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Magma
magma: G := TransitiveGroup(46, 6);
Group action invariants
| Degree $n$: | $46$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $6$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2\times F_{23}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,39,21,15,13,44,7,42,37,6,25,2,40,22,16,14,43,8,41,38,5,26)(3,10,12,28,17,30,33,20,46,23,31,4,9,11,27,18,29,34,19,45,24,32)(35,36), (1,42,46,28,40,32,21,44,13,34,35,4,9,6,24,11,19,30,7,38,17,15)(2,41,45,27,39,31,22,43,14,33,36,3,10,5,23,12,20,29,8,37,18,16)(25,26) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $11$: $C_{11}$ $22$: 22T1 x 3 $44$: 44T2 $506$: $F_{23}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 23: $F_{23}$
Low degree siblings
46T6Siblings 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 $ | $22$ | $23$ | $( 1, 3, 5, 7, 9,12,13,16,17,19,21,24,25,27,29,31,33,35,37,40,41,43,46) ( 2, 4, 6, 8,10,11,14,15,18,20,22,23,26,28,30,32,34,36,38,39,42,44,45)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,37, 5,27, 9, 7,17,13,33,25,19)( 4,38, 6,28,10, 8,18,14,34,26,20) (11,44,22,39,42,32,36,15,23,30,45)(12,43,21,40,41,31,35,16,24,29,46)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3, 5, 9,17,33,19,37,27, 7,13,25)( 4, 6,10,18,34,20,38,28, 8,14,26) (11,22,42,36,23,45,44,39,32,15,30)(12,21,41,35,24,46,43,40,31,16,29)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3, 9,33,37, 7,25, 5,17,19,27,13)( 4,10,34,38, 8,26, 6,18,20,28,14) (11,42,23,44,32,30,22,36,45,39,15)(12,41,24,43,31,29,21,35,46,40,16)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,33, 7, 5,19,13, 9,37,25,17,27)( 4,34, 8, 6,20,14,10,38,26,18,28) (11,23,32,22,45,15,42,44,30,36,39)(12,24,31,21,46,16,41,43,29,35,40)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3, 7,19, 9,25,27,33, 5,13,37,17)( 4, 8,20,10,26,28,34, 6,14,38,18) (11,32,45,42,30,39,23,22,15,44,36)(12,31,46,41,29,40,24,21,16,43,35)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,19,25,33,13,17, 7, 9,27, 5,37)( 4,20,26,34,14,18, 8,10,28, 6,38) (11,45,30,23,15,36,32,42,39,22,44)(12,46,29,24,16,35,31,41,40,21,43)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,25,13, 7,27,37,19,33,17, 9, 5)( 4,26,14, 8,28,38,20,34,18,10, 6) (11,30,15,32,39,44,45,23,36,42,22)(12,29,16,31,40,43,46,24,35,41,21)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,13,27,19,17, 5,25, 7,37,33, 9)( 4,14,28,20,18, 6,26, 8,38,34,10) (11,15,39,45,36,22,30,32,44,23,42)(12,16,40,46,35,21,29,31,43,24,41)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,27,17,25,37, 9,13,19, 5, 7,33)( 4,28,18,26,38,10,14,20, 6, 8,34) (11,39,36,30,44,42,15,45,22,32,23)(12,40,35,29,43,41,16,46,21,31,24)$ |
