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Magma
magma: G := TransitiveGroup(46, 27);
Group action invariants
| Degree $n$: | $46$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2\times M_{23}$ | ||
| 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: | (1,34,26)(2,33,25)(3,15,11,24,7,36)(4,16,12,23,8,35)(5,27,14)(6,28,13)(9,44,42,20,39,37)(10,43,41,19,40,38)(17,31)(18,32)(21,45)(22,46), (1,12,46,13,17,2,11,45,14,18)(3,8,5,10,29,4,7,6,9,30)(15,19,44,21,27,16,20,43,22,28)(23,42,32,36,25,24,41,31,35,26)(33,34)(37,38)(39,40) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $10200960$: $M_{23}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 23: $M_{23}$
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
| Label | Cycle Type | Size | Order | Representative |
| $1^{46}$ | $1$ | $1$ | $()$ | |
| $2^{23}$ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44) (45,46)$ | |
| $2^{16},1^{14}$ | $3795$ | $2$ | $( 3,24)( 4,23)( 5,42)( 6,41)( 7,39)( 8,40)( 9,15)(10,16)(19,43)(20,44)(21,30) (22,29)(25,32)(26,31)(27,36)(28,35)$ | |
| $2^{23}$ | $3795$ | $2$ | $( 1, 2)( 3,23)( 4,24)( 5,41)( 6,42)( 7,40)( 8,39)( 9,16)(10,15)(11,12)(13,14) (17,18)(19,44)(20,43)(21,29)(22,30)(25,31)(26,32)(27,35)(28,36)(33,34)(37,38) (45,46)$ | |
| $4^{8},2^{4},1^{6}$ | $318780$ | $4$ | $( 1,17)( 2,18)( 3,31,24,26)( 4,32,23,25)( 5,27,42,36)( 6,28,41,35) ( 7,22,39,29)( 8,21,40,30)( 9,44,15,20)(10,43,16,19)(11,37)(12,38)$ | |
| $4^{8},2^{7}$ | $318780$ | $4$ | $( 1,18)( 2,17)( 3,32,24,25)( 4,31,23,26)( 5,28,42,35)( 6,27,41,36) ( 7,21,39,30)( 8,22,40,29)( 9,43,15,19)(10,44,16,20)(11,38)(12,37)(13,14) (33,34)(45,46)$ | |
| $8^{4},4^{2},2^{2},1^{2}$ | $1275120$ | $8$ | $( 1,11,17,37)( 2,12,18,38)( 3, 5,31,27,24,42,26,36)( 4, 6,32,28,23,41,25,35) ( 7,44,22,15,39,20,29, 9)( 8,43,21,16,40,19,30,10)(33,45)(34,46)$ | |
| $8^{4},4^{2},2^{3}$ | $1275120$ | $8$ | $( 1,12,17,38)( 2,11,18,37)( 3, 6,31,28,24,41,26,35)( 4, 5,32,27,23,42,25,36) ( 7,43,22,16,39,19,29,10)( 8,44,21,15,40,20,30, 9)(13,14)(33,46)(34,45)$ | |
| $14^{3},2^{2}$ | $728640$ | $14$ | $( 1, 6,11,30,14,32,17, 2, 5,12,29,13,31,18)( 3, 8,39,38,20,25,34, 4, 7,40,37, 19,26,33)( 9,23,15,28,44,45,22,10,24,16,27,43,46,21)(35,36)(41,42)$ | |
| $7^{6},1^{4}$ | $728640$ | $7$ | $( 1, 5,11,29,14,31,17)( 2, 6,12,30,13,32,18)( 3, 7,39,37,20,26,34) ( 4, 8,40,38,19,25,33)( 9,24,15,27,44,46,22)(10,23,16,28,43,45,21)$ | |
