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Magma
magma: G := TransitiveGroup(45, 1);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{45}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $45$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,5,7,12,14,18,20,23,27,28,32,35,38,40,43,3,6,9,10,15,17,21,24,25,29,33,36,37,41,44,2,4,8,11,13,16,19,22,26,30,31,34,39,42,45) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $5$: $C_5$ $9$: $C_9$ $15$: $C_{15}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: $C_5$
Degree 9: $C_9$
Degree 15: $C_{15}$
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, 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 $ | $1$ | $1$ | $()$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,15,14)(16,17,18)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,38,39)(40,42,41) (43,45,44)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,11,12)(13,14,15)(16,18,17)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,39,38)(40,41,42) (43,44,45)$ | |
| $ 45 $ | $1$ | $45$ | $( 1, 4, 9,12,13,17,20,22,25,28,31,36,38,42,44, 3, 5, 8,10,14,16,21,23,26,29, 32,34,37,40,45, 2, 6, 7,11,15,18,19,24,27,30,33,35,39,41,43)$ | |
| $ 45 $ | $1$ | $45$ | $( 1, 5, 7,12,14,18,20,23,27,28,32,35,38,40,43, 3, 6, 9,10,15,17,21,24,25,29, 33,36,37,41,44, 2, 4, 8,11,13,16,19,22,26,30,31,34,39,42,45)$ | |
| $ 45 $ | $1$ | $45$ | $( 1, 6, 8,12,15,16,20,24,26,28,33,34,38,41,45, 3, 4, 7,10,13,18,21,22,27,29, 31,35,37,42,43, 2, 5, 9,11,14,17,19,23,25,30,32,36,39,40,44)$ | |
| $ 45 $ | $1$ | $45$ | $( 1, 7,14,20,27,32,38,43, 6,10,17,24,29,36,41, 2, 8,13,19,26,31,39,45, 5,12, 18,23,28,35,40, 3, 9,15,21,25,33,37,44, 4,11,16,22,30,34,42)$ | |
| $ 45 $ | $1$ | $45$ | $( 1, 8,15,20,26,33,38,45, 4,10,18,22,29,35,42, 2, 9,14,19,25,32,39,44, 6,12, 16,24,28,34,41, 3, 7,13,21,27,31,37,43, 5,11,17,23,30,36,40)$ | |
| $ 45 $ | $1$ | $45$ | $( 1, 9,13,20,25,31,38,44, 5,10,16,23,29,34,40, 2, 7,15,19,27,33,39,43, 4,12, 17,22,28,36,42, 3, 8,14,21,26,32,37,45, 6,11,18,24,30,35,41)$ | |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,10,19,28,37)( 2,12,21,30,38)( 3,11,20,29,39)( 4,14,24,31,40) ( 5,15,22,32,41)( 6,13,23,33,42)( 7,17,26,35,44)( 8,18,25,34,43) ( 9,16,27,36,45)$ | |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,11,21,28,39, 2,10,20,30,37, 3,12,19,29,38)( 4,15,23,31,41, 6,14,22,33,40, 5,13,24,32,42)( 7,16,25,35,45, 8,17,27,34,44, 9,18,26,36,43)$ | |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,12,20,28,38, 3,10,21,29,37, 2,11,19,30,39)( 4,13,22,31,42, 5,14,23,32,40, 6,15,24,33,41)( 7,18,27,35,43, 9,17,25,36,44, 8,16,26,34,45)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,13,25,38, 5,16,29,40, 7,19,33,43,12,22,36, 3,14,26,37, 6,18,30,41, 9,20, 31,44,10,23,34, 2,15,27,39, 4,17,28,42, 8,21,32,45,11,24,35)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,14,27,38, 6,17,29,41, 8,19,31,45,12,23,35, 3,15,25,37, 4,16,30,42, 7,20, 32,43,10,24,36, 2,13,26,39, 5,18,28,40, 9,21,33,44,11,22,34)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,15,26,38, 4,18,29,42, 9,19,32,44,12,24,34, 3,13,27,37, 5,17,30,40, 8,20, 