15, 8,16,20,22,23,12,18,21,11) (25,36,30,27,37,42,33,40,32,28,26)(29,38,31,39,43,45,46,35,41,44,34)$ | |
| $ 11, 11, 11, 11, 1, 1 $ | $529$ | $11$ | $( 2, 7,14,10, 9, 3,13, 4,19,17, 5)( 6, 8,20,23,18,11,15,16,22,12,21) (25,30,37,33,32,26,36,27,42,40,28)(29,31,43,46,41,34,38,39,45,35,44)$ | |
| $ 11, 11, 11, 11, 1, 1 $ | $529$ | $11$ | $( 2,14, 9,13,19, 5, 7,10, 3, 4,17)( 6,20,18,15,22,21, 8,23,11,16,12) (25,37,32,36,42,28,30,33,26,27,40)(29,43,41,38,45,44,31,46,34,39,35)$ | |
| $ 11, 11, 11, 11, 1, 1 $ | $529$ | $11$ | $( 2, 9,19, 7, 3,17,14,13, 5,10, 4)( 6,18,22, 8,11,12,20,15,21,23,16) (25,32,42,30,26,40,37,36,28,33,27)(29,41,45,31,34,35,43,38,44,46,39)$ | |
| $ 11, 11, 11, 11, 1, 1 $ | $529$ | $11$ | $( 2,19, 3,14, 5, 4, 9, 7,17,13,10)( 6,22,11,20,21,16,18, 8,12,15,23) (25,42,26,37,28,27,32,30,40,36,33)(29,45,34,43,44,39,41,31,35,38,46)$ | |
| $ 11, 11, 11, 11, 1, 1 $ | $529$ | $11$ | $( 2, 3, 5, 9,17,10,19,14, 4, 7,13)( 6,11,21,18,12,23,22,20,16, 8,15) (25,26,28,32,40,33,42,37,27,30,36)(29,34,44,41,35,46,45,43,39,31,38)$ | |
| $ 11, 11, 11, 11, 1, 1 $ | $529$ | $11$ | $( 2, 5,17,19, 4,13, 3, 9,10,14, 7)( 6,21,12,22,16,15,11,18,23,20, 8) (25,28,40,42,27,36,26,32,33,37,30)(29,44,35,45,39,38,34,41,46,43,31)$ | |
| $ 11, 11, 11, 11, 1, 1 $ | $529$ | $11$ | $( 2,17, 4, 3,10, 7, 5,19,13, 9,14)( 6,12,16,11,23, 8,21,22,15,18,20) (25,40,27,26,33,30,28,42,36,32,37)(29,35,39,34,46,31,44,45,38,41,43)$ | |
| $ 11, 11, 11, 11, 1, 1 $ | $529$ | $11$ | $( 2, 4,10, 5,13,14,17, 3, 7,19, 9)( 6,16,23,21,15,20,12,11, 8,22,18) (25,27,33,28,36,37,40,26,30,42,32)(29,39,46,44,38,43,35,34,31,45,41)$ | |
| $ 11, 11, 11, 11, 1, 1 $ | $529$ | $11$ | $( 2,10,13,17, 7, 9, 4, 5,14, 3,19)( 6,23,15,12, 8,18,16,21,20,11,22) (25,33,36,40,30,32,27,28,37,26,42)(29,46,38,35,31,41,39,44,43,34,45)$ | |
| $ 22, 22, 2 $ | $529$ | $22$ | $( 1,26, 9,41,13,37,15,35,16,34, 5,45,11,39,14,36, 4,46,22,28, 8,42) ( 2,25,21,29,19,31,18,32, 6,44,23,27,20,30, 7,43,12,38, 3,24,10,40)(17,33)$ | |
| $ 22, 22, 2 $ | $529$ | $22$ | $( 1,25,11,43,18,28,16,29,10,32,15,41, 7,45, 6,34, 3,24,17,40,13,42) ( 2,36,14,30, 4,35,20,27,22,26, 5,46,23,37, 8,33, 9,44,12,31,21,38)(19,39)$ | |