| $ 11, 11, 11, 11, 1, 1 $ | $23$ | $11$ | $( 3,17,37,13, 5,33,27,25, 9,19, 7)( 4,18,38,14, 6,34,28,26,10,20, 8) (11,36,44,15,22,23,39,30,42,45,32)(12,35,43,16,21,24,40,29,41,46,31)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,39,21,15,13,44, 7,42,37, 6,25, 2,40,22,16,14,43, 8,41,38, 5,26) ( 3,10,12,28,17,30,33,20,46,23,31, 4, 9,11,27,18,29,34,19,45,24,32)(35,36)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,42, 5,20,12,10,43,18,46,30,25, 2,41, 6,19,11, 9,44,17,45,29,26) ( 3, 8,31,38,27,14,21,23,35,15,33, 4, 7,32,37,28,13,22,24,36,16,34)(39,40)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,32, 5, 4,17,11, 7,36,24,15,25, 2,31, 6, 3,18,12, 8,35,23,16,26) ( 9,22,29,20,43,14,40,42,27,34,37,10,21,30,19,44,13,39,41,28,33,38)(45,46)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,15, 3,34,27,20,40,36,46,44,25, 2,16, 4,33,28,19,39,35,45,43,26)( 5, 6) ( 7,23,29,38,17,22,12,14,31,10,41, 8,24,30,37,18,21,11,13,32, 9,42)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,11, 3,38,19,15, 9,23,21,42,25, 2,12, 4,37,20,16,10,24,22,41,26) ( 5,18,35,39,46,32,33,14,29, 8,43, 6,17,36,40,45,31,34,13,30, 7,44)(27,28)$ |
| $ 46 $ | $22$ | $46$ | $( 1,23,46,22,43,20,41,18,40,15,37,14,35,11,33,10,31, 8,29, 6,27, 4,25, 2,24, 45,21,44,19,42,17,39,16,38,13,36,12,34, 9,32, 7,30, 5,28, 3,26)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44) (45,46)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,22,35,32,19,30,13,11, 5,34,25, 2,21,36,31,20,29,14,12, 6,33,26) ( 3,28, 7,39,43,10,46,15,17,23,41, 4,27, 8,40,44, 9,45,16,18,24,42)(37,38)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,44,27,10,29,18,16, 8,21,32,25, 2,43,28, 9,30,17,15, 7,22,31,26) ( 3, 6,13,45,35,42,19,23,40,11,37, 4, 5,14,46,36,41,20,24,39,12,38)(33,34)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,45,41,34,17,32,13,23,43,38,25, 2,46,42,33,18,31,14,24,44,37,26)( 3, 4) ( 5, 8,12,20,35,22,40,30, 9,15,27, 6, 7,11,19,36,21,39,29,10,16,28)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,36,29, 4,13,42, 9,39,31,28,25, 2,35,30, 3,14,41,10,40,32,27,26) ( 5,38, 7,15,19,22,46,11,17,44,33, 6,37, 8,16,20,21,45,12,18,43,34)(23,24)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,30,27,38,33, 8,46,39,24,11,25, 2,29,28,37,34, 7,45,40,23,12,26) ( 3,20,31,18,41,14,16, 6, 9,36,43, 4,19,32,17,42,13,15, 5,10,35,44)(21,22)$ |
| $ 22, 22, 1, 1 $ | $23$ | $22$ | $( 3,21,17,24,37,40,13,29, 5,41,33,46,27,31,25,12, 9,35,19,43, 7,16) ( 4,22,18,23,38,39,14,30, 6,42,34,45,28,32,26,11,10,36,20,44, 8,15)$ |
| $ 22, 22, 1, 1 $ | $23$ | $22$ | $( 3,40,33,12, 7,24, 5,31,19,21,13,46, 9,16,37,41,25,43,17,29,27,35) ( 4,39,34,11, 8,23, 6,32,20,22,14,45,10,15,38,42,26,44,18,30,28,36)$ |
| $ 22, 22, 1, 1 $ | $23$ | $22$ | $( 3,41,19,40,25,21,33,43,13,12,17,46, 7,29, 9,24,27,16, 5,35,37,31) ( 4,42,20,39,26,22,34,44,14,11,18,45, 8,30,10,23,28,15, 6,36,38,32)$ |
| $ 22, 22, 1, 1 $ | $23$ | $22$ | $( 3,35,27,29,17,43,25,41,37,16, 9,46,13,21,19,31, 5,24, 7,12,33,40) ( 4,36,28,30,18,44,26,42,38,15,10,45,14,22,20,32, 6,23, 8,11,34,39)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $23$ | $2$ | $( 3,46)( 4,45)( 5,43)( 6,44)( 7,41)( 8,42)( 9,40)(10,39)(11,38)(12,37)(13,35) (14,36)(15,34)(16,33)(17,31)(18,32)(19,29)(20,30)(21,27)(22,28)(23,26)(24,25)$ |
| $ 22, 22, 1, 1 $ | $23$ | $22$ | $( 3,16, 7,43,19,35, 9,12,25,31,27,46,33,41, 5,29,13,40,37,24,17,21) ( 4,15, 8,44,20,36,10,11,26,32,28,45,34,42, 6,30,14,39,38,23,18,22)$ |
| $ 22, 22, 1, 1 $ | $23$ | $22$ | $( 3,43, 9,31,33,29,37,21, 7,35,25,46, 5,40,17,16,19,12,27,41,13,24) ( 4,44,10,32,34,30,38,22, 8,36,26,45, 6,39,18,15,20,11,28,42,14,23)$ |