33,45,10,22,35, 2,14,25,39, 6,16,28,41, 7,21,31,43,11,23,36)$ | |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,16,33, 3,18,31, 2,17,32)( 4,21,35, 5,19,36, 6,20,34)( 7,22,37, 9,23,39, 8, 24,38)(10,27,42,11,25,40,12,26,41)(13,29,43,14,30,44,15,28,45)$ | |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,17,31, 3,16,32, 2,18,33)( 4,20,36, 5,21,34, 6,19,35)( 7,24,39, 9,22,38, 8, 23,37)(10,26,40,11,27,41,12,25,42)(13,28,44,14,29,45,15,30,43)$ | |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,18,32, 3,17,33, 2,16,31)( 4,19,34, 5,20,35, 6,21,36)( 7,23,38, 9,24,37, 8, 22,39)(10,25,41,11,26,42,12,27,40)(13,30,45,14,28,43,15,29,44)$ | |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,19,37,10,28)( 2,21,38,12,30)( 3,20,39,11,29)( 4,24,40,14,31) ( 5,22,41,15,32)( 6,23,42,13,33)( 7,26,44,17,35)( 8,25,43,18,34) ( 9,27,45,16,36)$ | |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,20,38,10,29, 2,19,39,12,28, 3,21,37,11,30)( 4,22,42,14,32, 6,24,41,13,31, 5,23,40,15,33)( 7,27,43,17,36, 8,26,45,18,35, 9,25,44,16,34)$ | |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,21,39,10,30, 3,19,38,11,28, 2,20,37,12,29)( 4,23,41,14,33, 5,24,42,15,31, 6,22,40,13,32)( 7,25,45,17,34, 9,26,43,16,35, 8,27,44,18,36)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,22,44,21,40,18,39,13,36,10,32, 7,30, 4,25, 3,23,45,19,41,17,38,14,34,11, 33, 9,28, 5,26, 2,24,43,20,42,16,37,15,35,12,31, 8,29, 6,27)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,23,43,21,41,16,39,14,35,10,33, 8,30, 5,27, 3,24,44,19,42,18,38,15,36,11, 31, 7,28, 6,25, 2,22,45,20,40,17,37,13,34,12,32, 9,29, 4,26)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,24,45,21,42,17,39,15,34,10,31, 9,30, 6,26, 3,22,43,19,40,16,38,13,35,11, 32, 8,28, 4,27, 2,23,44,20,41,18,37,14,36,12,33, 7,29, 5,25)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,25, 5,29, 7,33,12,36,14,37,18,41,20,44,23, 2,27, 4,28, 8,32,11,35,13,38, 16,40,19,43,22, 3,26, 6,30, 9,31,10,34,15,39,17,42,21,45,24)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,26, 4,29, 9,32,12,34,13,37,17,40,20,45,22, 2,25, 6,28, 7,31,11,36,15,38, 18,42,19,44,24, 3,27, 5,30, 8,33,10,35,14,39,16,41,21,43,23)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,27, 6,29, 8,31,12,35,15,37,16,42,20,43,24, 2,26, 5,28, 9,33,11,34,14,38, 17,41,19,45,23, 3,25, 4,30, 7,32,10,36,13,39,18,40,21,44,22)$ | |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,28,10,37,19)( 2,30,12,38,21)( 3,29,11,39,20)( 4,31,14,40,24) ( 5,32,15,41,22)( 6,33,13,42,23)( 7,35,17,44,26)( 8,34,18,43,25) ( 9,36,16,45,27)$ | |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,29,12,37,20, 2,28,11,38,19, 3,30,10,39,21)( 4,32,13,40,22, 6,31,15,42,24, 5,33,14,41,23)( 7,36,18,44,27, 8,35,16,43,26, 9,34,17,45,25)$ | |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,30,11,37,21, 3,28,12,39,19, 2,29,10,38,20)( 4,33,15,40,23, 5,31,13,41,24, 6,32,14,42,22)( 7,34,16,44,25, 9,35,18,45,26, 8,36,17,43,27)$ | |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,31,16, 2,33,17, 3,32,18)( 4,36,21, 6,35,20, 5,34,19)( 7,39,22, 8,37,24, 9, 38,23)(10,40,27,12,42,26,11,41,25)(13,44,29,15,43,28,14,45,30)$ | |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,32,17, 2,31,18, 3,33,16)( 4,34,20, 6,36,19, 5,35,21)( 7,38,24, 8,39,23, 9, 37,22)(10,41,26,12,40,25,11,42,27)(13,45,28,15,44,30,14,43,29)$ | |