| $ 22, 22, 2 $ | $529$ | $22$ | $( 1,36,19,43,21,31,11,45,15,44,18,26, 3,24, 9,34, 2,30,14,27,23,42)( 4,41) ( 5,35,22,25, 6,29,17,32, 8,40, 7,46,12,39,10,28,20,37,16,38,13,33)$ | |
| $ 22, 22, 2 $ | $529$ | $22$ | $( 1,27,18,36,17,26,13,32,20,33, 2,37,22,30,10,25, 8,28,23,40,14,42)( 3,24) ( 4,34, 7,41,19,46,21,43, 6,31,15,29, 5,44,11,35,12,45,16,39, 9,38)$ | |
| $ 22, 22, 2 $ | $529$ | $22$ | $( 1,40, 9,45,12,44,16,35, 6,46, 8,30, 3,24, 4,39,13,36, 2,32,18,42) ( 5,31,22,33,14,28,11,29, 7,38,17,27,15,43,20,26,19,34,10,37,21,41)(23,25)$ | |
| $ 46 $ | $253$ | $46$ | $( 1,37,23,32,22,27,21,45,20,40,19,35,18,30,17,25,16,43,15,38,14,33,13,28,12, 46,11,41,10,36, 9,31, 8,26, 7,44, 6,39, 5,34, 4,29, 3,24, 2,42)$ | |
| $ 46 $ | $253$ | $46$ | $( 1,24, 9,41,17,35, 2,29,10,46,18,40, 3,34,11,28,19,45, 4,39,12,33,20,27, 5, 44,13,38,21,32, 6,26,14,43,22,37, 7,31,15,25,23,42, 8,36,16,30)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $23$ | $2$ | $( 1,33)( 2,38)( 3,43)( 4,25)( 5,30)( 6,35)( 7,40)( 8,45)( 9,27)(10,32)(11,37) (12,42)(13,24)(14,29)(15,34)(16,39)(17,44)(18,26)(19,31)(20,36)(21,41)(22,46) (23,28)$ | |
| $ 22, 22, 2 $ | $529$ | $22$ | $( 1,28,15,46,20,36, 7,39, 4,45,21,34, 9,35, 8,37, 6,41, 2,26,17,42) ( 3,24,19,38, 5,43,23,30,13,27,16,44,22,32,11,31,12,29,14,25,18,40)(10,33)$ | |
| $ 22, 22, 2 $ | $529$ | $22$ | $( 1,32,21,44,20,25,12,34,17,37,11,38, 9,46,16,41, 3,24,14,26,10,42) ( 2,28, 6,35,15,45,18,33,19,29, 4,43,22,40, 5,39, 7,31,23,36,13,30)( 8,27)$ | |
| $ 22, 22, 2 $ | $529$ | $22$ | $( 1,33, 4,31,20,28,21,35,11,34,19,44, 8,36, 3,24, 7,29,13,25,22,42) ( 2,40,17,30, 5,38,10,27, 6,45,23,26,14,32,12,41, 9,43,16,46,15,39)(18,37)$ | |
| $ 22, 22, 2 $ | $529$ | $22$ | $( 1,30,19,45, 8,32,16,31, 6,38, 7,35,23,33, 3,24, 5,41,14,37,20,42) ( 2,27,12,43,11,46,18,25,15,34,13,40, 4,44,21,39,17,28,22,36,10,26)( 9,29)$ | |
| $ 22, 22, 2 $ | $529$ | $22$ | $( 1,42)( 2,33, 7,34,14,40,10,30, 9,39, 3,24,13,26, 4,38,19,41,17,36, 5,29) ( 6,43, 8,25,20,32,23,28,18,27,11,44,15,31,16,45,22,37,12,35,21,46)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $11638=2 \cdot 11 \cdot 23^{2}$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 11638.7 | magma: IdentifyGroup(G);
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| Character table: | not computed |
magma: CharacterTable(G);