| $7^{6},1^{4}$ | $728640$ | $7$ | $( 1,17,31,14,29,11, 5)( 2,18,32,13,30,12, 6)( 3,34,26,20,37,39, 7) ( 4,33,25,19,38,40, 8)( 9,22,46,44,27,15,24)(10,21,45,43,28,16,23)$ | |
| $14^{3},2^{2}$ | $728640$ | $14$ | $( 1,18,31,13,29,12, 5, 2,17,32,14,30,11, 6)( 3,33,26,19,37,40, 7, 4,34,25,20, 38,39, 8)( 9,21,46,43,27,16,24,10,22,45,44,28,15,23)(35,36)(41,42)$ | |
| $14^{2},7^{2},2^{2}$ | $728640$ | $14$ | $( 1, 3, 5, 7,11,39,29,37,14,20,31,26,17,34)( 2, 4, 6, 8,12,40,30,38,13,19,32, 25,18,33)( 9,44,24,46,15,22,27)(10,43,23,45,16,21,28)(35,41)(36,42)$ | |
| $14^{3},2^{2}$ | $728640$ | $14$ | $( 1, 4, 5, 8,11,40,29,38,14,19,31,25,17,33)( 2, 3, 6, 7,12,39,30,37,13,20,32, 26,18,34)( 9,43,24,45,15,21,27,10,44,23,46,16,22,28)(35,42)(36,41)$ | |
| $14^{3},2^{2}$ | $728640$ | $14$ | $( 1, 8,29,19,17, 4,11,38,31,33, 5,40,14,25)( 2, 7,30,20,18, 3,12,37,32,34, 6, 39,13,26)( 9,45,27,23,22,43,15,10,46,28,24,21,44,16)(35,42)(36,41)$ | |
| $14^{2},7^{2},2^{2}$ | $728640$ | $14$ | $( 1, 7,29,20,17, 3,11,37,31,34, 5,39,14,26)( 2, 8,30,19,18, 4,12,38,32,33, 6, 40,13,25)( 9,46,27,24,22,44,15)(10,45,28,23,21,43,16)(35,41)(36,42)$ | |
| $6^{6},2^{5}$ | $56672$ | $6$ | $( 1, 2)( 3,12,42, 4,11,41)( 5,19, 7, 6,20, 8)( 9,30,14,10,29,13)(15,16) (17,38,44,18,37,43)(21,31,28,22,32,27)(23,24)(25,26)(33,46,35,34,45,36)(39,40)$ | |
| $3^{12},1^{10}$ | $56672$ | $3$ | $( 3,11,42)( 4,12,41)( 5,20, 7)( 6,19, 8)( 9,29,14)(10,30,13)(17,37,44) (18,38,43)(21,32,28)(22,31,27)(33,45,35)(34,46,36)$ | |
| $5^{8},1^{6}$ | $680064$ | $5$ | $( 1,39,24,15,26)( 2,40,23,16,25)( 5,37, 9,46,22)( 6,38,10,45,21) ( 7,17,14,34,27)( 8,18,13,33,28)(19,43,30,35,32)(20,44,29,36,31)$ | |
| $10^{4},2^{3}$ | $680064$ | $10$ | $( 1,40,24,16,26, 2,39,23,15,25)( 3, 4)( 5,38, 9,45,22, 6,37,10,46,21) ( 7,18,14,33,27, 8,17,13,34,28)(11,12)(19,44,30,36,32,20,43,29,35,31)(41,42)$ | |
| $30,10,6$ | $680064$ | $30$ | $( 1,23,26,40,15, 2,24,25,39,16)( 3,41,11, 4,42,12)( 5,13,31,38,34,19, 9,28,44, 45, 7,30,22,18,36, 6,14,32,37,33,20,10,27,43,46, 8,29,21,17,35)$ | |
| $15^{2},5^{2},3^{2}$ | $680064$ | $15$ | $( 1,24,26,39,15)( 2,23,25,40,16)( 3,42,11)( 4,41,12)( 5,14,31,37,34,20, 9,27, 44,46, 7,29,22,17,36)( 6,13,32,38,33,19,10,28,43,45, 8,30,21,18,35)$ | |
| $15^{2},5^{2},3^{2}$ | $680064$ | $15$ | $( 1,24,26,39,15)( 2,23,25,40,16)( 3,11,42)( 4,12,41)( 5,29,27,37,36, 7, 9,31, 17,46,20,14,22,44,34)( 6,30,28,38,35, 8,10,32,18,45,19,13,21,43,33)$ | |