| $ 9, 9, 9, 9, 9 $ | $1$ | $9$ | $( 1,33,18, 2,32,16, 3,31,17)( 4,35,19, 6,34,21, 5,36,20)( 7,37,23, 8,38,22, 9, 39,24)(10,42,25,12,41,27,11,40,26)(13,43,30,15,45,29,14,44,28)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,34,22,11,44,33,21, 9,40,28,18, 5,39,26,13, 2,36,24,10,43,32,20, 7,42,30, 16, 4,37,25,15, 3,35,23,12,45,31,19, 8,41,29,17, 6,38,27,14)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,35,24,11,45,32,21, 8,42,28,17, 4,39,27,15, 2,34,23,10,44,31,20, 9,41,30, 18, 6,37,26,14, 3,36,22,12,43,33,19, 7,40,29,16, 5,38,25,13)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,36,23,11,43,31,21, 7,41,28,16, 6,39,25,14, 2,35,22,10,45,33,20, 8,40,30, 17, 5,37,27,13, 3,34,24,12,44,32,19, 9,42,29,18, 4,38,26,15)$ | |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,37,28,19,10)( 2,38,30,21,12)( 3,39,29,20,11)( 4,40,31,24,14) ( 5,41,32,22,15)( 6,42,33,23,13)( 7,44,35,26,17)( 8,43,34,25,18) ( 9,45,36,27,16)$ | |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,38,29,19,12, 3,37,30,20,10, 2,39,28,21,11)( 4,42,32,24,13, 5,40,33,22,14, 6,41,31,23,15)( 7,43,36,26,18, 9,44,34,27,17, 8,45,35,25,16)$ | |
| $ 15, 15, 15 $ | $1$ | $15$ | $( 1,39,30,19,11, 2,37,29,21,10, 3,38,28,20,12)( 4,41,33,24,15, 6,40,32,23,14, 5,42,31,22,13)( 7,45,34,26,16, 8,44,36,25,17, 9,43,35,27,18)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,40,36,30,23,17,11, 5,43,37,31,27,21,13, 7, 3,41,34,28,24,16,12, 6,44,39, 32,25,19,14, 9, 2,42,35,29,22,18,10, 4,45,38,33,26,20,15, 8)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,41,35,30,24,18,11, 6,45,37,32,26,21,14, 8, 3,42,36,28,22,17,12, 4,43,39, 33,27,19,15, 7, 2,40,34,29,23,16,10, 5,44,38,31,25,20,13, 9)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,42,34,30,22,16,11, 4,44,37,33,25,21,15, 9, 3,40,35,28,23,18,12, 5,45,39, 31,26,19,13, 8, 2,41,36,29,24,17,10, 6,43,38,32,27,20,14, 7)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,43,41,39,35,33,30,27,24,19,18,15,11, 7, 6, 2,45,40,37,34,32,29,26,23,21, 16,14,10, 8, 5, 3,44,42,38,36,31,28,25,22,20,17,13,12, 9, 4)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,44,40,39,36,32,30,25,23,19,17,14,11, 9, 5, 2,43,42,37,35,31,29,27,22,21, 18,13,10, 7, 4, 3,45,41,38,34,33,28,26,24,20,16,15,12, 8, 6)$ | |
| $ 45 $ | $1$ | $45$ | $( 1,45,42,39,34,31,30,26,22,19,16,13,11, 8, 4, 2,44,41,37,36,33,29,25,24,21, 17,15,10, 9, 6, 3,43,40,38,35,32,28,27,23,20,18,14,12, 7, 5)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $45=3^{2} \cdot 5$ | magma: Order(G);
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| Cyclic: | yes | magma: IsCyclic(G);
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| Abelian: | yes | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | $1$ | ||
| Label: | 45.1 | magma: IdentifyGroup(G);
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| Character table: | 45 x 45 character table |
magma: CharacterTable(G);