| $ 22, 22, 1, 1 $ | $23$ | $22$ | $( 3,12, 5,21, 9,41,17,35,33,24,19,46,37,43,27,40, 7,31,13,16,25,29) ( 4,11, 6,22,10,42,18,36,34,23,20,45,38,44,28,39, 8,32,14,15,26,30)$ |
| $ 22, 22, 1, 1 $ | $23$ | $22$ | $( 3,29,25,16,13,31, 7,40,27,43,37,46,19,24,33,35,17,41, 9,21, 5,12) ( 4,30,26,15,14,32, 8,39,28,44,38,45,20,23,34,36,18,42,10,22, 6,11)$ |
| $ 22, 22, 1, 1 $ | $23$ | $22$ | $( 3,31,37,35, 5,16,27,24, 9,29, 7,46,17,12,13,43,33,21,25,40,19,41) ( 4,32,38,36, 6,15,28,23,10,30, 8,45,18,11,14,44,34,22,26,39,20,42)$ |
| $ 22, 22, 1, 1 $ | $23$ | $22$ | $( 3,24,13,41,27,12,19,16,17,40, 5,46,25,35, 7,21,37,29,33,31, 9,43) ( 4,23,14,42,28,11,20,15,18,39, 6,45,26,36, 8,22,38,30,34,32,10,44)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,39,43,42,19, 8,13,34,24, 6,37,22,29,26,27, 4,16,10,35,45,17,32) ( 2,40,44,41,20, 7,14,33,23, 5,38,21,30,25,28, 3,15, 9,36,46,18,31)(11,12)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,42, 3,23,27,38,40,22,46,14,25,10,16, 8,33,30,19,18,35,11,43,32) ( 2,41, 4,24,28,37,39,21,45,13,26, 9,15, 7,34,29,20,17,36,12,44,31)( 5, 6)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $23$ | $2$ | $( 1,32)( 2,31)( 3,30)( 4,29)( 5,28)( 6,27)( 7,26)( 8,25)( 9,23)(10,24)(11,21) (12,22)(13,20)(14,19)(15,17)(16,18)(33,45)(34,46)(35,44)(36,43)(37,42)(38,41) (39,40)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,15,33,44,24,18,29, 6, 7, 4,12,42,27,10,46,20,25,14,37,36,40,32) ( 2,16,34,43,23,17,30, 5, 8, 3,11,41,28, 9,45,19,26,13,38,35,39,31)(21,22)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,11,24,20, 5,26, 3,42,13, 8, 9,39,29,18,21,36,16,38,46,28,33,32) ( 2,12,23,19, 6,25, 4,41,14, 7,10,40,30,17,22,35,15,37,45,27,34,31)(43,44)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,23,13, 6,17,45, 3,44,29,28, 7,38,16,26,33,22,40,36,41,10,12,32) ( 2,24,14, 5,18,46, 4,43,30,27, 8,37,15,25,34,21,39,35,42, 9,11,31)(19,20)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,22,24,38,43,39,12,45, 7,18,41,26, 5, 4,35,30,33,15,27,20, 9,32) ( 2,21,23,37,44,40,11,46, 8,17,42,25, 6, 3,36,29,34,16,28,19,10,31)(13,14)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,44,21,15, 5,20,27,26,37,11,29,14,17,39,46,10,41,34,35,23, 3,32) ( 2,43,22,16, 6,19,28,25,38,12,30,13,18,40,45, 9,42,33,36,24, 4,31)( 7, 8)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,45, 5,34,41,18,43,11,16, 4,40,23,25,20,37,30, 7,28,13,10,21,32) ( 2,46, 6,33,42,17,44,12,15, 3,39,24,26,19,38,29, 8,27,14, 9,22,31)(35,36)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,36,21,44,16,14, 3,45,25,18,24, 8,19,34,12,39,41, 6, 9,30,37,32) ( 2,35,22,43,15,13, 4,46,26,17,23, 7,20,33,11,40,42, 5,10,29,38,31)(27,28)$ |
| $ 22, 22, 2 $ | $23$ | $22$ | $( 1,30, 9,44,46,38,24,34,40,15,19, 4,21,42, 7, 6,13,28,17,11,35,32) ( 2,29,10,43,45,37,23,33,39,16,20, 3,22,41, 8, 5,14,27,18,12,36,31)(25,26)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $1012=2^{2} \cdot 11 \cdot 23$ | 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: | 1012.7 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);