| $30,10,6$ | $680064$ | $30$ | $( 1,23,26,40,15, 2,24,25,39,16)( 3,12,42, 4,11,41)( 5,30,27,38,36, 8, 9,32,17, 45,20,13,22,43,34, 6,29,28,37,35, 7,10,31,18,46,19,14,21,44,33)$ | |
| $11^{4},1^{2}$ | $927360$ | $11$ | $( 1, 9,46,31, 5,24,14,17,11,22,37)( 2,10,45,32, 6,23,13,18,12,21,38) ( 3, 7,39,29,36,27,20,15,42,34,44)( 4, 8,40,30,35,28,19,16,41,33,43)$ | |
| $22^{2},2$ | $927360$ | $22$ | $( 1,10,46,32, 5,23,14,18,11,21,37, 2, 9,45,31, 6,24,13,17,12,22,38) ( 3, 8,39,30,36,28,20,16,42,33,44, 4, 7,40,29,35,27,19,15,41,34,43)(25,26)$ | |
| $22^{2},2$ | $927360$ | $22$ | $( 1,38,22,12,17,13,24, 6,31,45, 9, 2,37,21,11,18,14,23, 5,32,46,10) ( 3,43,34,41,15,19,27,35,29,40, 7, 4,44,33,42,16,20,28,36,30,39, 8)(25,26)$ | |
| $11^{4},1^{2}$ | $927360$ | $11$ | $( 1,37,22,11,17,14,24, 5,31,46, 9)( 2,38,21,12,18,13,23, 6,32,45,10) ( 3,44,34,42,15,20,27,36,29,39, 7)( 4,43,33,41,16,19,28,35,30,40, 8)$ | |
| $6^{4},3^{4},2^{4},1^{2}$ | $850080$ | $6$ | $( 1,31,22,20,14,17)( 2,32,21,19,13,18)( 3,44)( 4,43)( 5,24,11)( 6,23,12) ( 7,46,34)( 8,45,33)( 9,29,39,36,42,26)(10,30,40,35,41,25)(15,37)(16,38)$ | |
| $6^{6},2^{5}$ | $850080$ | $6$ | $( 1,32,22,19,14,18)( 2,31,21,20,13,17)( 3,43)( 4,44)( 5,23,11, 6,24,12) ( 7,45,34, 8,46,33)( 9,30,39,35,42,25)(10,29,40,36,41,26)(15,38)(16,37)(27,28)$ | |
| $46$ | $443520$ | $46$ | $( 1,43,14, 6,24,12, 9,32,27,18,26,21,42,45,15,40,36,30, 7,33, 3,19,37, 2,44, 13, 5,23,11,10,31,28,17,25,22,41,46,16,39,35,29, 8,34, 4,20,38)$ | |
| $23^{2}$ | $443520$ | $23$ | $( 1,44,14, 5,24,11, 9,31,27,17,26,22,42,46,15,39,36,29, 7,34, 3,20,37) ( 2,43,13, 6,23,12,10,32,28,18,25,21,41,45,16,40,35,30, 8,33, 4,19,38)$ | |
| $23^{2}$ | $443520$ | $23$ | $( 1,37,20, 3,34, 7,29,36,39,15,46,42,22,26,17,27,31, 9,11,24, 5,14,44) ( 2,38,19, 4,33, 8,30,35,40,16,45,41,21,25,18,28,32,10,12,23, 6,13,43)$ | |
| $46$ | $443520$ | $46$ | $( 1,38,20, 4,34, 8,29,35,39,16,46,41,22,25,17,28,31,10,11,23, 5,13,44, 2,37, 19, 3,33, 7,30,36,40,15,45,42,21,26,18,27,32, 9,12,24, 6,14,43)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $20401920=2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 23$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 20401920.a | magma: IdentifyGroup(G);
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| Character table: | 34 x 34 character table |
magma: CharacterTable(G);